 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/19/b/solution

Suppress the held-fixed subscripts only within this matrix display. Collecting the preceding operator coefficients gives

\partial_{W} \partial_{θ} \partial_{ϕ} = tan θ cos ϕ W sec^{2} θ cos ϕ - W tan θ sin ϕ tan θ sin ϕ W sec^{2} θ sin ϕ W tan θ cos ϕ 100 \partial_{x} \partial_{y} \partial_{z} .

J = \partial (x, y, z) / \partial (W, θ, ϕ)

uses the usual convention of coordinate functions as rows, the displayed coefficient matrix is

J^{T}

W^{2} sec^{2} θ tan θ > 0

on the given domain, so the inverse operator transformation exists everywhere there.

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