91Ó°ÊÓ

$$\begin{gathered} \mathrm{w}=\left({\mathrm{x}}^2+\mathrm{z}\right){\mathrm{e}}^{\mathrm{y}} \\ \mathrm{x}=\mathrm{ÒÏsinÏ•cosθ} \\ y=\mathrm{ÒÏsinÏ•sinθ} \\ \mathrm{z}=\mathrm{ÒÏcosÏ•} \end{gathered}$$

By Theorem 12.34, we have

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

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

So we have

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

Use these above values in (1) we get,

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

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

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

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