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potential

Potential of vector field

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.

Input Arguments

 V Vector of symbolic expressions or functions. X Vector of symbolic variables with respect to which you compute the potential. Y Vector of symbolic 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.

Examples

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:

ans =
x
y
z*exp(z)

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

If a vector field is not gradient, potential returns NaN:

potential([x*y, y], [x y])
ans =
NaN

collapse all

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 $V\text{​}\left(X\right)=\nabla P\text{ }\left(X\right)$.

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:

$P\text{​}\text{ }\left(X\right)=\underset{0}{\overset{1}{\int }}\left(X\text{​}-\text{​}\text{ }Y\right)\text{\hspace{0.17em}}\cdot V\text{​}\left(Y\text{ }+\text{ }\text{\hspace{0.17em}}\lambda \left(X\text{​}-\text{​}\text{ }Y\right)\right)\text{\hspace{0.17em}}d\lambda$

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.