Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 2 / c

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- Solution c

Solution

The curl is

\nabla \times F = (\partial_{y} z^{2} - \partial_{z} 0, \partial_{z} 0 - \partial_{x} z^{2}, \partial_{x} 0 - \partial_{y} 0) = 0 .

To establish that

F

is a conservative vector field, construct a potential of a conservative vector field directly:

ϕ (x, y, z) = \frac{z ^{3}}{3} + C, \nabla ϕ = (0, 0, z^{2}) = F .

This gradient calculation is valid on all of

R^{3}

, and hence also on the cone. The explicit potential of a conservative vector field supplies the required global justification.

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