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:

Compute the gradient of a function specified in polar coordinates:

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Divergence

Calculate the divergence of a vector field.

Compute the divergence of a vector field:

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Curl

Calculate the curl of a vector field.

Compute the curl (rotor) of a vector field:

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Laplacian

Find the Laplacian of a function in various coordinate systems.

Compute the Laplacian of a function:

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Jacobian

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

Compute a Jacobian determinant:

Compute a Jacobian matrix:

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Hessian

Calculate the Hessian matrix and determinant of a multivariate function.

Compute a Hessian determinant:

Compute a Hessian matrix:

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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:

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