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Equation of sphere in n-dimensional space is:

[tex]x^2_1+x^2_2+...+x^2_n=R^2[/tex]

We serch volume as [tex]V=C_nR^n[/tex]. Why? Perhaps its analogy with [tex]CR^3[/tex].

Now we calculate this integral

[tex]I=\int^{\infty}_{-\infty}dx_1\int^{\infty}_{-\infty}dx_2\int^{\infty}_{-\infty}dx_3...\int^{\infty}_{-\infty}dx_ne^{-a(x^2_1+x^2_2+...+x^2_n)}[/tex]

Why we do this?

And we get [tex](\frac{\pi}{a})^{\frac{N}{2}}[/tex]

And then

[tex]I=\int dV_n e^{-ar^2}[/tex]

we get

[tex]V_n=\frac{(\pi)^{\frac{N}{2}}}{\Gamma(\frac{N}{2}+1)}R^n[/tex]

Can someone tell me idea of all this. Thanks

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# Volume of n-dimensional sphere equation

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