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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)

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

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`

.

`curl`

| `diff`

| `divergence`

| `gradient`

| `jacobian`

| `hessian`

| `laplacian`

| `vectorPotential`