Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 9 / a / i

Table of contents
- Solution i

Solution

Solve the two algebraic constraints for the squared coordinates:

x^{2} = \frac{z ^{2} + u ^{2}}{3}, y^{2} = \frac{z ^{2} - 2 u ^{2}}{3} .

On any nonsingular real branch, implicit differentiation at fixed

u

gives

(\partial x / \partial z)_{u} = z / (3 x)

and

(\partial y / \partial z)_{u} = z / (3 y)

. Differentiating

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

with the other original coordinate fixed gives

(\partial z / \partial x)_{y} = 2 x / z

and

(\partial z / \partial y)_{x} = y / z

. Therefore

(\frac{\partial x}{\partial z})_{u} (\frac{\partial z}{\partial x})_{y} + (\frac{\partial y}{\partial z})_{u} (\frac{\partial z}{\partial y})_{x} = \frac{z}{3 x} \frac{2 x}{z} + \frac{z}{3 y} \frac{y}{z} = \frac{2}{3} + \frac{1}{3} = 1.

This is the multivariable chain rule applied to

z (x (z, u), y (z, u)) = z

. These formulas assume the relevant denominators are nonzero and the chosen variables are valid local coordinates.

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