 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/6/a/solution

For

r > 0

r_{x} = x / r

r_{y} = y / r

and

r_{z} = z / r

. Applying the chain rule to the outer sine and then the radial profile gives the Cartesian coordinates of the gradient:

\nabla sin [f (r)] = \frac{f ^{'} ( r ) cos [ f ( r )]}{r} (x, y, z) .

At the origin these coordinate formulas require a differentiable radial extension; differentiability of

f

as a one-variable profile alone does not guarantee it.

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