potential

Potential of vector field

Syntax

potential(V,X)

potential(V,X,Y)

Description

potential(V,X) computes the potential of the vector field V with respect to the vector X in Cartesian coordinates. The vector field V must be a gradient field.

potential(V,X,Y) computes the potential of vector field V with respect to X using Y as 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 the gradient function 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 that P([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, potential returns NaN:

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

ans =
NaN

Input Arguments

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, potential returns P(X) such that P(Y) = 0. Otherwise, the potential is only defined up to some additive constant.

More About

Scalar Potential of Gradient Vector Field

The potential of a gradient vector field V(X) = [v₁(x₁,x₂,...),v₂(x₁,x₂,...),...] is the scalar P(X) such that $$V\text{}\left(X\right)=\nabla P\left(X\right)$$.

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

$$\left(\frac{\partial v_i}{\partial x_j}\right)=\left(\frac{\partial v_j}{\partial x_i}\right)$$

The potential function represents the potential in its integral form:

$$P\text{}\left(X\right)={\displaystyle \underset{0}{\overset{1}{\int }}\left(X\text{}-\text{}Y\right)\cdot V\text{}\left(Y+\lambda \left(X\text{}-\text{}Y\right)\right)d\lambda }$$

Tips

• If potential cannot verify that V is a gradient field, it returns NaN.

• Returning NaN does not prove that V is not a gradient field. For performance reasons, potential sometimes does not sufficiently simplify partial derivatives, and therefore, it cannot verify that the field is gradient.

• If Y is a scalar, then potential expands it into a vector of the same length as X with all elements equal to Y.