Problem 64
For which real values of \(a\) and \(b\) is \(|x|^{a}|\log | x||^{b}\) integrable over \(\left\\{x \in \mathbb{R}^{n}\right.\) : \(\left.|x|<\frac{1}{2}\right\\}\) ? Over \(\left\\{x \in \mathbb{R}^{n}:|x|>2\right\\}\) ?
Problem 65
Define \(G: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) by \(G\left(r, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)=\left(x_{1}, \ldots, x_{n}\right)\) where \(x_{1}=r \cos \phi_{1}, \quad x_{2}=r \sin \phi_{1} \cos \phi_{2}, \quad x_{3}=r \sin \phi_{1} \sin \phi_{2} \cos \phi_{3}, \ldots\), \(x_{n-1}=r \sin \phi_{1} \cdots \sin \phi_{n-2} \cos \theta, \quad x_{n}=r \sin \phi_{1} \cdots \sin \phi_{n-2} \sin \theta .\) a. \(G\) maps \(\mathbb{R}^{n}\) onto \(\mathbb{R}^{n}\), and \(\left|G\left(r, \phi_{1}, \ldots, \phi_{n-2,} \theta\right)\right|=|r|\). b. \(\operatorname{det} D_{\left(r, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)} G=r^{n-1} \sin ^{n-2} \phi_{1} \sin ^{n-3} \phi_{2} \cdots \sin \phi_{n-2}\). c. Let \(\Omega=(0, \infty) \times(0, \pi)^{n-2} \times(0,2 \pi)\). Then \(G \mid \Omega\) is a diffeomorphism and \(m\left(\mathbb{R}^{n} \backslash G(\Omega)\right)=0\). d. Let \(F\left(\phi_{1}, \ldots, \phi_{n-2}, \theta\right)=G\left(1, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)\) and \(\Omega^{\prime}=(0, \pi)^{n-2} \times\) \((0,2 \pi)\). Then \(\left(F \mid \Omega^{\prime}\right)^{-1}\) defines a coordinate system on \(S^{n-1}\) except on a \(\sigma\)-null set, and the measure \(\sigma\) is given in these coordinates by $$ d \sigma\left(\phi_{1}, \ldots \phi_{n-2}, \theta\right)=\sin ^{n-2} \phi_{1} \sin ^{n-3} \phi_{2} \cdots \sin \phi_{n-2} d \phi_{1} \cdots d \phi_{n-2} d \theta $$
