Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 18C / b

Solution

ζ = v_{x} - u_{y} .

Differentiate the

v

x

and the

u

y

, then subtract. The mixed pressure derivatives cancel, giving

ζ_{t} = (- f u - g η_{y})_{x} - (f v - g η_{x})_{y} = - f (u_{x} + v_{y}) .

η_{t} = - h (u_{x} + v_{y})

q = ζ - \frac{f}{h} η

satisfies

q_{t} = ζ_{t} - \frac{f}{h} η_{t} = - f (u_{x} + v_{y}) + f (u_{x} + v_{y}) = 0.

Thus

\frac{\partial q}{\partial t} = 0 .

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