 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/18/a/solution

Regard

u = u (s, t (s, v))

. Applying the multivariable chain rule while holding

v

fixed gives

(\frac{\partial u}{\partial s})_{v} = (\frac{\partial u}{\partial s})_{t} + (\frac{\partial u}{\partial t})_{s} (\frac{\partial t}{\partial s})_{v} .

Each subscript records the variable held fixed. A differentiable local coordinate change from

(s, t)

(s, v)

requires

(\partial v / \partial t)_{s} \neq = 0

; the question's notation

t = t (s, v)

presupposes such a branch.

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