We can use the result (1.126) to derive an expression for th
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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 ra
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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 = 2?D/2 ?(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 ?(1) = 1 and ?(3/2) = ??/2 to show that (1.143) and (1.144) reduce to the usual expressions for D = 2 and D = 3.
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