What calculus rule lets you differentiate under the integral sign?
https://en.wikipedia.org/wiki/Leibniz_integral_rule
Leibniz integral rule - Wikipedia
2005-08-28
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<div><div>
<p>This article is about the integral rule. For the convergence test for alternating series, see <a href="/wiki/Alternating_series_test">Alternating series test</a>.</p>
<p>In <a href="/wiki/Calculus">calculus</a>, the <b>Leibniz integral rule</b> for differentiation under the integral sign, named after <a href="/wiki/Gottfried_Leibniz">Gottfried Leibniz</a>, states that for an <a href="/wiki/Integral">integral</a> of the form
</p><div><p></p></div><p>
where and the integrands are <a href="/wiki/Function_(mathematics)">functions</a> dependent on the derivative of this integral is expressible as
</p><div><p></p></div><p>
where the <a href="/wiki/Partial_derivative">partial derivative</a> indicates that inside the integral, only the variation of with is considered in taking the derivative.<sup><a href="#cite_note-1">[1]</a></sup></p><p>In the special case where the functions and are constants and with values that do not depend on this simplifies to:
</p><div><p></p></div>
<p>If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:
</p><div><p></p></div>
<p>This important result may, under certain conditions, be used to interchange the integral and partial differential <a href="/wiki/Operator_(mathematics)">operators</a>, and is particularly useful in the differentiation of <a href="/wiki/Integral_transform">integral transforms</a>. An example of such is the <a href="/wiki/Moment_generating_function">moment generating function</a> in <a href="/wiki/Probability_theory">probability theory</a>, a variation of the <a href="/wiki/Laplace_transform">Laplace transform</a>, which can be differentiated to generate the <a href="/wiki/Moment_(mathematics)">moments</a> of a <a href="/wiki/Random_variable">random variable</a>. Whether Leibniz's integral rule applies is essentially a question about the interchange of <a href="/wiki/Limit_(mathematics)">limits</a>.
</p>
<h2>General form: differentiation under the integral sign[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=1">edit</a>]</h2>
<p>Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous.<sup><a href="#cite_note-Talvila-2">[2]</a></sup> This formula is the general form of the Leibniz integral rule and can be derived using the <a href="/wiki/Fundamental_theorem_of_calculus">fundamental theorem of calculus</a>. The (first) fundamental theorem of calculus is just the particular case of the above formula where , , and .
</p><p>If both upper and lower limits are taken as constants, then the formula takes the shape of an <a href="/wiki/Operator_(mathematics)">operator</a> equation:
</p><div><p></p></div><p>
where is the <a href="/wiki/Partial_derivative">partial derivative</a> with respect to and is the integral operator with respect to over a fixed <a href="/wiki/Interval_(mathematics)">interval</a>. That is, it is related to the <a href="/wiki/Symmetry_of_second_derivatives">symmetry of second derivatives</a>, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.
</p><p>The following three basic theorems on the <a href="/wiki/Interchange_of_limiting_operations">interchange of limits</a> are essentially equivalent:
</p>
<ul><li>the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);</li>
<li>the change of order of partial derivatives;</li>
<li>the change of order of integration (integration under the integral sign; i.e., <a href="/wiki/Fubini%27s_theorem">Fubini's theorem</a>).</li></ul>
<h2>Three-dimensional, time-dependent case[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=2">edit</a>]</h2>
<p>See also: <a href="#Higher_dimensions">§ Higher dimensions</a></p>
<div><p><a href="/wiki/File:Vector_field_on_a_surface.svg"></a></p><div><p>Figure 1: A vector field <b>F</b>(<b>r</b>, <i>t</i>) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity <b>v</b> over which the field is integrated.</p></div></div>
<p>A Leibniz integral rule for a <a href="#Higher_dimensions">two dimensional surface</a> moving in three dimensional space is<sup><a href="#cite_note-3">[3]</a></sup><sup><a href="#cite_note-Flanders-4">[4]</a></sup>
</p><div><p></p></div>
<p>where:
</p>
<ul><li><b>F</b>(<b>r</b>, <i>t</i>) is a vector field at the spatial position <b>r</b> at time t,</li>
<li>Σ is a surface bounded by the closed curve ∂Σ,</li>
<li><i>d</i><b>A</b> is a vector element of the surface Σ,</li>
<li><i>d</i><b>s</b> is a vector element of the curve ∂Σ,</li>
<li><b>v</b> is the velocity of movement of the region Σ,</li>
<li>∇⋅ is the vector <a href="/wiki/Divergence">divergence</a>,</li>
<li>× is the <a href="/wiki/Vector_cross_product">vector cross product</a>,</li>
<li>The double integrals are <a href="/wiki/Surface_integral">surface integrals</a> over the surface Σ, and the <a href="/wiki/Line_integral">line integral</a> is over the bounding curve ∂Σ.</li></ul>
<h2> Higher dimensions[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=3">edit</a>]</h2>
<p>The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of <a href="/wiki/Fluid_dynamics">fluid dynamics</a> as the <a href="/wiki/Reynolds_transport_theorem">Reynolds transport theorem</a>:
</p><div><p></p></div>
<p>where is a scalar function, <i>D</i>(<i>t</i>) and ∂<i>D</i>(<i>t</i>) denote a time-varying connected region of <b>R</b><sup>3</sup> and its boundary, respectively, is the Eulerian velocity of the boundary (see <a href="/wiki/Lagrangian_and_Eulerian_coordinates">Lagrangian and Eulerian coordinates</a>) and <i>d</i><b>Σ</b> = <b>n</b> <i>dS</i> is the unit normal component of the <a href="/wiki/Surface_integral">surface</a> <a href="/wiki/Volume_element">element</a>.
</p><p>The general statement of the Leibniz integral rule requires concepts from <a href="/wiki/Differential_geometry_and_topology">differential geometry</a>, specifically <a href="/wiki/Differential_forms">differential forms</a>, <a href="/wiki/Exterior_derivative">exterior derivatives</a>, <a href="/wiki/Wedge_product">wedge products</a> and <a href="/wiki/Interior_product">interior products</a>. With those tools, the Leibniz integral rule in <i>n</i> dimensions is<sup><a href="#cite_note-Flanders-4">[4]</a></sup>
</p><div><p></p></div><p>
where Ω(<i>t</i>) is a time-varying domain of integration, <i>ω</i> is a <i>p</i>-form, is the vector field of the velocity, denotes the <a href="/wiki/Interior_product">interior product</a> with , <i>d</i><sub><i>x</i></sub><i>ω</i> is the <a href="/wiki/Exterior_derivative">exterior derivative</a> of <i>ω</i> with respect to the space variables only and is the time derivative of <i>ω</i>.
</p><p>However, all of these identities can be derived from a most general statement about Lie derivatives:
</p><div><p></p></div>
<p>Here, the ambient manifold on which the differential form lives includes both space and time.
</p>
<p>Something remarkable about this form, is that it can account for the case when changes its shape and size over time, since such deformations are fully determined by .
</p>
<h2>Measure theory statement[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=4">edit</a>]</h2>
<p>Let be an open subset of , and be a <a href="/wiki/Measure_space">measure space</a>. Suppose satisfies the following conditions:<sup><a href="#cite_note-5">[5]</a></sup><sup><a href="#cite_note-6">[6]</a></sup><sup><a href="#cite_note-Talvila-2">[2]</a></sup>
</p>
<ol><li> is a Lebesgue-integrable function of for each .</li>
<li>For <a href="/wiki/Almost_all">almost all</a> , the partial derivative exists for all .</li>
<li>There is an integrable function such that for all and almost every .</li></ol>
<p>Then, for all ,
</p><div><p></p></div>
<p>The proof relies on the <a href="/wiki/Dominated_convergence_theorem">dominated convergence theorem</a> and the <a href="/wiki/Mean_value_theorem">mean value theorem</a> (details below).
</p>
<h2>Proofs[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=5">edit</a>]</h2>
<h3>Proof of basic form[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=6">edit</a>]</h3>
<p>We first prove the case of constant limits of integration <i>a</i> and <i>b</i>.
</p><p>We use <a href="/wiki/Fubini%27s_theorem">Fubini's theorem</a> to change the order of integration. For every x and h, such that <i>h</i> &gt; 0 and both x and <i>x</i> +<i>h</i> are within [<i>x</i><sub>0</sub>,<i>x</i><sub>1</sub>], we have:
</p><div><p></p></div>
<p>Note that the integrals at hand are well defined since is continuous at the closed rectangle and thus also uniformly continuous there; thus its integrals by either <i>dt</i> or <i>dx</i> are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).
</p><p>Therefore:
</p><div><p></p></div>
<p>Where we have defined:
</p><div><p></p></div><p>
(we may replace <i>x</i><sub>0</sub> here by any other point between <i>x</i><sub>0</sub> and <i>x</i>)
</p><p><i>F</i> is differentiable with derivative , so we can take the limit where h approaches zero. For the left hand side this limit is:
</p><div><p></p></div>
<p>For the right hand side, we get:
</p><div><p></p></div><p>
And we thus prove the desired result:
</p><div><p></p></div>
<h4>Another proof using the bounded convergence theorem[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=7">edit</a>]</h4>
<p>If the integrals at hand are <a href="/wiki/Lebesgue_integral">Lebesgue integrals</a>, we may use the <a href="/wiki/Bounded_convergence_theorem">bounded convergence theorem</a> (valid for these integrals, but not for <a href="/wiki/Riemann_integral">Riemann integrals</a>) in order to show that the limit can be passed through the integral sign.
</p><p>Note that this proof is weaker in the sense that it only shows that <i>f<sub>x</sub></i>(<i>x</i>,<i>t</i>) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if <i>f</i>(<i>x</i>,<i>t</i>) is Riemann integrable, then so is <i>f<sub>x</sub></i>(<i>x</i>,<i>t</i>) (and thus is obviously also Lebesgue integrable).
</p><p>Let
</p>
<table>
<tbody><tr>
<td><div><p></p></div>
</td>
<td>
</td>
<td><p><b>(1)</b></p>
</td></tr></tbody></table>
<p>By the definition of the derivative,
</p>
<table>
<tbody><tr>
<td><div><p></p></div>
</td>
<td>
</td>
<td><p><b>(2)</b></p>
</td></tr></tbody></table>
<p>Substitute equation (<b><a href="#math_1">1</a></b>) into equation (<b><a href="#math_2">2</a></b>). The difference of two integrals equals the integral of the difference, and 1/<i>h</i> is a constant, so
</p><div><p></p></div>
<p>We now show that the limit can be passed through the integral sign.
</p><p>We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the <a href="/wiki/Dominated_convergence_theorem">dominated convergence theorem</a>). For each <i>δ</i> &gt; 0, consider the <a href="/wiki/Difference_quotient">difference quotient</a>
</p><div><p></p></div><p>
For <i>t</i> fixed, the <a href="/wiki/Mean_value_theorem">mean value theorem</a> implies there exists <i>z</i> in the interval [<i>x</i>, <i>x</i> + <i>δ</i>] such that
</p><div><p></p></div><p>
Continuity of <i>f</i><sub><i>x</i></sub>(<i>x</i>, <i>t</i>) and compactness of the domain together imply that <i>f</i><sub><i>x</i></sub>(<i>x</i>, <i>t</i>) is bounded. The above application of the mean value theorem therefore gives a uniform (independent of ) bound on . The difference quotients converge pointwise to the partial derivative <i>f</i><sub><i>x</i></sub> by the assumption that the partial derivative exists.
