 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016/8/c/solution

By Stokes theorem, it suffices to calculate the boundary line integral. On the

x z

y = 0

and

d y = 0

, so

V \cdot d r = e^{x} d x + z^{2} d z = d (e^{x} + \frac{z ^{3}}{3}) .

\iint_{S} (\nabla \times V) \cdot \hat{n} d S = 0.

The argument applies to any admissible spanning surface, including a curved one; the surface itself need not lie in the

x z

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