Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 6

Table of contents
- a 6
  - Solution a
- b 6
  - Solution b

a

Solution

The potential is smooth everywhere, so its mixed partial derivatives commute. The curl of a gradient therefore vanishes:

\nabla \times F = \nabla \times \nabla ϕ = 0 .

b

Solution

The fundamental theorem for line integrals evaluates this conservative vector field integral by the change in potential. At the initial point

ϕ = 0

. At the final point

z = 0

, so the exponential factor is one, and

\int_{C} F \cdot d r = ϕ (C_{end}) - ϕ (C_{start}) = sin \frac{π}{8} cos \frac{π}{4} = \frac{2}{2} sin \frac{π}{8} .

The detailed shape of the path does not affect this line integral.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version