</p><p>The above argument shows that for every sequence {<i>δ<sub>n</sub></i>} → 0, the sequence is uniformly bounded and converges pointwise to <i>f</i><sub><i>x</i></sub>. The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {<i>δ<sub>n</sub></i>} → 0. Therefore, the limit as <i>δ</i> → 0 may be passed through the integral sign.
</p>
<h3>Variable limits form[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=8">edit</a>]</h3>
<p>For a <a href="/wiki/Continuous_function">continuous</a> <a href="/wiki/Real-valued_function">real valued function</a> <i>g</i> of one <a href="/wiki/Function_of_a_real_variable">real variable</a>, and real valued <a href="/wiki/Differentiable_function">differentiable</a> functions and of one real variable,
</p><div><p></p></div>
<p>This follows from the <a href="/wiki/Chain_rule">chain rule</a> and the <a href="/wiki/Fundamental_theorem_of_calculus#First_part">First Fundamental Theorem of Calculus</a>. Define
</p><div><p></p></div><p>
and
</p><div><p></p></div><p>
(The lower limit just has to be some number in the domain of )
</p><p>Then, can be written as a <a href="/wiki/Function_composition">composition</a>: . The <a href="/wiki/Chain_rule#Statement">Chain Rule</a> then implies that
</p><div><p></p></div><p>
By the <a href="/wiki/Fundamental_theorem_of_calculus#First_part">First Fundamental Theorem of Calculus</a>, . Therefore, substituting this result above, we get the desired equation:
</p><div><p></p></div>
<p><b>Note:</b> This form can be particularly useful if the expression to be differentiated is of the form:
</p><div><p></p></div><p>
Because does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the <a href="/wiki/Product_rule">Product rule</a>, i.e.,
</p><div><p></p></div>
<h3>General form with variable limits[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=9">edit</a>]</h3>
<p>Set
</p><div><p></p></div><p>
where <i>a</i> and <i>b</i> are functions of <i>α</i> that exhibit increments Δ<i>a</i> and Δ<i>b</i>, respectively, when <i>α</i> is increased by Δ<i>α</i>. Then,
</p><div><p></p></div>
<p>A form of the <a href="/wiki/Mean_value_theorem">mean value theorem</a>, , where <i>a</i> &lt; <i>ξ</i> &lt; <i>b</i>, may be applied to the first and last integrals of the formula for Δ<i>φ</i> above, resulting in
</p><div><p></p></div>
<p>Divide by Δ<i>α</i> and let Δ<i>α</i> → 0. Notice <i>ξ</i><sub>1</sub> → <i>a</i> and <i>ξ</i><sub>2</sub> → <i>b</i>. We may pass the limit through the integral sign:
</p><div><p></p></div><p>
again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule,
</p><div><p></p></div>
<h3>Alternative proof of the general form with variable limits, using the chain rule[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=10">edit</a>]</h3>
<p>The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the <a href="#Proof_of_basic_form">basic form</a> of Leibniz's Integral Rule, the <a href="/wiki/Chain_rule#Higher_dimensions">multivariable chain rule</a>, and the <a href="/wiki/Fundamental_theorem_of_calculus#First_part">First Fundamental Theorem of Calculus</a>. Suppose is defined in a rectangle in the plane, for and . Also, assume and the partial derivative are both continuous functions on this rectangle. Suppose are <a href="/wiki/Differentiable_function#Differentiability_of_real_functions_of_one_variable">differentiable</a> real valued functions defined on , with values in (i.e. for every ). Now, set
</p><div><p></p></div><p>
and 
</p><div><p></p></div>
<p>Then, by properties of <a href="/wiki/Integral#Conventions">Definite Integrals</a>, we can write
</p><div><p></p></div>
<p>Since the functions are all differentiable (see the remark at the end of the proof), by the <a href="/wiki/Chain_rule#Higher_dimensions">Multivariable Chain Rule</a>, it follows that is differentiable, and its derivative is given by the formula:
</p><div><p></p></div> <p>Now, note that for every , and for every , we have that , because when taking the partial derivative with respect to of , we are keeping fixed in the expression ; thus the <a href="#Proof_of_basic_form">basic form</a> of Leibniz's Integral Rule with constant limits of integration applies. Next, by the <a href="/wiki/Fundamental_theorem_of_calculus#First_part">First Fundamental Theorem of Calculus</a>, we have that ; because when taking the partial derivative with respect to of , the first variable is fixed, so the fundamental theorem can indeed be applied.
</p><p>Substituting these results into the equation for above gives:
</p><div><p></p></div><p>
as desired.
</p><p>There is a technical point in the proof above which is worth noting: applying the Chain Rule to requires that already be <a href="/wiki/Differentiable_function#Differentiability_in_higher_dimensions">differentiable</a>. This is where we use our assumptions about . As mentioned above, the partial derivatives of are given by the formulas and . Since is continuous, its integral is also a continuous function,<sup><a href="#cite_note-7">[7]</a></sup> and since is also continuous, these two results show that both the partial derivatives of are continuous. Since continuity of partial derivatives implies differentiability of the function,<sup><a href="#cite_note-8">[8]</a></sup> is indeed differentiable.
</p>
<h3>Three-dimensional, time-dependent form[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=11">edit</a>]</h3>
<p>See also: <a href="#Higher_dimensions">§ Higher dimensions</a></p>
<p>At time <i>t</i> the surface Σ in <a href="#Three-dimensional,_time-dependent_case">Figure 1</a> contains a set of points arranged about a centroid . The function can be written as
</p><div><p></p></div><p>
with independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at . For a rigidly translating surface, the limits of integration are then independent of time, so:
</p><div><p></p></div><p>
where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only:
</p><div><p></p></div><p>
with the velocity of motion of the surface defined by
</p><div><p></p></div>
<p>This equation expresses the <a href="/wiki/Material_derivative">material derivative</a> of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see <a href="/wiki/Curl_(mathematics)#Identities">article on curl</a>)
</p><div><p></p></div><p>
and that <a href="/wiki/Stokes_theorem#Kelvin–Stokes_theorem">Stokes theorem</a> equates the surface integral of the curl over Σ with a line integral over ∂Σ:
</p><div><p></p></div>
<p>The sign of the line integral is based on the <a href="/wiki/Right-hand_rule">right-hand rule</a> for the choice of direction of line element <i>d</i><b>s</b>. To establish this sign, for example, suppose the field <b>F</b> points in the positive <i>z</i>-direction, and the surface Σ is a portion of the <i>xy</i>-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive <i>z</i>-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along <i>z</i>-axis). Then the integral on the left-hand side determines a <i>positive</i> flux of <b>F</b> through Σ. Suppose Σ translates in the positive <i>x</i>-direction at velocity <b>v</b>. An element of the boundary of Σ parallel to the <i>y</i>-axis, say <i>d</i><b>s</b>, sweeps out an area <b>v</b><i>t</i> × <i>d</i><b>s</b> in time <i>t</i>. If we integrate around the boundary ∂Σ in a counterclockwise sense, <b>v</b><i>t</i> × <i>d</i><b>s</b> points in the negative <i>z</i>-direction on the left side of ∂Σ (where <i>d</i><b>s</b> points downward), and in the positive <i>z</i>-direction on the right side of ∂Σ (where <i>d</i><b>s</b> points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of <b>F</b> is increasing on the right of ∂Σ and decreasing on the left. However, the <a href="/wiki/Dot_product">dot product</a> <b>v</b> × <b>F</b> ⋅ <i>d</i><b>s</b> = −<b>F</b> × <b>v</b> ⋅ <i>d</i><b>s</b> = −<b>F</b> ⋅ <b>v</b> × <i>d</i><b>s</b>. Consequently, the sign of the line integral is taken as negative.
</p><p>If <b>v</b> is a constant,
</p><div><p></p></div><p>
which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.
</p>
<h3>Alternative derivation[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=12">edit</a>]</h3>
<p><b>Lemma.</b> One has:
</p><div><p></p></div>
<p><b>Proof.</b> From the <a href="/wiki/Fundamental_theorem_of_calculus#Proof_of_the_first_part">proof of the fundamental theorem of calculus</a>,
</p><div><p></p></div><p>
and
</p><div><p></p></div>
<p>Suppose <i>a</i> and <i>b</i> are constant, and that <i>f</i>(<i>x</i>) involves a parameter <i>α</i> which is constant in the integration but may vary to form different integrals. Assume that <i>f</i>(<i>x</i>, <i>α</i>) is a continuous function of <i>x</i> and <i>α</i> in the compact set {(<i>x</i>, <i>α</i>) : <i>α</i><sub>0</sub> ≤ <i>α</i> ≤ <i>α</i><sub>1</sub> and <i>a</i> ≤ <i>x</i> ≤ <i>b</i>}, and that the partial derivative <i>f</i><sub><i>α</i></sub>(<i>x</i>, <i>α</i>) exists and is continuous. If one defines:
</p><div><p></p></div><p>
then may be differentiated with respect to <i>α</i> by differentiating under the integral sign, i.e.,
</p><div><p></p></div>
<p>By the <a href="/wiki/Heine%E2%80%93Cantor_theorem">Heine–Cantor theorem</a> it is uniformly continuous in that set. In other words, for any <i>ε</i> &gt; 0 there exists Δ<i>α</i> such that for all values of <i>x</i> in [<i>a</i>, <i>b</i>],
</p><div><p></p></div>
<p>On the other hand,
</p><div><p></p></div>
<p>Hence <i>φ</i>(<i>α</i>) is a continuous function.
</p><p>Similarly if exists and is continuous, then for all <i>ε</i> &gt; 0 there exists Δ<i>α</i> such that:
</p><div><p></p></div>
<p>Therefore,
</p><div><p></p></div><p>
where
</p><div><p></p></div>
<p>Now, <i>ε</i> → 0 as Δ<i>α</i> → 0, so
</p><div><p></p></div>
<p>This is the formula we set out to prove.
</p><p>Now, suppose
</p><div><p></p></div><p>
where <i>a</i> and <i>b</i> are functions of <i>α</i> which take increments Δ<i>a</i> and Δ<i>b</i>, respectively, when <i>α</i> is increased by Δ<i>α</i>. Then,
</p><div><p></p></div>
<p>A form of the <a href="/wiki/Mean_value_theorem">mean value theorem</a>, where <i>a</i> &lt; <i>ξ</i> &lt; <i>b</i>, can be applied to the first and last integrals of the formula for Δ<i>φ</i> above, resulting in
</p><div><p></p></div>
<p>Dividing by Δ<i>α</i>, letting Δ<i>α</i> → 0, noticing <i>ξ</i><sub>1</sub> → <i>a</i> and <i>ξ</i><sub>2</sub> → <i>b</i> and using the above derivation for
</p><div><p></p></div><p>
yields
</p><div><p></p></div>
<p>This is the general form of the Leibniz integral rule.
</p>
<h2>Examples[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=13">edit</a>]</h2>
<h3>Example 1: Fixed limits[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=14">edit</a>]</h3>
<p>Consider the function
</p><div><p></p></div>
<p>The function under the integral sign is not continuous at the point (<i>x</i>, <i>α</i>) = (0, 0), and the function <i>φ</i>(<i>α</i>) has a discontinuity at <i>α</i> = 0 because <i>φ</i>(<i>α</i>) approaches ±<i>π</i>/2 as <i>α</i> → 0<sup>±</sup>.
</p><p>If we differentiate <i>φ</i>(<i>α</i>) with respect to <i>α</i> under the integral sign, we get
</p><div><p></p></div><p>
which is, of course, true for all values of <i>α</i> except <i>α</i> = 0. This may be integrated (with respect to <i>α</i>) to find
</p><div><p></p></div>
<h3>Example 2: Variable limits[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=15">edit</a>]</h3>
