 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-resit/9/a/solution

(A (x, y), B (x, y), 0)

, with no dependence on

z

. Its curl is

\nabla \times V = i \partial_{x} A j \partial_{y} B k \partial_{z} 0 = (\frac{\partial B}{\partial x} - \frac{\partial A}{\partial y}) k .

Thus, for continuously differentiable

A

and

B

, the necessary and sufficient pointwise condition is

\frac{\partial B}{\partial x} = \frac{\partial A}{\partial y} .

To deduce global path independence from this local condition also requires a suitable domain, for example a simply connected domain.

 New to topics? Read the docs here!