Question Number 215535 by MrGaster last updated on 10/Jan/25
$$\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \leq{R}^{\mathrm{2}} } \underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} \frac{\partial{f}}{\partial{x}_{{i}} }\underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} =? \\ $$
Answered by MrGaster last updated on 10/Jan/25
$$\int_{\mathrm{0}} ^{{R}} {rdr}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} −{r}^{\mathrm{2}\:} } \underset{\mathrm{1}\leq{i}\leq{n}} {\sum}\frac{{x}_{{i}} }{{r}}\:\frac{\partial{f}}{\partial{x}_{{i}} }{dS} \\ $$$$=\int_{\mathrm{0}} ^{{R}} {r}\:{dr}\:\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} ={r}^{\mathrm{2}} } \langle{n},\bigtriangledown{f}\rangle{dS} \\ $$$$=\int_{\mathrm{0}} ^{{R}} {r}\:{dr}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \leq{r}^{\mathrm{2}} } \bigtriangledown^{\mathrm{2}} {f}\underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} =\int_{\mathrm{0}} ^{{R}} {rdr}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \leq{r}^{\mathrm{2}} } {g}\left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \right)\underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} \\ $$$$\mathrm{Let}\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} ={u}^{\mathrm{2}} \Rightarrow\underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} ={d}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} ={u}^{\mathrm{2}} } \underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} =\frac{\mathrm{2}\pi^{\frac{{n}}{\mathrm{2}}} {u}^{{n}−\mathrm{1}} }{\Gamma\left(\frac{{n}}{\mathrm{2}}\right)}{du} \\ $$$$\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \leq{R}^{\mathrm{2}} } \underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} \frac{\partial{f}}{\partial{x}_{{i}} }\underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} =\frac{\mathrm{2}\pi^{\frac{{n}}{\mathrm{2}}} }{\Gamma\left(\frac{{n}}{\mathrm{2}}\right)}\int_{\mathrm{0}} ^{{R}} {rdr}\int_{\mathrm{0}} ^{{r}} {g}\left({u}^{\mathrm{2}} \right){u}^{{n}−\mathrm{1}} {du} \\ $$