Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 9 / a

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- Solution a

Solution

Regard the planar vector field as

(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.

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