91Ó°ÊÓ

In given exercises we have to find a potential function for the given vector field.

$\mathbf{G}(x,y,z)=(y{e}^{xy+z},x{e}^{xy+z},e^{xy+z})$

$$\begin{gathered} g(x,y,z)=∫y{e}^{xy+z}dx+B+C \\ g(x,y,z)=∫y{e}^{xy}·e^{z}dx+B+C \\ \mathrm{Let}u=xy \\ \frac{du}{dx}=y \\ du=ydx \\ g(x,y,z)=e^z∫e^{u}dx+B+C \\ g(x,y,z)=e^z{e}^u+\mathrm{Î\pm }+B+C \\ g(x,y,z)=e^z{e}^{xy}+\mathrm{Î\pm }+B+C \\ g(x,y,z)=e^{xy+z}+\mathrm{Î\pm }+B+C \end{gathered}$$

where $\mathrm{Î\pm }$ is an arbitrary constant and B is the integral with respect to y of the terms in localid="1650474562962" $G_2(x,y,z)$ in which the factor x does not appear.

$$\begin{gathered} B=∫x{e}^{xy+z}dy \\ B=∫x{e}^{xy}·e^{z}dy \\ \mathrm{Let}u=xy \\ \frac{du}{dy}=x \\ du=xdy \\ B=e^z∫e^{u}dy \\ B=e^z{e}^u+\mathrm{β} \\ B=e^z{e}^{xy}+\mathrm{β} \\ B=e^{xy+z}+\mathrm{β} \end{gathered}$$

whereβ is an arbitrary constant.

$$\begin{gathered} C=∫e^{xy}·e^{z}dz \\ C=e^{xy}∫e^{z}dz \\ C=e^{xy}{e}^z+\mathrm{γ} \\ C=e^{xy+z}+\mathrm{γ} \end{gathered}$$

where$\mathrm{γ}$ is an arbitrary constant.

Setting the constants equal to zero since they do not affect the gradient of $g(x,y,z)$

We have,

$g(x,y,z)=e^{xy+z}$