Harman Patil


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The gradient (or gradient vector field) of a scalar function f(x1, x2, x3,... xn) is denoted f or 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,

( f ( x ) ) v = D v f ( x ) .


f = f x 1 e 1 + + f x n e n


Cartesian coordinates

In the three-dimensional Cartesian coordinate system, the gradient is given by:

f = f x i + f y j + f z k


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


f = 2 i + 6 y j cos ( z ) k .


Cylindrical and spherical coordinates

In cylindrical coordinates, the gradient is given by:

f ( ρ , φ , z ) = f ρ e ρ + 1 ρ f φ e φ + f z e z

where φ is the azimuthal or azmuth angle, z is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions.

In spherical coordinates, the gradient is given by:

f ( r , θ , φ ) = f r e r + 1 r f θ e θ + 1 r sin θ f φ e φ

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


Linear approximation to a function

The gradient of a function f from the Euclidean space n to at any particular point x0 in n characterizes the best linear approximation to f at x0. The approximation is as follows:

f ( x ) f ( x 0 ) + ( f ) x 0 ( x x 0 )

for x close to x0, where (∇f )x0 is the gradient of f computed at x0, and the dot denotes the dot product on n. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f at x0.

Differential or (exterior) derivative

The best linear approximation to a differentiable function

f : R n R

at a point x in n is a linear map from n to which is often denoted by dfx or Df(x) and called the differential or (total) derivative of f at x. The gradient is therefore related to the differential by the formula

( f ) x v = d f x ( v )


If n is viewed as the space of (dimension n) column vectors (of real numbers), then one can regard df as the row vector with components

( f x 1 , , f x n )


( f ) i = d f i T .

Gradient as a derivative

Let U be an open set in n. If the function f : U → ℝ is differentiable, then the differential of f is the (Fréchet) derivative of f. Thus f is a function from U to the space such that

lim h 0 f ( x + h ) f ( x ) f ( x ) h h = 0

where · is the dot product.



The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a ∈ ℝn, and α and β are two constants, then αf + βg is differentiable at a, and moreover

( α f + β g ) ( a ) = α f ( a ) + β g ( a ) .

Product rule

If f and g are real-valued functions differentiable at a point a ∈ ℝn, then the product rule asserts that the product fg is differentiable at a, and

( f g ) ( a ) = f ( a ) g ( a ) + g ( a ) f ( a ) .

Chain rule

Suppose that f : A → ℝ is a real-valued function defined on a subset A of n, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → ℝn maps a subset I ⊂ ℝ into n. If g is differentiable at a point cI such that g(c) = a, then

( f g ) ( c ) = f ( a ) g ( c ) ,

where ∘ is the composition operator: ( f ∘ g)(x) = f(g(x)).

More generally, if instead I ⊂ ℝk, then the following holds:

( f g ) ( c ) = ( D g ( c ) ) T ( f ( a ) )

where (Dg)T denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that h : I → ℝ is a real valued function on a subset I of , and that h is differentiable at the point f(a) ∈ I. Then

( h f ) ( a ) = h ( f ( a ) ) f ( a ) .

Level sets




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 is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

Conservative vector fields and the gradient theorem


Riemannian manifolds

For any smooth function f on a Riemannian manifold (M,g), the gradient of f is the vector field f such that for any vector field X,

g ( f , X ) = X f , i.e., g x ( ( f ) x , X x ) = ( X f ) ( x )

where gx( , ) denotes the inner product of tangent vectors at x defined by the metric g and Xf (sometimes denoted X( f )) is the function that takes any point xM to the directional derivative of f in the direction X, evaluated at x. In other words, in a coordinate chart φ from an open subset of M to an open subset of n, (∂Xf )(x) is given by:

j = 1 n X j ( φ ( x ) ) x j ( f φ 1 ) | φ ( x )


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

f = g i k f x k x i .

Generalizing the case M = ℝn, the gradient of a function is related to its exterior derivative, since

( X f ) ( x ) = ( d f ) x ( X x ) .

More precisely, the gradient f is the vector field associated to the differential 1-form df using the musical isomorphism

= g : T M T M

(called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on n is a special case of this in which the metric is the flat metric given by the dot product.

Gradient of a vector

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.

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

f = g j k f i x j e i e k

where the Einstein summation notation is used and the product of the vectors ei, ek is a dyadic tensor of type (2,0), or the Jacobian matrix

f i x j = ( f 1 , f 2 , f 3 ) ( x 1 , x 2 , x 3 ) .

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

f = g j k ( f i x j + Γ i j l f l ) e i e k

where gjk are the components of the metric tensor and the ei are the coordinate vectors.

Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor:

a f b = g a c c f b

where c is the connection.


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