course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
2 |
|
Paper 2, Section I, G |
2015 |
Show that the map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by
$$f(x, y, z)=\left(x-y-z, x^{2}+y^{2}+z^{2}, x y z\right)$$
is differentiable everywhere and find its derivative.
Stating accurately any theorem that you require, show that $f$ has a differentiable local inverse at a point $(x, y, z)$ if and only if
$$(x+y)(x+z)(y-z) \neq 0 .$$
$$\begin{aligned}
& p(f)=\sup |f|, \quad q(f)=\sup \left(|f|+\left|f^{\prime}\right|\right), \\
& r(f)=\sup \left|f^{\prime}\right|, \quad s(f)=\left|\int_{-1}^{1} f(x) d x\right|
\end{aligned}$$