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Potential of vector field
potential(V,X)
potential(V,X,Y)
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.
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
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