<p>An example with variable limits:
</p><div><p></p></div>
<h2>Applications[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=16">edit</a>]</h2>
<h3>Evaluating definite integrals[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=17">edit</a>]</h3>
<p>The formula
</p><div><p></p></div><p>
can be of use when evaluating certain definite integrals. When used in this context, the Leibniz integral rule for differentiating under the integral sign is also known as Feynman's trick for integration.
</p>
<h4>Example 3[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=18">edit</a>]</h4>
<p>Consider
</p><div><p></p></div>
<p>Now,
</p><div><p></p></div>
<p>As varies from to , we have
</p><div><p></p></div>
<p>Hence,
</p><div><p></p></div>
<p>Therefore,
</p><div><p></p></div>
<p>Integrating both sides with respect to , we get:
</p><div><p></p></div>
<p> follows from evaluating :
</p><div><p></p></div>
<p>To determine in the same manner, we should need to substitute in a value of greater than 1 in . This is somewhat inconvenient. Instead, we substitute , where . Then,
</p><div><p></p></div>
<p>Therefore, 
</p><p>The definition of is now complete:
</p><div><p></p></div>
<p>The foregoing discussion, of course, does not apply when , since the conditions for differentiability are not met.
</p>
<h4>Example 4[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=19">edit</a>]</h4>
<div><p></p></div>
<p>First we calculate:
</p><div><p></p></div>
<p>The limits of integration being independent of , we have:
</p><div><p></p></div>
<p>On the other hand:
</p><div><p></p></div>
<p>Equating these two relations then yields
</p><div><p></p></div>
<p>In a similar fashion, pursuing yields
</p><div><p></p></div>
<p>Adding the two results then produces
</p><div><p></p></div><p>
which computes as desired.
</p><p>This derivation may be generalized. Note that if we define
</p><div><p></p></div><p>
it can easily be shown that
</p><div><p></p></div>
<p>Given , this integral reduction formula can be used to compute all of the values of for . Integrals like and may also be handled using the <a href="/wiki/Weierstrass_substitution">Weierstrass substitution</a>.
</p>
<h4>Example 5[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=20">edit</a>]</h4>
<p>Here, we consider the integral
</p><div><p></p></div>
<p>Differentiating under the integral with respect to , we have
</p><div><p></p></div>
<p>Therefore:
</p><div><p></p></div>
<p>But by definition so and
</p><div><p></p></div>
<h4>Example 6[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=21">edit</a>]</h4>
<p>Here, we consider the integral
</p><div><p></p></div>
<p>We introduce a new variable <i>φ</i> and rewrite the integral as
</p><div><p></p></div>
<p>When <i>φ</i> = 1 this equals the original integral. However, this more general integral may be differentiated with respect to :
</p><div><p></p></div>
<p>Now, fix <i>φ</i>, and consider the vector field on defined by . Further, choose the <a href="/wiki/Curve_orientation">positive oriented</a> parametrization of the <a href="/wiki/Unit_circle">unit circle</a> given by , , so that . Then the final integral above is precisely
</p><div><p></p></div><p>
the line integral of over . By <a href="/wiki/Green%27s_theorem">Green's Theorem</a>, this equals the double integral
</p><div><p></p></div><p>
where is the closed <a href="/wiki/Unit_disk">unit disc</a>. Its integrand is identically 0, so is likewise identically zero. This implies that <i>f</i>(<i>φ</i>) is constant. The constant may be determined by evaluating at :
</p><div><p></p></div>
<p>Therefore, the original integral also equals .
</p>
<h4>Other problems to solve[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=22">edit</a>]</h4>
<p>There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter :
</p><div><p></p></div>
<p>The first integral, the <a href="/wiki/Dirichlet_integral">Dirichlet integral</a>, is absolutely convergent for positive <i>α</i> but only conditionally convergent when . Therefore, differentiation under the integral sign is easy to justify when , but proving that the resulting formula remains valid when requires some careful work.
</p>
<h3>Infinite series[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=23">edit</a>]</h3>
<p>The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as <a href="/wiki/Counting_measure">counting measure</a>. An example of an application is the fact that power series are differentiable in their radius of convergence.<sup>[<i><a href="/wiki/Wikipedia:Citation_needed">citation needed</a></i>]</sup>
</p>
<h2>In popular culture[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=24">edit</a>]</h2>
<p>Differentiation under the integral sign is mentioned in the late <a href="/wiki/Physicist">physicist</a> <a href="/wiki/Richard_Feynman">Richard Feynman</a>'s best-selling memoir <i><a href="/wiki/Surely_You%27re_Joking,_Mr._Feynman!">Surely You're Joking, Mr. Feynman!</a></i> in the chapter "A Different Box of Tools". He describes learning it, while in <a href="/wiki/High_school">high school</a>, from an old text, <i>Advanced Calculus</i> (1926), by <a href="/wiki/Frederick_S._Woods">Frederick S. Woods</a> (who was a professor of mathematics in the <a href="/wiki/Massachusetts_Institute_of_Technology">Massachusetts Institute of Technology</a>). The technique was not often taught when Feynman later received his formal education in <a href="/wiki/Calculus">calculus</a>, but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at <a href="/wiki/Princeton_University">Princeton University</a>:
</p>
<blockquote>
<p>One thing I never did learn was <a href="/wiki/Methods_of_contour_integration">contour integration</a>. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: <a href="https://books.google.com/books/about/Advanced_calculus.html?id=94MZAQAAIAAJ">"Advanced Calculus"</a>, by Woods. Bader knew I had studied <a href="https://archive.org/details/calulusforthepra000526mbp">"Calculus for the Practical Man"</a> a little bit, so he gave me the real works—it was for a junior or senior course in college. It had <a href="/wiki/Fourier_series">Fourier series</a>, <a href="/wiki/Bessel_function">Bessel functions</a>, <a href="/wiki/Determinant">determinants</a>, <a href="/wiki/Elliptic_function">elliptic functions</a>—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or <a href="/wiki/Princeton_University">Princeton</a> had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.
</p>
</blockquote>
<h2>See also[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=25">edit</a>]</h2>
<ul><li><a href="/wiki/Chain_rule">Chain rule</a></li>
<li><a href="/wiki/Differentiation_of_integrals">Differentiation of integrals</a></li>
<li><a href="/wiki/Leibniz_rule_(generalized_product_rule)">Leibniz rule (generalized product rule)</a></li>
<li><a href="/wiki/Reynolds_transport_theorem">Reynolds transport theorem</a>, a generalization of Leibniz rule</li></ul>
<h2>References[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=26">edit</a>]</h2>
<div><ol>
<li><b><a href="#cite_ref-1">^</a></b> <cite>Protter, Murray H.; Morrey, Charles B. Jr. (1985). <a href="https://books.google.com/books?id=3lTmBwAAQBAJ&amp;pg=PA421">"Differentiation under the Integral Sign"</a>. <i>Intermediate Calculus</i> (Second ed.). New York: Springer. pp. 421–426. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-1086-3">10.1007/978-1-4612-1086-3</a>. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96058-6"><bdi>978-0-387-96058-6</bdi></a>.</cite>
</li>
<li>^ <a href="#cite_ref-Talvila_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Talvila_2-1"><sup><i><b>b</b></i></sup></a> <cite>Talvila, Erik (June 2001). <a href="https://www.jstor.org/stable/2695709">"Necessary and Sufficient Conditions for Differentiating under the Integral Sign"</a>. <i>American Mathematical Monthly</i>. <b>108</b> (6): 544–548. <a href="/wiki/ArXiv_(identifier)">arXiv</a>:<a href="//arxiv.org/abs/math/0101012">math/0101012</a>. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.2307%2F2695709">10.2307/2695709</a>. <a href="/wiki/JSTOR_(identifier)">JSTOR</a> <a href="//www.jstor.org/stable/2695709">2695709</a>. Retrieved 16 April 2022.</cite>
</li>
<li><b><a href="#cite_ref-3">^</a></b> <cite>Abraham, Max; Becker, Richard (1950). <i>Classical Theory of Electricity and Magnetism</i> (2nd ed.). London: Blackie &amp; Sons. pp. 39–40.</cite>
</li>
<li>^ <a href="#cite_ref-Flanders_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Flanders_4-1"><sup><i><b>b</b></i></sup></a> <cite><a href="/wiki/Harley_Flanders">Flanders, Harly</a> (June–July 1973). <a href="http://sgpwe.izt.uam.mx/files/users/uami/jdf/proyectos/Derivar_inetegral.pdf">"Differentiation under the integral sign"</a> (PDF). <i><a href="/wiki/American_Mathematical_Monthly">American Mathematical Monthly</a></i>. <b>80</b> (6): 615–627. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.2307%2F2319163">10.2307/2319163</a>. <a href="/wiki/JSTOR_(identifier)">JSTOR</a> <a href="//www.jstor.org/stable/2319163">2319163</a>.</cite>
</li>
<li><b><a href="#cite_ref-5">^</a></b> <cite>Cheng, Steve (22 March 2013). <a href="https://planetmath.org/differentiationundertheintegralsign">"Differentiation under the integral sign"</a>. <i>PlanetMath.org</i>. PlanetMath.org, Ltd. Retrieved 16 April 2022.</cite>
</li>
<li><b><a href="#cite_ref-6">^</a></b> <cite>Cheng, Steve (6 September 2010). Differentiation under the integral sign with weak derivatives (Report). CiteSeerX. <a href="/wiki/CiteSeerX_(identifier)">CiteSeerX</a> <a href="//citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.525.2529">10.1.1.525.2529</a>.</cite>
</li>
<li><b><a href="#cite_ref-7">^</a></b> <cite>Spivak, Michael (1994). <a href="https://archive.org/details/calculus00spiv_191"><i>Calculus</i></a> (3 ed.). Houston, Texas: Publish or Perish, Inc. pp. <a href="https://archive.org/details/calculus00spiv_191/page/n280">267</a>–268. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-89-8"><bdi>978-0-914098-89-8</bdi></a>.</cite>
</li>
<li><b><a href="#cite_ref-8">^</a></b> <cite>Spivak, Michael (1965). <i>Calculus on Manifolds</i>. Addison-Wesley Publishing Company. p. 31. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-9021-6"><bdi>978-0-8053-9021-6</bdi></a>.</cite>
</li>
</ol></div>
<h2>Further reading[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=27">edit</a>]</h2>
<ul><li><cite>Amazigo, John C.; Rubenfeld, Lester A. (1980). <a href="https://archive.org/details/advancedcalculus0000amaz/page/155">"Single Integrals: Leibnitz's Rule; Numerical Integration"</a>. <i>Advanced Calculus and its Applications to the Engineering and Physical Sciences</i>. New York: Wiley. pp. <a href="https://archive.org/details/advancedcalculus0000amaz/page/155">155–165</a>. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-04934-4"><bdi>0-471-04934-4</bdi></a>.</cite></li>
<li><cite><a href="/wiki/Wilfred_Kaplan">Kaplan, Wilfred</a> (1973). "Integrals Depending on a Parameter—Leibnitz's Rule". <i>Advanced Calculus</i> (2nd ed.). Reading: Addison-Wesley. pp. 285–288.</cite></li></ul>
<h2>External links[<a href="/w/index.php?title=Leibniz_integral_rule&amp;action=edit&amp;section=28">edit</a>]</h2>
<ul><li><cite>Harron, Rob. <a href="https://math.hawaii.edu/~rharron/teaching/MAT203/LeibnizRule.pdf">"The Leibniz Rule"</a> (PDF). <i>MAT-203</i>.</cite></li></ul>
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 Contents

