A vector field F in ℝ3 is a conservative flux field, or a solenoidal field, if there is a vector field G such that curlG = F. Given a conservative flux vector field F, the general solution of curlG = F is the sum of a particular solution and the gradient of an arbitrary functions.
The vpotential command finds a particular vector field G such that curlG = F if F is a conservative flux field, and signals an error otherwise. Specifically, vpotential returns the solution G with zero as the first component.
vpotential is the reciprocal function of curl.
Example.
Input:
Output:
⎡ ⎢ ⎢ ⎣ | 0,−2 x y z,− |
| +4 x z+3 y | ⎤ ⎥ ⎥ ⎦ |
In ℝ3, a vector field F is a curl if and only if its divergence is zero. In time-independent electro-magnetism, F = B is the magnetic field and G = A is the potential vector.