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Vector potential of vector field

`vectorPotential(`

computes the vector potential of the vector
field
`V`

,`X`

)`V`

with respect to the vector `X`

in Cartesian
coordinates. The vector field `V`

and the vector `X`

are both three-dimensional.

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

:

syms x y z vectorPotential([x^2*y, -1/2*y^2*x, -x*y*z], [x y z])

ans = -(x*y^2*z)/2 -x^2*y*z 0

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

:

syms x y z f(x,y,z) = 2*y^3 - 4*x*y; g(x,y,z) = 2*y^2 - 16*z^2+18; h(x,y,z) = -32*x^2 - 16*x*y^2; A = vectorPotential([f; g; h], [x y z])

A(x, y, z) = z*(2*y^2 + 18) - (16*z^3)/3 + (16*x*y*(y^2 + 6*x))/3 2*y*z*(- y^2 + 2*x) 0

To check whether the vector potential exists for a particular vector field, compute the divergence of that vector field:

syms x y z V = [x^2 2*y z]; divergence(V, [x y z])

ans = 2*x + 3

If the divergence is not equal to 0, the vector potential does
not exist. In this case, `vectorPotential`

returns
the vector with all three components equal to `NaN`

:

vectorPotential(V, [x y z])

ans = NaN NaN NaN