potential

Potential of vector field

Syntax

potential(V,X)

potential(V,X,Y)

Description

potential(V,X)computes the potential of the vector fieldVwith respect to the vectorXin Cartesian coordinates. The vector fieldVmust be a gradient field.

example

potential(V,X,Y)computes the potential of vector fieldVwith respect toXusingYas base point for the integration.

Examples

Compute Potential of Vector Field

Compute the potential of this vector field with respect to the vector[x, y, z]:

syms x y z P = potential([x, y, z*exp(z)], [x y z])

P = x^2/2 + y^2/2 + exp(z)*(z - 1)

Use thegradientfunction to verify the result:

simplify(gradient(P, [x y z]))

ans = x y z*exp(z)

Specify Integration Base Point

Compute the potential of this vector field specifying the integration base point as[0 0 0]:

syms x y z P = potential([x, y, z*exp(z)], [x y z], [0 0 0])

P = x^2/2 + y^2/2 + exp(z)*(z - 1) + 1

Verify thatP([0 0 0]) = 0:

subs(P, [x y z], [0 0 0])

ans = 0

Test Potential for Field Without Gradient

If a vector field is not gradient,potentialreturnsNaN:

potential([x*y, y], [x y])

ans = NaN

Input Arguments

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`V`—Vector field
3-D symbolic vector of symbolic expressions or functions(default)

Vector field, specified as a 3-D vector of symbolic expressions or functions.

`X`—Input
vector of three symbolic variables

Input, specified as a vector of three symbolic variables with respect to which you compute the potential.

`Y`—Input
symbolic vector

Input, specified as a symbolic vector of variables, expressions, or numbers that you want to use as a base point for the integration. If you use this argument,potentialreturnsP(X)such thatP(Y) = 0. Otherwise, the potential is only defined up to some additive constant.

More About

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Scalar Potential of Gradient Vector Field

The potential of a gradient vector fieldV(X) = [v₁(x₁,x₂,...),v₂(x₁,x₂,...),...]is the scalarP(X)such that $V (X) = \nabla P (X)$ .

The vector field is gradient if and only if the corresponding Jacobian is symmetrical:

$(\frac{\partial v_{i}}{\partial x_{j}}) = (\frac{\partial v_{j}}{\partial x_{i}})$

Thepotentialfunction represents the potential in its integral form:

$P (X) = \int_{0}^{1} (X - Y) \cdot V (Y + λ (X - Y)) d λ$

Tips

Ifpotentialcannot verify thatV是一个梯度场,我t returnsNaN.
ReturningNaNdoes not prove thatVis not a gradient field. For performance reasons,potentialsometimes does not sufficiently simplify partial derivatives, and therefore, it cannot verify that the field is gradient.
IfYis a scalar, thenpotentialexpands it into a vector of the same length asXwith all elements equal toY.

Version History

Introduced in R2012a