[Solved] Let f: R^3 \rightarrow R be differentiable. Making the change from rectangular coordinates (x, y, z) to spherical coordinates (r, θ, \varphi)
Question: Let \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}\) be differentiable. Making the change from rectangular coordinates \((x, y, z)\) to spherical coordinates \((r, \theta, \varphi)\) where
\[\begin{aligned} x &=r \sin \theta \cos \varphi \\ y &=r \sin \theta \sin \varphi \\ z &=r \cos \theta \end{aligned}\]
compute \(\frac{\partial f}{\partial r}(r, \theta, \varphi), \frac{\partial f}{\partial \theta}(r, \theta, \varphi)\) and \(\frac{\partial f}{\partial \varphi}(r, \theta, \varphi)\) in terms of \(\frac{\partial f}{\partial x}(x, y, z), \frac{\partial f}{\partial y}(x, y, z)\) and \(\frac{\partial f}{\partial z}(x, y, z) .\) More formally, if \(\Psi\) denotes the function (with a suitable domain) given by
\[\Psi(r, \theta, \varphi)=(r \sin \theta \cos \varphi, r \sin \theta \sin \varphi, r \cos \theta)\]calculate \(\mathrm{D}(f \circ \Psi)(r, \theta, \varphi)\)
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![[All Steps] Let F: R^2 \rightarrow R^4](/images/solutions/MC-solution-library-72397.jpg "[All Steps] Let F: R^2 \rightarrow R^4 be the function given by")
[All Steps] Let F: R^2 \rightarrow R^4 be the function given by
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