Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 13 / a

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- Solution a

Solution

Parametrize the path by

r (t) = (x (t), y (t), z (t))

, from

t = a

to

t = b

. For the conservative vector field

F = \nablaΦ

, the chain rule gives

F \cdot \frac{d r}{d t} = Φ_{x} \frac{d x}{d t} + Φ_{y} \frac{d y}{d t} + Φ_{z} \frac{d z}{d t} = \frac{d}{d t} Φ (r (t)) .

Integrating proves the fundamental theorem for line integrals:

\int_{C} F \cdot d r = Φ (r (b)) - Φ (r (a)) .

Only the endpoints matter, so the integral is independent of the connecting path.

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