 Solution

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

F = ψ B

, with

B

\nabla \times (ψ B) = \nabla ψ \times B

B \cdot \oint_{C} ψ d r = \iint_{S} \hat{n} \cdot (\nabla ψ \times B) d S = B \cdot \iint_{S} (\hat{n} \times \nabla ψ) d S .

Since these scalar products agree for every

B

\oint_{C} ψ d r = \iint_{S} (\hat{n} \times \nabla ψ) d S .

In particular, the ordering of the cross product is normal crossed with gradient; reversing it would change the sign.

 New to topics? Read the docs here!