2.1 Using the following hints, show that for d 1, ????d = 2????d2(d2) is the surface area of the unit sphere in d. (a)

2.1 Using the following hints, show that for d ≥ 1, ????d = 2????d∕2∕Γ(d∕2) is the surface area of the unit sphere in ℝd.

(a) (d = 1). Note ????1 = 2 by direct observation.

(b) (d = 2). Using the fact that the length of a curve y = y(x) between x = a and x = b is given by

∫

b a

{1 + (dy∕dx)

2}1∕2 dx, express the perimeter of the unit circle as

????2 = 2???? = 2 ∫

1

−1

(1 − s 2

)

−1∕2 ds.

(c) (d ≥ 3). Suppose by induction that ????d−1 takes the required form.

Express ????d as the volume of a surface of rotation about the x[d] axis in

ℝd to get

????d = ∫

1

−1

{????d−1(1 − s 2

)

(d−2)∕2}(1 − s 2

)

−1∕2 ds.

Use the integral for the beta function

∫

1 0

????????−1(1 − ????)

????−1d???? = B(????, ????) = Γ(????)Γ(????)∕Γ(???? + ????)

for ????, ???? > 0, to simplify this expression.