 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/12/b/solution

The notation

(\partial f / \partial x)_{y}

x

while

y

is held fixed:

(\frac{\partial f}{\partial x})_{y} = lim_{h \to 0} \frac{f ( x + h , y ) - f ( x , y )}{h} .

df = f_{x} d x + f_{y} d y

. If both coefficients vanish, the tangent plane is horizontal and every directional derivative vanishes: the surface is flat to first order at that point. It need not be constant in a neighbourhood;

f = x^{2} + y^{2}

at the origin is a counterexample. Existence of coordinate partials alone, without differentiability, would not justify this tangent-plane conclusion.

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