A-point-in-rectangular-coordinates-x-y-z-can-be-represented-in-spherical-coordinates-r-by-x-r-sin-sin-y-sin-sin-z-sin-0-2pi-0-pi-a-Calculate-the-Jacobian-o

Question Number 168339 by MikeH last updated on 09/Apr/22

A point in rectangular coordinates

(x, y, z) can be represented in spherical

coordinates (r, θ, φ) by :

x = r \sin θ \sin φ, y = \sin θ \sin φ,

z = \sin φ, 0 ⩽ θ ⩽ 2 π, 0 ⩽ φ ⩽ π

(a) Calculate the Jacobian of the

transformation \frac{\partial (x, y, z)}{\partial (r, θ, φ)}

(b) Calculate the volume of the region

delimited by the sphere :

S = {x, y, z \in R^{3}, x^{2} + y^{2} + z^{2} ⩽ R^{2}, R > 0}