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 * (Top)
 * 1 General form: differentiation under the integral sign
 * 2 Three-dimensional, time-dependent case
 * 3 Higher dimensions
 * 4 Measure theory statement
 * 5 Proofs
 + 5.1 Proof of basic form
 o 5.1.1 Another proof using the bounded convergence theorem
 + 5.2 Variable limits form
 + 5.3 General form with variable limits
 + 5.4 Alternative proof of the general form with variable limits, using the chain rule
 + 5.5 Three-dimensional, time-dependent form
 + 5.6 Alternative derivation
 * 6 Examples
 + 6.1 Example 1: Fixed limits
 + 6.2 Example 2: Variable limits
 * 7 Applications
 + 7.1 Evaluating definite integrals
 o 7.1.1 Example 3
 o 7.1.2 Example 4
 o 7.1.3 Example 5
 o 7.1.4 Example 6
 o 7.1.5 Other problems to solve
 + 7.2 Infinite series
 * 8 In popular culture
 * 9 See also
 * 10 References
 * 11 Further reading
 * 12 External links
 Toggle the table of contents

 Leibniz integral rule

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 From Wikipedia, the free encyclopedia
 This article is about the integral rule. For the convergence test for alternating series, see Alternating series test.
 This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed.
 Find sources: "Leibniz integral rule" – news · newspapers · books · scholar · JSTOR ( October 2016 ) (Learn how and when to remove this template message) 
 
 Part of a series of articles about 
 Calculus 
 * Fundamental theorem 
 * Leibniz integral rule 
 * Limits of functions 
 * Continuity 
 * Mean value theorem 
 * Rolle's theorem 
 Differential 
 Definitions 
 * Derivative (generalizations) 
 * Differential 
 + infinitesimal 
 + of a function 
 + total 
 Concepts 
 * Differentiation notation 
 * Second derivative 
 * Implicit differentiation 
 * Logarithmic differentiation 
 * Related rates 
 * Taylor's theorem 
 Rules and identities 
 * Sum 
 * Product 
 * Chain 
 * Power 
 * Quotient 
 * L'Hôpital's rule 
 * Inverse 
 * General Leibniz 
 * Faà di Bruno's formula 
 * Reynolds 
  
 Integral 
 * Lists of integrals 
 * Integral transform 
 Definitions 
 * Antiderivative 
 * Integral (improper) 
 * Riemann integral 
 * Lebesgue integration 
 * Contour integration 
 * Integral of inverse functions 
 Integration by 
 * Parts 
 * Discs 
 * Cylindrical shells 
 * Substitution (trigonometric, tangent half-angle, Euler)
 * Euler's formula 
 * Partial fractions 
 * Changing order 
 * Reduction formulae 
 * Differentiating under the integral sign 
 * Risch algorithm 
  
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 In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

 where and the integrands are functions dependent on the derivative of this integral is expressible as
 where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative.[1]

 In the special case where the functions and are constants and with values that do not depend on this simplifies to:

 If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:

 This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.

 General form: differentiation under the integral sign [ edit ]

 Theorem — Let be a function such that both and its partial derivative are continuous in and in some region of the -plane, including , . Also suppose that the functions and are both continuous and both have continuous derivatives for . Then, for ,

 Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous.[2] This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where , , and .

 If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:

 where is the partial derivative with respect to and is the integral operator with respect to over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.

 The following three basic theorems on the interchange of limits are essentially equivalent:

 * the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);
 * the change of order of partial derivatives;
 * the change of order of integration (integration under the integral sign; i.e., Fubini's theorem).

 Three-dimensional, time-dependent case [ edit ]

 See also: § Higher dimensions
 Figure 1: A vector field F(r, t) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.

 A Leibniz integral rule for a two dimensional surface moving in three dimensional space is[3][4]

 where:

 * F ( r , t ) is a vector field at the spatial position r at time t ,
 * Σ is a surface bounded by the closed curve ∂Σ ,
 * dA is a vector element of the surface Σ ,
 * ds is a vector element of the curve ∂Σ ,
 * v is the velocity of movement of the region Σ ,
 * ∇⋅ is the vector divergence,
 * × is the vector cross product,
 * The double integrals are surface integrals over the surface Σ , and the line integral is over the bounding curve ∂Σ .

 Higher dimensions [ edit ]

 The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:

 where is a scalar function, D ( t ) and ∂ D ( t ) denote a time-varying connected region of R3 and its boundary, respectively, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.

 The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in n dimensions is[4]

 where Ω( t ) is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, denotes the interior product with , dxω is the exterior derivative of ω with respect to the space variables only and is the time derivative of ω.

 However, all of these identities can be derived from a most general statement about Lie derivatives:

 Here, the ambient manifold on which the differential form lives includes both space and time.

 * is the region of integration (a submanifold) at a given instant (it does not depend on , since its parametrization as a submanifold defines its position in time),
 * is the Lie derivative,
 * is the spacetime vector field obtained from adding the unitary vector field in the direction of time to the purely spatial vector field from the previous formulas (i.e, is the spacetime velocity of ),
 * is a diffeomorphism from the one-parameter group generated by the flow of , and
 * is the image of under such diffeomorphism.

 Something remarkable about this form, is that it can account for the case when changes its shape and size over time, since such deformations are fully determined by .

 Measure theory statement [ edit ]

 Let be an open subset of , and be a measure space. Suppose satisfies the following conditions:[5][6][2]

 1. is a Lebesgue-integrable function of for each .
 2. For almost all , the partial derivative exists for all .
 3. There is an integrable function such that for all and almost every .

 Then, for all ,

 The proof relies on the dominated convergence theorem and the mean value theorem (details below).

 Proofs [ edit ]

 Proof of basic form [ edit ]

 We first prove the case of constant limits of integration a and b.

 We use Fubini's theorem to change the order of integration. For every x and h , such that h > 0 and both x and x + h are within [ x0 , x1 ] , we have:

 Note that the integrals at hand are well defined since is continuous at the closed rectangle and thus also uniformly continuous there; thus its integrals by either dt or dx are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).

 Therefore:

 Where we have defined:

 (we may replace x0 here by any other point between x0 and x)

 F is differentiable with derivative , so we can take the limit where h approaches zero. For the left hand side this limit is:

 For the right hand side, we get:

 And we thus prove the desired result:

 Another proof using the bounded convergence theorem [ edit ]

 If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign.

