[Math] How to create a vector field whose Curl and Divergence are zero at any point

Question

What is the mathematical procedure to derive a vector field whose curl and divergence are zero at any point at any time?

Edit: Please explain it by solving the differential equations of curl and divergence.

Answer 1

Take any harmonic function $\phi$, and set your vector field to be $v = \nabla \phi$, the gradient.

Edit On any simply connected domain (if your space is just the entire Euclidean space, for example), any curl free vector field can be written as the gradient of a function. You can see this explicitly by verifying that given a curl free vector field $v$, the function $\phi(x) = \int_{C} v(s)\cdot ds$ where $C$ is a curve starting at the origin and ending at the point $x$ is independent of the choice of the curve $C$, and hence $\phi$ is well-defined. (This is generally treated in any introductory textbook on vector calculus, and is a special instance of Stokes' theorem.) Once you have that, plugging $v = \nabla \phi$ into the divergence equation you automatically get that $\nabla^2\phi = 0$.