The gradient in three-dimensional Cartesian coordinates:

The gradient using an orthonormal basis for three-dimensional cylindrical coordinates:

The gradient in two dimensions:

Use del to enter ∇ and to enter the list of subscripted variables:

Use grad to enter the template ∇_; press to move between inputs:

The gradient of a vector field in Cartesian coordinates, the Jacobian matrix:

Compute the Hessian of a scalar function:

In a curvilinear coordinate system, a vector with constant components may have a nonzero gradient:

Gradient specifying metric, coordinate system, and parameters:

Grad works on curved spaces:

The gradient of the norm TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3] with respect to TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3] is the unit vector in the direction of TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3] :

The gradient of TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3] with respect to itself is SymbolicIdentityArray [{n}]:

The power rule for vectors gives a matrix whose diagonal follows the normal power rule:

Write out this result in three dimensions:

The gradient of the coordinates with respect to themselves:

This is the identity matrix in n dimensions:

The gradient of the norm in n dimensions:

Specialize to the result in three dimensions:

Compute the force from a potential function:

Find the critical points of a function of two variables:

Compute the signs of and the Hessian determinant at the three critical points:

By the second derivative test, the first two points—red and blue in the plot—are minima, and the third—green in the plot—is a saddle point:

Find the critical points of a function of three variables:

Compute the Hessian matrix of :

When the eigenvalues of a critical point all have the same sign, the point is a local extremum; if there are both positive and negative values, it is a saddle point:

Since the first two points have all positive eigenvalues, they are local minima; the global minimum is found by evaluating at those two points:

For this function, any three of the critical points are linearly dependent, so they all lie in a single plane:

Compute the normal to that plane:

Visualize the function, with the minima yellow and the non-extreme critical points cyan:

Compute the gradients at the origin of a vector-valued function up to order 6:

View the value of the function and its first three derivatives at the origin:

Compute the third-order Maclaurin series of f by hand:

Programmatically compute the sixth-order Maclaurin series:

Grad [f,vars] is effectively equivalent to D [f,{vars}]:

Grad adds a new tensor slot at the end, corresponding to a new innermost dimension:

To add the new slot at the beginning, use Transpose to exchange the first and last slots:

This is relevant when working with directional derivatives of arrays:

Compute Grad in a Euclidean coordinate chart c by transforming to and then back from Cartesian coordinates:

The result is the same as directly computing Grad [f,{x₁,…,x_n},c]:

The gradient of an array equals the gradient of its components only in Cartesian coordinates:

If chart is defined with metric g, expressed in the orthonormal basis, Grad [g,{x₁,…,x_n},chart] is zero:

Grad preserves the structure of SymmetrizedArray objects:

The gradient has an additional dimension but the same symmetry as the input:

The normal vectors to the level contours of a function equal the normalized gradient of the function:

Create an interactive contour plot that displays the normal at a point:

View expressions for the gradient of a scalar function in different coordinate systems: