 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability-resit/4/b/iii/solution

Apply the multivariable chain rule to a differentiable function

f (s, t)

on the chosen local branch of the parabolic coordinates:

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial s} \frac{\partial s}{\partial x} + \frac{\partial f}{\partial t} \frac{\partial t}{\partial x} .

Substituting part (ii) gives

\frac{\partial f}{\partial x} = \frac{t f _{s} + s f _{t}}{s ^{2} + t ^{2}}, (s, t) \neq = (0, 0) .

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