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In mathematics, the gradient is a generalization of the usual concept of derivative to the functions of several variables. If f(x1, ..., xn) is a differentiable function of several variables, also called "scalar field", its gradient is the vector of the n partial derivatives of f. It is thus a vector-valued function also called vector field.

Similarly to the usual derivative, the gradient represents the slope of and the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function and its magnitude is the slope of the graph in that direction. The components of the gradient are the non-constant coefficients of the equation of the tangent space to the graph.

Let f be differentiable function defined on a Euclidean space. It becomes a differentiable function of several variables as soon as one chooses an orthonormal frame. The gradient does not depend of the choice of this orthonormal frame. It follows that one may speak of the gradient of f without choosing explicitly a frame.

The Jacobian is the generalization of the gradient for vector valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.


Gradient of the 2-d function f(x,y)= xe^{-x^2 - y^2} is plotted as blue arrows over the pseudocolor plot of the function.

Consider a room in which the temperature is given by a scalar field, T, so at each point (x,y,z) the temperature is T(x,y,z). (We will assume that the temperature does not change over time.) At each point in the room, the gradient of T at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.






The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, ..., xn) is denoted ∇f or \vec{\nabla} f where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,

(\nabla f(x))\cdot \mathbf{v} = D_{\mathbf v}f(x).


 \nabla f  = \frac{\partial f}{\partial x_1 }\mathbf{e}_1 + \cdots + \frac{\partial f}{\partial x_n }\mathbf{e}_n



\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y}  \mathbf{j} +
\frac{\partial f}{\partial z} \mathbf{k}


f(x,y,z)= \ 2x+3y^2-\sin(z)


\nabla f= 
 = 2\mathbf{i}+ 6y\mathbf{j} -\cos(z)\mathbf{k}.


Gradient and the derivative or differential[edit]

Linear approximation to a function[edit]


 f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)

for x close to x0, where (\nabla f)_{x_0} is the gradient of f computed at x0, and the dot denotes the dot product on ℝn. This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of f at x0.

Differential or (exterior) derivative[edit]

The best linear approximation to a function



 (\nabla f)_x\cdot v = \mathrm d f_x(v)





(\nabla f)_i = \mathrm{d} f^\mathsf{T}_i.

Gradient as a derivative[edit]


\lim_{h\to 0} \frac{\|f(x+h)-f(x) -\nabla f(x)\cdot h\|}{\|h\|} = 0

where ⋅ is the dot product.




\nabla\left(\alpha f+\beta g\right)(a) = \alpha \nabla f(a) + \beta\nabla g (a).
Product rule

If f and g are real-valued functions differentiable at a point aRn, then the product rule asserts that the product (fg)(x) = f(x)g(x) of the functions f and g is differentiable at a, and

\nabla (fg)(a) = f(a)\nabla g(a) + g(a)\nabla f(a).
Chain rule


(f\circ g)'(c) = \nabla f(a)\cdot g'(c),

where ∘ is the composition operator : (g ∘ f )(x) = g(f(x)). More generally, if instead IRk, then the following holds:

\nabla (f\circ g)(c) = (Dg(c))^\mathsf{T} (\nabla f(a))

where (Dg)T denotes the transpose Jacobian matrix.


\nabla (h\circ f)(a) = h'(f(a))\nabla f(a).

Further properties and applications[edit]

Level sets[edit]




Similarly, an affine algebraic hypersurface may be defined by an equation F(x1, ..., xn) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

Conservative vector fields and the gradient theorem[edit]


Riemannian manifolds[edit]


g(\nabla f, X) = \partial_X f, \qquad \text{i.e.,}\quad g_x((\nabla f)_x, X_x ) = (\partial_X f) (x)


\sum_{j=1}^n X^{j} (\varphi(x)) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Big|_{\varphi(x)},


So, the local form of the gradient takes the form:

 \nabla f= g^{ik}\frac{\partial f}{\partial x^{k}}\frac{\partial}{\partial x^{i}}.


(\partial_X f) (x) = df_x(X_x)\ .


\sharp=\sharp^g\colon T^*M\to TM


Cylindrical and spherical coordinates[edit]

In cylindrical coordinates, the gradient is given by (Schey 1992, pp. 139–142):

\nabla f(\rho, \phi, z) = 
\frac{\partial f}{\partial \rho}\mathbf{e}_\rho+
\frac{1}{\rho}\frac{\partial f}{\partial \phi}\mathbf{e}_\phi+
\frac{\partial f}{\partial z}\mathbf{e}_z


In spherical coordinates (Schey 1992, pp. 139–142):

\nabla f(r, \theta, \phi) = 
\frac{\partial f}{\partial r}\mathbf{e}_r+
\frac{1}{r}\frac{\partial f}{\partial \theta}\mathbf{e}_\theta+
\frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi}\mathbf{e}_\phi

where ϕ is the azimuth angle and θ is the zenith angle.


Gradient of a vector[edit]

In rectangular coordinates, the gradient of a vector f = (f1, f2, f3) is defined by

\nabla \mathbf{f}=\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}{{\mathbf{e}}_{i}}{{\mathbf{e}}_{j}}

or the Jacobian matrix

\frac{\partial ({{f}_{1}},{{f}_{2}},{{f}_{3}})}{\partial ({{x}_{1}},{{x}_{2}},{{x}_{3}})}.

In curvilinear coordinates, the gradient involves Christoffel symbols.

See also[edit]


  • Korn, Theresa M.; Korn, Granino Arthur (2000), Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, New York: Dover Publications, pp. 157–160, ISBN 0-486-41147-8, OCLC 43864234 .
  • Dubrovin, B.A.; A.T. Fomenko, S.P. Novikov (1991), Modern Geometry--Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (2nd ed.), Springer, pp. 14–17, ISBN 978-0-387-97663-1 

External links[edit]