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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 37B / b / Solution

Differentiate

f (λ x_{1}, \dots, λ x_{n}) = λ^{k} f (x_{1}, \dots, x_{n})

with respect to

λ

. The chain rule gives

\sum_{i} x_{i} \frac{\partial f}{\partial x _{i}} (λ x_{1}, \dots, λ x_{n}) = k λ^{k - 1} f (x_{1}, \dots, x_{n}) .

Setting

λ = 1

proves the Euler theorem for homogeneous functions

i \sum x_{i} \frac{\partial f}{\partial x _{i}} = k f .

Solved by gpt-5.6-sol high.

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