 Note that this proof is weaker in the sense that it only shows that fx(x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx(x,t) (and thus is obviously also Lebesgue integrable).

 Let

 ( 1 )
 
 
 

 By the definition of the derivative,

 ( 2 )
 
 
 

 Substitute equation (1) into equation (2). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so

 We now show that the limit can be passed through the integral sign.

 We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each δ > 0, consider the difference quotient

 For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that
 Continuity of fx(x, t) and compactness of the domain together imply that fx(x, t) is bounded. The above application of the mean value theorem therefore gives a uniform (independent of ) bound on . The difference quotients converge pointwise to the partial derivative fx by the assumption that the partial derivative exists.

 The above argument shows that for every sequence {δn} → 0, the sequence is uniformly bounded and converges pointwise to fx. The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {δn} → 0. Therefore, the limit as δ → 0 may be passed through the integral sign.

 Variable limits form [ edit ]

 For a continuous real valued function g of one real variable, and real valued differentiable functions and of one real variable,

 This follows from the chain rule and the First Fundamental Theorem of Calculus. Define

 and
 (The lower limit just has to be some number in the domain of )

 Then, can be written as a composition: . The Chain Rule then implies that

 By the First Fundamental Theorem of Calculus, . Therefore, substituting this result above, we get the desired equation:

 Note: This form can be particularly useful if the expression to be differentiated is of the form:

 Because does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e.,

 General form with variable limits [ edit ]

 Set

 where a and b are functions of α that exhibit increments Δa and Δb, respectively, when α is increased by Δα. Then,

 A form of the mean value theorem, , where a < ξ < b, may be applied to the first and last integrals of the formula for Δφ above, resulting in

 Divide by Δα and let Δα → 0. Notice ξ1 → a and ξ2 → b. We may pass the limit through the integral sign:

 again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule,

 Alternative proof of the general form with variable limits, using the chain rule [ edit ]

 The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the First Fundamental Theorem of Calculus. Suppose is defined in a rectangle in the plane, for and . Also, assume and the partial derivative are both continuous functions on this rectangle. Suppose are differentiable real valued functions defined on , with values in (i.e. for every ). Now, set

 and

 Then, by properties of Definite Integrals, we can write

 Since the functions are all differentiable (see the remark at the end of the proof), by the Multivariable Chain Rule, it follows that is differentiable, and its derivative is given by the formula:

 Now, note that for every , and for every , we have that , because when taking the partial derivative with respect to of , we are keeping fixed in the expression ; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the First Fundamental Theorem of Calculus, we have that ; because when taking the partial derivative with respect to of , the first variable is fixed, so the fundamental theorem can indeed be applied.

 Substituting these results into the equation for above gives:

 as desired.

 There is a technical point in the proof above which is worth noting: applying the Chain Rule to requires that already be differentiable. This is where we use our assumptions about . As mentioned above, the partial derivatives of are given by the formulas and . Since is continuous, its integral is also a continuous function,[7] and since is also continuous, these two results show that both the partial derivatives of are continuous. Since continuity of partial derivatives implies differentiability of the function,[8] is indeed differentiable.

 Three-dimensional, time-dependent form [ edit ]

 See also: § Higher dimensions

 At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid . The function can be written as

 with independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at . For a rigidly translating surface, the limits of integration are then independent of time, so:
 where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only:
 with the velocity of motion of the surface defined by

 This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on curl)

 and that Stokes theorem equates the surface integral of the curl over Σ with a line integral over ∂Σ :

 The sign of the line integral is based on the right-hand rule for the choice of direction of line element ds. To establish this sign, for example, suppose the field F points in the positive z-direction, and the surface Σ is a portion of the xy-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive z-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along z-axis). Then the integral on the left-hand side determines a positive flux of F through Σ. Suppose Σ translates in the positive x-direction at velocity v. An element of the boundary of Σ parallel to the y-axis, say ds, sweeps out an area vt × ds in time t. If we integrate around the boundary ∂Σ in a counterclockwise sense, vt × ds points in the negative z-direction on the left side of ∂Σ (where ds points downward), and in the positive z-direction on the right side of ∂Σ (where ds points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of F is increasing on the right of ∂Σ and decreasing on the left. However, the dot product v × F ⋅ ds = − F × v ⋅ ds = − F ⋅ v × ds. Consequently, the sign of the line integral is taken as negative.

 If v is a constant,

 which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.

 Alternative derivation [ edit ]

 Lemma. One has:

 Proof. From the proof of the fundamental theorem of calculus,

 and

 Suppose a and b are constant, and that f(x) involves a parameter α which is constant in the integration but may vary to form different integrals. Assume that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α0 ≤ α ≤ α1 and a ≤ x ≤ b}, and that the partial derivative fα(x, α) exists and is continuous. If one defines:

 then may be differentiated with respect to α by differentiating under the integral sign, i.e.,

 By the Heine–Cantor theorem it is uniformly continuous in that set. In other words, for any ε > 0 there exists Δα such that for all values of x in [a, b],

 On the other hand,

 Hence φ(α) is a continuous function.

 Similarly if exists and is continuous, then for all ε > 0 there exists Δα such that:

 Therefore,

 where

 Now, ε → 0 as Δα → 0, so

 This is the formula we set out to prove.

 Now, suppose

 where a and b are functions of α which take increments Δa and Δb, respectively, when α is increased by Δα. Then,

 A form of the mean value theorem, where a < ξ < b, can be applied to the first and last integrals of the formula for Δφ above, resulting in

 Dividing by Δα, letting Δα → 0, noticing ξ1 → a and ξ2 → b and using the above derivation for

 yields

 This is the general form of the Leibniz integral rule.

 Examples [ edit ]

 Example 1: Fixed limits [ edit ]

 Consider the function

 The function under the integral sign is not continuous at the point (x, α) = (0, 0), and the function φ(α) has a discontinuity at α = 0 because φ(α) approaches ±π/2 as α → 0±.

 If we differentiate φ(α) with respect to α under the integral sign, we get

 which is, of course, true for all values of α except α = 0. This may be integrated (with respect to α) to find

 Example 2: Variable limits [ edit ]

 An example with variable limits:

 Applications [ edit ]

 Evaluating definite integrals [ edit ]

 The formula

 can be of use when evaluating certain definite integrals. When used in this context, the Leibniz integral rule for differentiating under the integral sign is also known as Feynman's trick for integration.

 Example 3 [ edit ]

 Consider

 Now,

 As varies from to , we have

 Hence,

 Therefore,

 Integrating both sides with respect to , we get:

 follows from evaluating :

 To determine in the same manner, we should need to substitute in a value of greater than 1 in . This is somewhat inconvenient. Instead, we substitute , where . Then,

 Therefore,

 The definition of is now complete:

 The foregoing discussion, of course, does not apply when , since the conditions for differentiability are not met.

 Example 4 [ edit ]

 First we calculate:

 The limits of integration being independent of , we have:

 On the other hand:

 Equating these two relations then yields

 In a similar fashion, pursuing yields

 Adding the two results then produces

 which computes as desired.

 This derivation may be generalized. Note that if we define

 it can easily be shown that

 Given , this integral reduction formula can be used to compute all of the values of for . Integrals like and may also be handled using the Weierstrass substitution.

 Example 5 [ edit ]

 Here, we consider the integral

 Differentiating under the integral with respect to , we have

 Therefore:

 But by definition so and

 Example 6 [ edit ]

 Here, we consider the integral

 We introduce a new variable φ and rewrite the integral as

 When φ = 1 this equals the original integral. However, this more general integral may be differentiated with respect to :

 Now, fix φ, and consider the vector field on defined by . Further, choose the positive oriented parametrization of the unit circle given by , , so that . Then the final integral above is precisely

 the line integral of over . By Green's Theorem, this equals the double integral
 where is the closed unit disc. Its integrand is identically 0, so is likewise identically zero. This implies that f(φ) is constant. The constant may be determined by evaluating at :

 Therefore, the original integral also equals .

 Other problems to solve [ edit ]

 There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter :

 The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when . Therefore, differentiation under the integral sign is easy to justify when , but proving that the resulting formula remains valid when requires some careful work.

 Infinite series [ edit ]

 The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.[ citation needed ]

 In popular culture [ edit ]

 Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman! in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:

 One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.

 See also [ edit ]

 * Mathematics portal
 * Chain rule
 * Differentiation of integrals
 * Leibniz rule (generalized product rule)
 * Reynolds transport theorem, a generalization of Leibniz rule

 References [ edit ]

 1. ^ Protter, Murray H.; Morrey, Charles B. Jr. (1985). "Differentiation under the Integral Sign". Intermediate Calculus (Second ed.). New York: Springer. pp. 421–426. doi:10.1007/978-1-4612-1086-3. ISBN 978-0-387-96058-6.
 2. ^ a b Talvila, Erik (June 2001). "Necessary and Sufficient Conditions for Differentiating under the Integral Sign". American Mathematical Monthly. 108 (6): 544–548. arXiv:math/0101012. doi:10.2307/2695709. JSTOR 2695709 . Retrieved 16 April 2022 .
 3. ^ Abraham, Max; Becker, Richard (1950). Classical Theory of Electricity and Magnetism (2nd ed.). London: Blackie & Sons. pp. 39–40.
 4. ^ a b Flanders, Harly (June–July 1973). "Differentiation under the integral sign" (PDF) . American Mathematical Monthly. 80 (6): 615–627. doi:10.2307/2319163. JSTOR 2319163.
 5. ^ Cheng, Steve (22 March 2013). "Differentiation under the integral sign". PlanetMath.org. PlanetMath.org, Ltd . Retrieved 16 April 2022 .
 6. ^ Cheng, Steve (6 September 2010). Differentiation under the integral sign with weak derivatives (Report). CiteSeerX. CiteSeerX 10.1.1.525.2529.
 7. ^ Spivak, Michael (1994). Calculus (3 ed.). Houston, Texas: Publish or Perish, Inc. pp. 267–268. ISBN 978-0-914098-89-8.
 8. ^ Spivak, Michael (1965). Calculus on Manifolds. Addison-Wesley Publishing Company. p. 31. ISBN 978-0-8053-9021-6.

 Further reading [ edit ]

 * Amazigo, John C.; Rubenfeld, Lester A. (1980). "Single Integrals: Leibnitz's Rule; Numerical Integration". Advanced Calculus and its Applications to the Engineering and Physical Sciences. New York: Wiley. pp. 155–165. ISBN 0-471-04934-4.
 * Kaplan, Wilfred (1973). "Integrals Depending on a Parameter—Leibnitz's Rule". Advanced Calculus (2nd ed.). Reading: Addison-Wesley. pp. 285–288.

 External links [ edit ]

 * Harron, Rob. "The Leibniz Rule" (PDF) . MAT-203.
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This article is about the integral rule. For the convergence test for alternating series, see Alternating series test.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
     
where and the integrands are functions dependent on the derivative of this integral is expressible as
     
where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative. [1]   In the special case where the functions and are constants and with values that do not depend on this simplifies to:
    
If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:
    
This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.
General form: differentiation under the integral sign[edit]
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. [2]  This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where , , and .
 If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
     
where is the partial derivative with respect to and is the integral operator with respect to over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.
 The following three basic theorems on the interchange of limits are essentially equivalent:
 the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);
the change of order of partial derivatives;
the change of order of integration (integration under the integral sign; i.e., Fubini's theorem). 
Three-dimensional, time-dependent case[edit]
See also: § Higher dimensions 
      Figure 1: A vector field F(r, t) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.  
A Leibniz integral rule for a two dimensional surface moving in three dimensional space is [3]   [4] 
    
where:
  F(r, t) is a vector field at the spatial position r at time t,
Σ is a surface bounded by the closed curve ∂Σ,
 d A is a vector element of the surface Σ,
 d s is a vector element of the curve ∂Σ,
 v is the velocity of movement of the region Σ,
∇⋅ is the vector divergence,
× is the vector cross product,
The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ. 
 Higher dimensions[edit]
The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:
    
where is a scalar function, D(t) and ∂D(t) denote a time-varying connected region of R 3 and its boundary, respectively, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and d Σ = n dS is the unit normal component of the surface element.
 The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in n dimensions is [4] 
     
where Ω(t) is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, denotes the interior product with , d  x  ω is the exterior derivative of ω with respect to the space variables only and is the time derivative of ω.
 However, all of these identities can be derived from a most general statement about Lie derivatives:
    
Here, the ambient manifold on which the differential form lives includes both space and time.
Something remarkable about this form, is that it can account for the case when changes its shape and size over time, since such deformations are fully determined by .
Measure theory statement[edit]
Let be an open subset of , and be a measure space. Suppose satisfies the following conditions: [5]   [6]   [2] 
  is a Lebesgue-integrable function of for each .
For almost all , the partial derivative exists for all .
There is an integrable function such that for all and almost every . 
Then, for all ,
    
The proof relies on the dominated convergence theorem and the mean value theorem (details below).
Proofs[edit]
Proof of basic form[edit]
We first prove the case of constant limits of integration a and b.
 We use Fubini's theorem to change the order of integration. For every x and h, such that h &gt; 0 and both x and x +h are within [x 0,x 1], we have:
    
Note that the integrals at hand are well defined since is continuous at the closed rectangle and thus also uniformly continuous there; thus its integrals by either dt or dx are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).
 Therefore:
    
Where we have defined:
     
(we may replace x 0 here by any other point between x 0 and x)
  F is differentiable with derivative , so we can take the limit where h approaches zero. For the left hand side this limit is:
    
For the right hand side, we get:
     
And we thus prove the desired result:
    
Another proof using the bounded convergence theorem[edit]
If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign.
 Note that this proof is weaker in the sense that it only shows that fx (x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx (x,t) (and thus is obviously also Lebesgue integrable).
 Let
 
    
  (1) 
   
By the definition of the derivative,
 
    
  (2) 
   
Substitute equation ( 1 ) into equation ( 2 ). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so
    
We now show that the limit can be passed through the integral sign.
 We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each δ &gt; 0, consider the difference quotient
     
For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that
     
Continuity of f  x (x, t) and compactness of the domain together imply that f  x (x, t) is bounded. The above application of the mean value theorem therefore gives a uniform (independent of ) bound on . The difference quotients converge pointwise to the partial derivative f  x  by the assumption that the partial derivative exists.
 The above argument shows that for every sequence {δn } → 0, the sequence is uniformly bounded and converges pointwise to f  x . The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {δn } → 0. Therefore, the limit as δ → 0 may be passed through the integral sign.
Variable limits form[edit]
For a continuous real valued function g of one real variable, and real valued differentiable functions and of one real variable,
    
This follows from the chain rule and the First Fundamental Theorem of Calculus. Define
     
and
     
(The lower limit just has to be some number in the domain of )
 Then, can be written as a composition: . The Chain Rule then implies that
     
By the First Fundamental Theorem of Calculus, . Therefore, substituting this result above, we get the desired equation:
    
 Note: This form can be particularly useful if the expression to be differentiated is of the form:
     
Because does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e.,
    
General form with variable limits[edit]
Set
     
where a and b are functions of α that exhibit increments Δa and Δb, respectively, when α is increased by Δα. Then,
    
A form of the mean value theorem, , where a &lt; ξ &lt; b, may be applied to the first and last integrals of the formula for Δφ above, resulting in
    
Divide by Δα and let Δα → 0. Notice ξ 1 → a and ξ 2 → b. We may pass the limit through the integral sign:
     
again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule,
    
Alternative proof of the general form with variable limits, using the chain rule[edit]
The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the First Fundamental Theorem of Calculus. Suppose is defined in a rectangle in the plane, for and . Also, assume and the partial derivative are both continuous functions on this rectangle. Suppose are differentiable real valued functions defined on , with values in (i.e. for every ). Now, set
     
and 
    
Then, by properties of Definite Integrals, we can write
    
Since the functions are all differentiable (see the remark at the end of the proof), by the Multivariable Chain Rule, it follows that is differentiable, and its derivative is given by the formula:
     Now, note that for every , and for every , we have that , because when taking the partial derivative with respect to of , we are keeping fixed in the expression ; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the First Fundamental Theorem of Calculus, we have that ; because when taking the partial derivative with respect to of , the first variable is fixed, so the fundamental theorem can indeed be applied.
 Substituting these results into the equation for above gives:
     
as desired.
 There is a technical point in the proof above which is worth noting: applying the Chain Rule to requires that already be differentiable. This is where we use our assumptions about . As mentioned above, the partial derivatives of are given by the formulas and . Since is continuous, its integral is also a continuous function, [7]  and since is also continuous, these two results show that both the partial derivatives of are continuous. Since continuity of partial derivatives implies differentiability of the function, [8]  is indeed differentiable.
Three-dimensional, time-dependent form[edit]
See also: § Higher dimensions 
At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid . The function can be written as
     
with independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at . For a rigidly translating surface, the limits of integration are then independent of time, so:
     
where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only:
     
with the velocity of motion of the surface defined by
    
This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on curl)
     
and that Stokes theorem equates the surface integral of the curl over Σ with a line integral over ∂Σ:
    
The sign of the line integral is based on the right-hand rule for the choice of direction of line element d s. To establish this sign, for example, suppose the field F points in the positive z-direction, and the surface Σ is a portion of the xy-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive z-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along z-axis). Then the integral on the left-hand side determines a positive flux of F through Σ. Suppose Σ translates in the positive x-direction at velocity v. An element of the boundary of Σ parallel to the y-axis, say d s, sweeps out an area v t × d s in time t. If we integrate around the boundary ∂Σ in a counterclockwise sense, v t × d s points in the negative z-direction on the left side of ∂Σ (where d s points downward), and in the positive z-direction on the right side of ∂Σ (where d s points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of F is increasing on the right of ∂Σ and decreasing on the left. However, the dot product v × F ⋅ d s = −F × v ⋅ d s = −F ⋅ v × d s. Consequently, the sign of the line integral is taken as negative.
 If v is a constant,
     
which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.
Alternative derivation[edit]
 Lemma. One has:
    
 Proof. From the proof of the fundamental theorem of calculus,
     
and
    
Suppose a and b are constant, and that f(x) involves a parameter α which is constant in the integration but may vary to form different integrals. Assume that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α 0 ≤ α ≤ α 1 and a ≤ x ≤ b}, and that the partial derivative f  α (x, α) exists and is continuous. If one defines:
     
then may be differentiated with respect to α by differentiating under the integral sign, i.e.,
    
By the Heine–Cantor theorem it is uniformly continuous in that set. In other words, for any ε &gt; 0 there exists Δα such that for all values of x in [a, b],
    
On the other hand,
    
Hence φ(α) is a continuous function.
 Similarly if exists and is continuous, then for all ε &gt; 0 there exists Δα such that:
    
Therefore,
     
where
    
Now, ε → 0 as Δα → 0, so
    
This is the formula we set out to prove.
 Now, suppose
     
where a and b are functions of α which take increments Δa and Δb, respectively, when α is increased by Δα. Then,
    
A form of the mean value theorem, where a &lt; ξ &lt; b, can be applied to the first and last integrals of the formula for Δφ above, resulting in
    
Dividing by Δα, letting Δα → 0, noticing ξ 1 → a and ξ 2 → b and using the above derivation for
     
yields
    
This is the general form of the Leibniz integral rule.
Examples[edit]
Example 1: Fixed limits[edit]
Consider the function
    
The function under the integral sign is not continuous at the point (x, α) = (0, 0), and the function φ(α) has a discontinuity at α = 0 because φ(α) approaches ±π/2 as α → 0±.
 If we differentiate φ(α) with respect to α under the integral sign, we get
     
which is, of course, true for all values of α except α = 0. This may be integrated (with respect to α) to find
    
Example 2: Variable limits[edit]
An example with variable limits:
    
Applications[edit]
Evaluating definite integrals[edit]
The formula
     
can be of use when evaluating certain definite integrals. When used in this context, the Leibniz integral rule for differentiating under the integral sign is also known as Feynman's trick for integration.
Example 3[edit]
Consider
    
Now,
    
As varies from to , we have
    
Hence,
    
Therefore,
    
Integrating both sides with respect to , we get:
    
 follows from evaluating :
    
To determine in the same manner, we should need to substitute in a value of greater than 1 in . This is somewhat inconvenient. Instead, we substitute , where . Then,
    
Therefore, 
 The definition of is now complete:
    
The foregoing discussion, of course, does not apply when , since the conditions for differentiability are not met.
Example 4[edit]
   
