Examples for

Vector Analysis

Vector analysis is the study of calculus over vector fields. Operators such as divergence, gradient and curl can be used to analyze the behavior of scalar- and vector-valued multivariate functions. Wolfram|Alpha can compute these operators along with others, such as the Laplacian, Jacobian and Hessian.
Gradient

Find the gradient of a multivariable function in various coordinate systems.

Compute the gradient of a function:

grad sin(x^2 y)
del z e^(x^2+y^2)
grad of a scalar field

Compute the gradient of a function specified in polar coordinates:

grad sqrt(r) cos(theta)
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Divergence

Calculate the divergence of a vector field.

Compute the divergence of a vector field:

div (x^2-y^2, 2xy)
div [x^2 sin y, y^2 sin xz, xy sin (cos z)]
divergence calculator
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Curl

Calculate the curl of a vector field.

Compute the curl (rotor) of a vector field:

curl [-y/(x^2+y^2), -x/(x^2+y^2), z]
rotor operator
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Laplacian

Find the Laplacian of a function in various coordinate systems.

Compute the Laplacian of a function:

Laplacian e^x sin y
Laplacian x^2+y^2+z^2
laplacian calculator
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Jacobian

Calculate the Jacobian matrix or determinant of a vector-valued function.

Compute a Jacobian determinant:

jacobian of (4x^2y, x-y^2)

Compute a Jacobian matrix:

(r sin(t), r cos(t)) what's the Jacobian matrix?
Jacobian matrix (r p sin(t), r p cos(t), r^2/p) w.r.t. r, t, and p
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Hessian

Calculate the Hessian matrix and determinant of a multivariate function.

Compute a Hessian determinant:

hessian of x^3 (y^2 - z)^2

Compute a Hessian matrix:

Hessian matrix 4x^2 - y^3
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Vector Analysis Identities

Explore identities involving vector functions and operators, such as div, grad and curl.

Calculate alternate forms of a vector analysis expression:

div (grad f)
curl (curl F)
grad (F . G)
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