1.18 (

) www We can use the result (1.126) to derive an expression for the surface area SD, and the volume VD, of a sphere of unit radius in D dimensions. To do this, consider the following result, which is obtained by transforming from Cartesian to polar coordinates

D i=1

 ∞

−∞

e

−x2 i dxi = SD

 ∞

0 e

−r2 rD−1 dr. (1.142)

Using the definition (1.141) of the Gamma function, together with (1.126), evaluate both sides of this equation, and hence show that SD =

2πD/2

Γ(D/2) . (1.143)

Next, by integrating with respect to radius from 0 to 1, show that the volume of the unit sphere in D dimensions is given by VD = SD D

. (1.144)

Finally, use the results Γ(1) = 1 and Γ(3/2) =

√

π/2 to show that (1.143) and

(1.144) reduce to the usual expressions for D = 2 and D = 3.