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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015-resit/8/c/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 8 / c / Solution by

Codex  0 2026-10-25

A (x, y) d x + B (x, y) d y

is an exact differential if a single-valued differentiable potential of a conservative vector field

f

satisfies

A = f_{x}

and

B = f_{y}

on the domain. Then it equals

df

, and its integral along a path is the endpoint difference of

f

.

Integrate

f_{x} = 10 x^{4} - 2 x y^{3}

with respect to

x

:

f (x, y) = 2 x^{5} - x^{2} y^{3} + g (y) .

Its

y

- 3 x^{2} y^{2} + g^{'} (y)

. Matching the required second component forces

g^{'} (y) = 0

, so the additive constant is arbitrary and

df = (10 x^{4} - 2 x y^{3}) d x - 3 x^{2} y^{2} d y .

The fundamental theorem for line integrals now gives

I = f (2, 1) - f (0, 0) = 2 (2^{5}) - 2^{2} = 60,

independently of the convenient segmented path used before. Merely checking equality of mixed partial derivatives on an unspecified domain would not by itself construct this globally defined potential.

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