13.7.7  Conservative flux field

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 takes two arguments:
  • F, a vector field in ℝ³, given as a list of three expressions depending on three variables.
  • vars, a list of the variable names.
• vpotential(Fvars) returns a solution of curlG=F whose first coordinate is zero if F is a conservative vector field, and signals an error otherwise.

vpotential is the reciprocal function of curl.

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.

Example