First we calculate:
    
The limits of integration being independent of , we have:
    
On the other hand:
    
Equating these two relations then yields
    
In a similar fashion, pursuing yields
    
Adding the two results then produces
     
which computes as desired.
 This derivation may be generalized. Note that if we define
     
it can easily be shown that
    
Given , this integral reduction formula can be used to compute all of the values of for . Integrals like and may also be handled using the Weierstrass substitution.
Example 5[edit]
Here, we consider the integral
    
Differentiating under the integral with respect to , we have
    
Therefore:
    
But by definition so and
    
Example 6[edit]
Here, we consider the integral
    
We introduce a new variable φ and rewrite the integral as
    
When φ = 1 this equals the original integral. However, this more general integral may be differentiated with respect to :
    
Now, fix φ, and consider the vector field on defined by . Further, choose the positive oriented parametrization of the unit circle given by , , so that . Then the final integral above is precisely
     
the line integral of over . By Green's Theorem, this equals the double integral
     
where is the closed unit disc. Its integrand is identically 0, so is likewise identically zero. This implies that f(φ) is constant. The constant may be determined by evaluating at :
    
Therefore, the original integral also equals .
Other problems to solve[edit]
There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter :
    
The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when . Therefore, differentiation under the integral sign is easy to justify when , but proving that the resulting formula remains valid when requires some careful work.
Infinite series[edit]
The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.[ citation needed ]
In popular culture[edit]
Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir  Surely You're Joking, Mr. Feynman!  in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:
One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.
See also[edit]
  Chain rule 
 Differentiation of integrals 
 Leibniz rule (generalized product rule) 
 Reynolds transport theorem, a generalization of Leibniz rule 
References[edit]
 
  ^  Protter, Murray H.; Morrey, Charles B. Jr. (1985). "Differentiation under the Integral Sign". Intermediate Calculus (Second ed.). New York: Springer. pp. 421–426. doi:10.1007/978-1-4612-1086-3. ISBN  978-0-387-96058-6 .
^    a       b    Talvila, Erik (June 2001). "Necessary and Sufficient Conditions for Differentiating under the Integral Sign". American Mathematical Monthly. 108 (6): 544–548. arXiv:math/0101012. doi:10.2307/2695709. JSTOR 2695709. Retrieved 16 April 2022.
  ^  Abraham, Max; Becker, Richard (1950). Classical Theory of Electricity and Magnetism (2nd ed.). London: Blackie &amp; Sons. pp. 39–40.
^    a       b     Flanders, Harly (June–July 1973). "Differentiation under the integral sign" (PDF).  American Mathematical Monthly . 80 (6): 615–627. doi:10.2307/2319163. JSTOR 2319163.
  ^  Cheng, Steve (22 March 2013). "Differentiation under the integral sign". PlanetMath.org. PlanetMath.org, Ltd. Retrieved 16 April 2022.
  ^  Cheng, Steve (6 September 2010). Differentiation under the integral sign with weak derivatives (Report). CiteSeerX. CiteSeerX 10.1.1.525.2529.
  ^  Spivak, Michael (1994).  Calculus  (3 ed.). Houston, Texas: Publish or Perish, Inc. pp. 267–268. ISBN  978-0-914098-89-8 .
  ^  Spivak, Michael (1965). Calculus on Manifolds. Addison-Wesley Publishing Company. p. 31. ISBN  978-0-8053-9021-6 .
 
Further reading[edit]
  Amazigo, John C.; Rubenfeld, Lester A. (1980). "Single Integrals: Leibnitz's Rule; Numerical Integration". Advanced Calculus and its Applications to the Engineering and Physical Sciences. New York: Wiley. pp. 155–165. ISBN  0-471-04934-4 . 
  Kaplan, Wilfred (1973). "Integrals Depending on a Parameter—Leibnitz's Rule". Advanced Calculus (2nd ed.). Reading: Addison-Wesley. pp. 285–288.  
External links[edit]
  Harron, Rob. "The Leibniz Rule" (PDF). MAT-203.
Leibniz integral rule - Wikipedia
https://en.wikipedia.org/wiki/Leibniz_integral_rule
https://mathworld.wolfram.com/LeibnizIntegralRule.html
Leibniz Integral Rule -- from Wolfram MathWorld
2003-12-13
Weisstein; Eric W
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<p>
The Leibniz integral rule gives a formula for differentiation of a <a href="/DefiniteIntegral.html">definite
integral</a> whose limits are functions of the differential variable,
</p>
<div>
<table>
<tbody><tr><td></td><td>
<p>
(1)
</p>
</td></tr>
</tbody></table>
</div>
<p>
It is sometimes known as differentiation under the integral sign.
</p>
<p>
This rule can be used to evaluate certain unusual definite integrals such as
</p>
<p>
for 
 (Woods 1926).
</p>
<p>
Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."
</p>
</div>
<hr />
<div>
<h2>See also</h2><p><a href="/Derivative.html">Derivative</a>, <a href="/Integral.html">Integral</a>,
<a href="/IntegrationUndertheIntegralSign.html">Integration Under the Integral Sign</a></p>
<h2>Explore with Wolfram|Alpha</h2>
<h2>References</h2><p><cite>Abramowitz, M. and Stegun, I. A. (Eds.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.</a></i>
 New York: Dover, p. 11, 1972.</cite><cite>Beyer, W. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1584882913/ref=nosim/ericstreasuretro">CRC
 Standard Mathematical Tables, 28th ed.</a></i> Boca Raton, FL: CRC Press, p. 232,
 1987.</cite><cite>Boros, G. and Moll, V. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521796369/ref=nosim/ericstreasuretro">Irresistible
 Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals.</a></i>
 Cambridge, England: Cambridge University Press, pp. 20-21, 2004.</cite><cite>Feynman,
 R. P. "A Different Set of Tools." In <i><a href="http://www.amazon.com/exec/obidos/ASIN/0393316041/ref=nosim/ericstreasuretro">'Surely
 You're Joking, Mr. Feynman!': Adventures of a Curious Character.</a></i> New York:
 W. W. Norton, 1997.</cite><cite>Hijab, O. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387949267/ref=nosim/ericstreasuretro">Introduction
 to Calculus and Classical Analysis.</a></i> New York: Springer-Verlag, p. 189,
 1997.</cite><cite>Kaplan, W. "Integrals Depending on a Parameter--Leibnitz's
 Rule." §4.9 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201578883/ref=nosim/ericstreasuretro">Advanced
 Calculus, 4th ed.</a></i> Reading, MA: Addison-Wesley, pp. 256-258, 1992.</cite><cite>Woods,
 F. S. "Differentiation of a Definite Integral." §60 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/B00085L67S/ref=nosim/ericstreasuretro">Advanced
 Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied
 Mathematics.</a></i> Boston, MA: Ginn, pp. 141-144, 1926.</cite></p><h2>Referenced
 on Wolfram|Alpha</h2><p><a href="http://www.wolframalpha.com/entities/mathworld/leibniz_integral_rule/su/i4/57/">Leibniz Integral Rule</a></p>
<h2>Cite this as:</h2>
<p>
<a href="/about/author.html">Weisstein, Eric W.</a> "Leibniz Integral Rule."
From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/LeibnizIntegralRule.html">https://mathworld.wolfram.com/LeibnizIntegralRule.html</a>
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 Leibniz Integral Rule

 Download Wolfram Notebook

 The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable,

 (1)
 

 It is sometimes known as differentiation under the integral sign.

 This rule can be used to evaluate certain unusual definite integrals such as

 (2)
 (3)
 

 for (Woods 1926).

 Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."

 See also

 Derivative, Integral, Integration Under the Integral Sign

 Explore with Wolfram|Alpha

 More things to try:

 * second fundamental theorem of calculus
 * 8:5 odds, bet 97 euros
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 References

 Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 232, 1987.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, pp. 20-21, 2004.Feynman, R. P. "A Different Set of Tools." In 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, 1997.Hijab, O. Introduction to Calculus and Classical Analysis. New York: Springer-Verlag, p. 189, 1997.Kaplan, W. "Integrals Depending on a Parameter--Leibnitz's Rule." §4.9 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 256-258, 1992.Woods, F. S. "Differentiation of a Definite Integral." §60 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 141-144, 1926.

 Referenced on Wolfram|Alpha

 Leibniz Integral Rule

 Cite this as:

 Weisstein, Eric W. "Leibniz Integral Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizIntegralRule.html

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The Leibniz integral rule gives a formula for differentiation of a definite
integral whose limits are functions of the differential variable,
    
(1)
 
 
It is sometimes known as differentiation under the integral sign.
This rule can be used to evaluate certain unusual definite integrals such as
for 
 (Woods 1926).
Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."
See also  Derivative, Integral,
Integration Under the Integral Sign 
Explore with Wolfram|Alpha
References  Abramowitz, M. and Stegun, I. A. (Eds.).  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. 
 New York: Dover, p. 11, 1972. Beyer, W. H.  CRC
 Standard Mathematical Tables, 28th ed.  Boca Raton, FL: CRC Press, p. 232,
 1987. Boros, G. and Moll, V.  Irresistible
 Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. 
 Cambridge, England: Cambridge University Press, pp. 20-21, 2004. Feynman,
 R. P. "A Different Set of Tools." In  'Surely
 You're Joking, Mr. Feynman!': Adventures of a Curious Character.  New York:
 W. W. Norton, 1997. Hijab, O.  Introduction
 to Calculus and Classical Analysis.  New York: Springer-Verlag, p. 189,
 1997. Kaplan, W. "Integrals Depending on a Parameter--Leibnitz's
 Rule." §4.9 in  Advanced
 Calculus, 4th ed.  Reading, MA: Addison-Wesley, pp. 256-258, 1992. Woods,
 F. S. "Differentiation of a Definite Integral." §60 in  Advanced
 Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied
 Mathematics.  Boston, MA: Ginn, pp. 141-144, 1926.  Referenced
 on Wolfram|Alpha  Leibniz Integral Rule 
Cite this as:
Weisstein, Eric W. "Leibniz Integral Rule."
From  MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizIntegralRule.html
Subject classifications
Leibniz Integral Rule -- from Wolfram MathWorld
https://mathworld.wolfram.com/LeibnizIntegralRule.html
https://mathgarage.wordpress.com/tag/leibniz-rule/
Leibniz Rule – mathgarage
2020-09-12
Author Kelvinella
<div><div>
<p>Leibniz Integral Rule states that given a function f(<em>x</em>, <em>t</em>) is continuous and continuously differentiable such as the partial derivatives exist and continuous and both the limit <em>a</em>(<em>x</em>) and <em>b</em>(<em>x</em>) are continuous and continuously differentiable, then</p>
<p></p>
<p>If <em>a</em>(<em>x</em>) and <em>b</em>(<em>x</em>) are both constant functions, then the above formula reduces to the simpler form:</p>
<p></p>
<p><strong><em>For example:</em></strong></p>
<figure></figure>
<p>Let , then</p>
<p></p>
<p>and hence,</p>
<p></p>
<p>To get rid of the constant <em>C</em>, notice that <em>I</em>(1) = 0, so</p>
<p></p>
<p></p>
<p>Therefore,</p>
<p></p>
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 Posts Tagged ‘Leibniz Rule’

