 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/3/a/solution

Stokes theorem equates the circulation of a continuously differentiable vector field around an oriented boundary to the flux of its curl across a spanning oriented surface:

\oint_{\partial S} A \cdot d r = \int_{S} (\nabla \times A) \cdot \hat{n} d S .

The boundary direction follows the right-hand rule relative to the normal vector. For the polynomial vector field, only the

z

component of the curl can be nonzero, and

\nabla \times A = (\frac{\partial ( 2 x y )}{\partial x} - \frac{\partial ( x ^{2} - y ^{2} )}{\partial y}) \hat{z} = 4 y \hat{z} .

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