91Ó°ÊÓ

$$\begin{gathered} \mathrm{w}=\left({\mathrm{x}}^2+\mathrm{z}\right){\mathrm{e}}^{\mathrm{y}} \\ \mathrm{x}=\mathrm{Òϲõ¾\pm ²ÔÏ•³¦´Ç²õθ} \\ y=\mathrm{Òϲõ¾\pm ²ÔÏ•²õ¾\pm ²Ôθ} \\ \mathrm{z}=\mathrm{Òϳ¦´Ç²õÏ•} \end{gathered}$$

By Theorem 12.34, we have

$\frac{∂w}{∂\mathrm{θ}}=\frac{∂w}{∂x}·\frac{∂x}{∂\mathrm{θ}}+\frac{∂w}{∂y}·\frac{∂y}{∂\mathrm{θ}}+\frac{∂w}{∂z}·\frac{∂z}{∂\mathrm{θ}}-------\left(1\right)$

So first we find $\frac{∂w}{∂x},\frac{∂x}{∂\mathrm{θ}},\frac{∂w}{∂y},\frac{∂y}{∂\mathrm{θ}},\frac{∂w}{∂z},\frac{∂z}{∂\mathrm{θ}}$

So we have

$$\begin{gathered} \frac{∂w}{∂x}=e^{y}2x \\ \frac{∂w}{∂y}=\left(x^2+z\right)e^y \\ \frac{∂w}{∂z}=e^y·1=e^y \\ \frac{∂x}{∂\mathrm{θ}}=-\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ} \\ \frac{∂y}{∂\mathrm{θ}}=\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ} \\ \frac{∂z}{∂\mathrm{θ}}=0 \end{gathered}$$

Use these above values in (1) we get,

$$\begin{gathered} \frac{∂w}{∂\mathrm{θ}}=\frac{∂w}{∂x}·\frac{∂x}{∂\mathrm{θ}}+\frac{∂w}{∂y}·\frac{∂y}{∂\mathrm{θ}}+\frac{∂w}{∂z}·\frac{∂z}{∂\mathrm{θ}} \\ \frac{∂w}{∂\mathrm{θ}}=2x{e}^y\left\{\left(-\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ}\right)+e^y\left(x^2+z\right)\mathrm{ÒÏ}\sin \mathrm{Ï•}\cos \mathrm{θ}+e^y\left(0\right)\right\} \\ \frac{∂w}{∂\mathrm{θ}}=\mathrm{ÒÏ}{e}^y\sin \mathrm{Ï•}\left\{-2x\sin \mathrm{θ}+\left(x^2+z\right)\cos \mathrm{θ}\right\} \end{gathered}$$

So from here, putting the value of $x\mathrm{and}y$in terms of $\mathrm{ÒÏ},\mathrm{Ï•},\mathrm{θ}$we get

$$\begin{gathered} \frac{∂w}{∂\mathrm{θ}}=\mathrm{ÒÏ}{e}^y\sin \mathrm{Ï•}\left\{-2x\sin \mathrm{θ}+\left(x^2+z\right)\cos \mathrm{θ}\right\} \\ \frac{∂w}{∂\mathrm{θ}}=\mathrm{ÒÏ}{e}^{\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ}}\sin \mathrm{Ï•}\left\{-2\sin \mathrm{Ï•}\cos \mathrm{θ}\sin \mathrm{θ}+{\mathrm{ÒÏ}}^2{\sin }^2\mathrm{Ï•}{\cos }^3\mathrm{θ}+\mathrm{ÒÏ}\cos \mathrm{Ï•}\cos \mathrm{θ}\right\} \end{gathered}$$

The value of$\frac{∂w}{∂\mathrm{θ}}=\mathrm{ÒÏ}{e}^{\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ}}\sin \mathrm{Ï•}\left\{-2\sin \mathrm{Ï•}\cos \mathrm{θ}\sin \mathrm{θ}+{\mathrm{ÒÏ}}^2{\sin }^2\mathrm{Ï•}{\cos }^3\mathrm{θ}+\mathrm{ÒÏ}\cos \mathrm{Ï•}\cos \mathrm{θ}\right\}$