 Integration using Leibniz Rule

 Posted: September 12, 2020 in Mathematics
 Tags: Integration, Leibniz Rule, math
 0

 Leibniz Integral Rule states that given a function f(x, t) is continuous and continuously differentiable such as the partial derivatives exist and continuous and both the limit a(x) and b(x) are continuous and continuously differentiable, then

 If a(x) and b(x) are both constant functions, then the above formula reduces to the simpler form:

 For example:

 Let , then

 and hence,

 To get rid of the constant C, notice that I(1) = 0, so

 Therefore,

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Leibniz Integral Rule states that given a function f(x, t) is continuous and continuously differentiable such as the partial derivatives exist and continuous and both the limit a(x) and b(x) are continuous and continuously differentiable, then
 
If a(x) and b(x) are both constant functions, then the above formula reduces to the simpler form:
 
  For example:  
 
Let , then
 
and hence,
 
To get rid of the constant C, notice that I(1) = 0, so
 
 
Therefore,
Leibniz Rule – mathgarage
https://mathgarage.wordpress.com/tag/leibniz-rule/
http://ht.ly/779Xe
integralCALC
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<p>
</p></div></div>||||I|||| @KristaKingMath created this 1 decade ago using Hootsuite
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https://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/
A Geometrical Theorem of Leibniz - Wolfram Demonstrations Project
2011-03-07
Download
<div><div>
 <h2>A Geometrical Theorem of Leibniz</h2>
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<p>Let P be a point in the plane of the triangle RST and C be the centroid of RST. Then .</p>
 
<div>
<p>
	Contributed by: <a href="http://demonstrations.wolfram.com/author.html?author=Jay+Warendorff">Jay Warendorff</a> (March 2011)
	<br />Open content licensed under <a href="http://www.wolfram.com/legal/terms/wolfram-demonstrations-project.html">CC BY-NC-SA</a>
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 <p>See also "<a href="http://arxiv.org/abs/0710.0342">A Generalization of a Leibniz Geometrical Theorem</a>".</p>
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 <a href="http://demonstrations.wolfram.com/author.html?author=Jay+Warendorff">Jay Warendorff</a>
 <a href="http://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/">"A Geometrical Theorem of Leibniz"</a><br />
 <a href="http://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/">http://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/</a><br />
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 Published: March 7 2011
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 A Geometrical Theorem of Leibniz

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 Let P be a point in the plane of the triangle RST and C be the centroid of RST. Then .

 Contributed by: Jay Warendorff (March 2011)
 Open content licensed under CC BY-NC-SA

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 See also "A Generalization of a Leibniz Geometrical Theorem".

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 Jay Warendorff "A Geometrical Theorem of Leibniz"
 http://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/
 Wolfram Demonstrations Project
 Published: March 7 2011

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A Geometrical Theorem of Leibniz
 
 
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Let P be a point in the plane of the triangle RST and C be the centroid of RST. Then .
 
	Contributed by: Jay Warendorff (March 2011)
	Open content licensed under CC BY-NC-SA
	
 
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 Details
 See also "A Generalization of a Leibniz Geometrical Theorem".
 
 
 
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   Triangle Centroid (Wolfram MathWorld )
 
 
 
 
 
 Permanent Citation
 
 Jay Warendorff
 "A Geometrical Theorem of Leibniz" 
 http://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/ 
 Wolfram Demonstrations Project 
 Published: March 7 2011
 
 
 
 
 
 
 
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 Take advantage of the Wolfram Notebook Emebedder for the recommended user experience.
 
 
 
 
 
  
  Related Demonstrations
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  Stewart's Theorem   Jay Warendorff 
  Routh's Theorem   Jay Warendorff 
  Miquel's Theorem   Jay Warendorff 
  Menelaus' Theorem   Jay Warendorff 
  Ceva's Theorem   Jay Warendorff 
  Viviani's Theorem   Jay Warendorff 
  Napoleon's Theorem   Jay Warendorff 
  Cross's Theorem   Jay Warendorff 
  Kosnita's Theorem   Jay Warendorff 
  Stengel's Theorem   Jay Warendorff 
 
 
 
 
Mixtilinear Incircles 
Jay Warendorff 
Total Areas of Alternating Subtriangles in a Regular Polygon with 2n Sides 
Jay Warendorff 
The Sum of Opposite Angles of a Quadrilateral in a Circle is 180 Degrees 
Jay Warendorff 
Comparing Sorting Algorithms on Rainbow-Colored Bar Charts 
Jay Warendorff 
Triangle Altitudes and Circumradius 
Jay Warendorff 
Another Variant of Kosnita's Theorem 
Jay Warendorff 
The Begonia Point 
Jay Warendorff 
An Application of the Gergonne-Euler Theorem 
Jay Warendorff 
The Knight's Tour 
Jay Warendorff 
The Excentral Triangle and a Related Hexagon 
Jay Warendorff 
Four Concyclic Points 
Jay Warendorff 
Cross's Theorem 
Jay Warendorff 
A Concurrency from Midpoints of Arcs of the Circumcircle 
Jay Warendorff 
A Concurrency of Lines through Points of Tangency with Excircles 
Jay Warendorff 
A Concurrency from Six Pedal Points 
Jay Warendorff 
Zeckendorf Representation of Integers 
Jay Warendorff 
A 2011 IMO Tangency Problem 
Jay Warendorff 
The Fuhrmann Circle 
Jay Warendorff 
A Chinese Knight's Tour 
Jay Warendorff 
The Triangles Formed by the Endpoints and Midpoints of Cevians 
Jay Warendorff 
 
 
 
 
 
 
 Related Topics
 
  Plane Geometry 
  Triangles 
 
  
 
  Browse all topics
A Geometrical Theorem of Leibniz - Wolfram Demonstrations Project
https://demonstrations.wolfram.com/AGeometricalTheoremOfLeibniz/
https://www.johndcook.com/blog/2021/07/09/starts-with-an-integral-sign/
Anything that starts with an integral sign
2021-07-09
John
<div><div>
	
<article>
	<div>
	<p>Ran across a fun quote this afternoon:</p>
<blockquote><p>There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.</p>
<p><i>Bulletin of the American Mathematical Society</i>, v. 69, p. 611, 1963.</p></blockquote>
<p>As much as mathematicians like to joke about the gaps between pure and applied math, as in the quote above, the gap is not as wide as supposed. Things like <a href="https://www.johndcook.com/blog/2015/10/20/impulse-response/">delta</a> “functions,” or <a href="https://www.johndcook.com/blog/2009/10/25/how-to-differentiate-a-non-differentiable-function/">differentiating non-differentiable functions</a>, or <a href="https://www.johndcook.com/blog/2018/09/06/sum-a-divergent-series/">summing divergent series</a> have a rigorous theory behind them.</p>
<p>It is true that these things were used in practice before theory caught up, but the theory caught up a long time ago. There are still instances where hand-waving calculation is ahead of theory (or just wrong), and there always will be, but distribution theory and asymptotic analysis provided a rigorous justification for a lot of seemingly non-rigorous math decades ago. This is well-known to some mathematicians and unknown to others.</p>
<p>That said, you still have to be careful. Some formal calculations lead to false results. They don’t just lack rigorous theory; they’re simply wrong.</p>
	</div>
	
</article>
	 
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 Anything that starts with an integral sign

 Posted on 9 July 2021 by John

 Ran across a fun quote this afternoon:

 There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.

 Bulletin of the American Mathematical Society, v. 69, p. 611, 1963.

 As much as mathematicians like to joke about the gaps between pure and applied math, as in the quote above, the gap is not as wide as supposed. Things like delta “functions,” or differentiating non-differentiable functions, or summing divergent series have a rigorous theory behind them.

 It is true that these things were used in practice before theory caught up, but the theory caught up a long time ago. There are still instances where hand-waving calculation is ahead of theory (or just wrong), and there always will be, but distribution theory and asymptotic analysis provided a rigorous justification for a lot of seemingly non-rigorous math decades ago. This is well-known to some mathematicians and unknown to others.

 That said, you still have to be careful. Some formal calculations lead to false results. They don’t just lack rigorous theory; they’re simply wrong.

 Categories : Math
 Tags : Integration
 Bookmark the permalink

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 Previous Post Technological boundary layer
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 One thought on “ Anything that starts with an integral sign ”

 1. Jonathan
 10 July 2021 at 08:21

 The people who abuse notation are either really bad or really really good at mathematics. As someone who is solidly middling, I’m irritated by the former and in awe of the latter.

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Ran across a fun quote this afternoon:
 There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.
 Bulletin of the American Mathematical Society, v. 69, p. 611, 1963. 
As much as mathematicians like to joke about the gaps between pure and applied math, as in the quote above, the gap is not as wide as supposed. Things like delta “functions,” or differentiating non-differentiable functions, or summing divergent series have a rigorous theory behind them.
It is true that these things were used in practice before theory caught up, but the theory caught up a long time ago. There are still instances where hand-waving calculation is ahead of theory (or just wrong), and there always will be, but distribution theory and asymptotic analysis provided a rigorous justification for a lot of seemingly non-rigorous math decades ago. This is well-known to some mathematicians and unknown to others.
That said, you still have to be careful. Some formal calculations lead to false results. They don’t just lack rigorous theory; they’re simply wrong.
Anything that starts with an integral sign
https://www.johndcook.com/blog/2021/07/09/starts-with-an-integral-sign/