Conservative flux field : vpotential

### 5.59.7 Conservative flux field : vpotential

vpotential takes two arguments : a vector field V in Rn with respect to n real variables and the vector of these variable names.
vpotential returns, if it is possible, a vector U such that curl(U)=V. When it is possible we say that V is a conservative flux field or a solenoidal field. The general solution is the sum of a particular solution and of the gradient of an arbitrary function, Xcas returns a particular solution with zero as first component.
vpotential is the reciprocal function of curl.
Input :

vpotential([2*x*y+3,x`^`2-4*z,-2*y*z],[x,y,z])

Output :

[0,(-(2*y))*z*x,-x`^`3/3-(-(4*z))*x+3*y]

In ℝ3, a vector field V is a rotational if and only if its divergence is zero
(divergence(V,[x,y,z])=0). In time-independent electro-magnetism, V= B is the magnetic field and U= A is the potential vector.