Polar coordinates in En (n ≥ 3) are introduced by the equationsa) Show that if x = (x1, .....

Polar coordinates in En (n ≥ 3) are introduced by the equations

a) Show that if x = (x1, ..., xn) and (xn−1, xn) is not (0, 0) or of the form (c, 0) with c > 0, then unique polar coordinates are assigned to x by the requirements r = (Hence the limits of integration for a multiple integral over the n−dimensional spherical region are 0 to a for r, 0 to π for each φi except 0 to 2π for φn−1.) [Hint: x cannot be 0, so . Choose φ1 as cos−1(x1/r), 0 ≤ φ1 ≤ π. Show that φ1 = 0 or φ1 = π cannot occur, since they imply |x| = r and hence x2 = x3 = = 0. Next choose φ2 = cos−1(x2/(r sin φ1)), 0 ≤ φ2 ≤ π, and show that φ2 = 0 or π cannot occur, etc.]

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b) Show that the corresponding Jacobian is [Hint: Use induction. For the step from n − 1 to n, use the polar coordinates ρ, φ2, ..., φn−1 in the subspace x1 = 0, so that x2 = ρ cos φ2, x3 = ρ sin φ2 cos φ3, ..., and x1, ρ, φ2, … φn−1 can be considered as cylindrical coordinates in En. Go from these cylindrical coordinates to full polar coordinates by the equations x1 = r cos φ1, ρ = r sin φ1. Show that

where Jn−1 = ρn−2 sin n−3 φ2 sin φn−2 by induction hypothesis. The conclusion now follows from (2.51).]