 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/8/c/solution

For constant

b

\nabla \times [(b \cdot r) b] = \nabla (b \cdot r) \times b = b \times b = 0.

Using part (b) and the order of the cross product,

\iint_{S} F \times d S = - \iint_{S} n \times F d S = - ∭_{V} \nabla \times F d V = 0.

Here

d S

inside a vector product denotes the outward vector area element

n d S

. Closedness is the condition that permits replacing the entire surface by the boundary integral of a volume.

 New to topics? Read the docs here!