Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 2 / Section B / 6 / c

Table of contents
- Solution c

Solution

Parametrize the straight line by

x (t) = (t, t, t)

,

0 \leq t \leq 1

. The line integral is

\int F \cdot d x = \int_{0}^{1} [(2 t^{2} + t) + (t^{2} + 2 t) + t] d t = \int_{0}^{1} (3 t^{2} + 4 t) d t = 3 .

The change in scalar potential is

ψ (1, 1, 1) - ψ (0, 0, 0) = - 3

, so the answer is

- Δ ψ

, as required by the fundamental theorem for line integrals for

F = - \nabla ψ

.

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