Page 99 - Vector Analysis
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§3.5 The Change of Variables Formula 95
if the radius of B is R, we must have
żż
f (x, y, z) d(x, y, z) = f (x, y, z) d(x, y, z)
B ψ((0,R)ˆ(0,2π)ˆ(0,π))
ż
= (f ˝ ψ)(ρ, θ, ϕ)ˇˇJψ(ρ, θ, ϕ)ˇˇ d(ρ, θ, ϕ)
(0,R)ˆ(0,2π)ˆ(0,π)
ż
= f (x0 + ρ cos θ sin ϕ, y0 + ρ sin θ sin ϕ, z0 + ρ cos ϕ) ρ2 sin ϕ d(r, θ, z) .
(0,R)ˆ(0,2π)ˆ(0,π)
