Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 20 / a / Solution

Parametrize the segment in its stated direction by

x (t) = (1 - t, t, t)

,

0 \leq t \leq 1

. Then

d x = (- 1, 1, 1) d t

, and substitution into the two vector fields gives

F (x (t)) = (2 t, 1, 1), G (x (t)) = (2 t^{2}, 1 - 2 t + 2 t^{2}, 1 - 2 t + 2 t^{2}) .

Taking each dot product with

d x

evaluates the line integrals:

\int F \cdot d x = \int_{0}^{1} (2 - 2 t) d t = 1, \int G \cdot d x = \int_{0}^{1} (2 - 4 t + 2 t^{2}) d t = \frac{2}{3} .

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