91Ó°ÊÓ

The force field is $\mathrm{F}=\mathrm{yi}+\mathrm{xj}+\mathrm{k}$

and $\mathrm{F}=-∇\mathrm{φ}.$

A force is said to be conservative if $\mathbf{∇}\mathbf{×}\mathbf{F}\mathbf{=}\mathbf{0}$.

The formula for the scalar potential is $\mathbf{W}\mathbf{=}\mathbf{∫}\mathbf{F}\mathbf{.}\mathbf{dr}\mathbf{.}$

The force is said to be conservative if $∇×F=0$.

Putthe values given below in the above equation.

$F=yi+xj+k$

The equation becomes as follows.

$$\begin{gathered} ∇×F=\left(\begin{array}{ccc}i& j& k\\ \frac{∂}{∂x}& \frac{∂}{∂y}& \frac{∂}{∂z}\\ y& x& 1\end{array}\right) \\ ∇×F=\left(\frac{∂}{∂y}\left(1\right)-\frac{∂}{∂z}\left(x\right)\right)i-\left(\frac{∂}{∂x}\left(1\right)-\frac{∂}{∂z}\left(y\right)\right)j+\left(\frac{∂}{∂x}\left(y\right)-\frac{∂}{∂y}\left(x\right)\right)k \\ ∇×F=0 \end{gathered}$$

The field is conservative.

The formula for the scalar potential is$\mathrm{W}=∫\mathrm{F}.\mathrm{dr}.$

$$\begin{gathered} \mathrm{W}=∫(\mathrm{yi}+\mathrm{xj}+\mathrm{k}).(\mathrm{dxi}+\mathrm{dyj}+\mathrm{dzk}) \\ \mathrm{W}=∫\mathrm{ydx}+\mathrm{xdy}+\mathrm{dz} \end{gathered}$$

$W_1$is from$(0,0,0)\mathrm{to}(\mathrm{x},0,0).$ .

$$\begin{gathered} y=0 \\ dy=0 \\ z=0 \\ dz=0 \end{gathered}$$

Substitute the above value in the equation mentioned below.

$$\begin{gathered} W=∫ydx+xdy+dz \\ {W}_1={∫}_0^{x}0×dx \\ {W}_1=0 \end{gathered}$$

$W_2$is from $(\mathrm{x},0,0)\mathrm{to}(\mathrm{x},0,\mathrm{z})$x is constant.

$\begin{array}{l}dx=0\\ y=0\\ dy=0\end{array}$

Substitute the above value in the equation mentioned below.

role="math" localid="1659336578414" $$\begin{gathered} W=∫ydx+xdy+dz \\ {W}_2={∫}_0^{z}dz \\ {W}_2={\left[z\right]}_0^z \\ {W}_2=z \end{gathered}$$

$W_3$is from $(x,0,z)to(x,y,z)$and x is constant, z is constant.

$\begin{array}{l}dx=0\\ dz=0\end{array}$

Substitute the above value in the equation mentioned below.

role="math" localid="1659336294517" $$\begin{gathered} W=∫ydx+xdy+dz \\ {W}_3={∫}_0^{y}xdy \\ {W}_3={\left[xy\right]}_0^y \\ {W}_3=xy \end{gathered}$$

The formula states the equation mentioned below.

$\begin{array}{l}W=W_1+W_2+W_3\\ W=0+z+xy\\ W=Z+xy\end{array}$

The formula states the equation mentioned below.

$F=∇W$

It is given that$F=-∇\mathrm{φ}.$.

By both the values of , $\mathrm{F},-∇\mathrm{φ}=∇\mathrm{W}.$

$\mathrm{φ}=-\mathrm{W}$

Put the value of in above equation.

$\begin{array}{l}\mathrm{φ}=-\left(z+xy\right)\\ \mathrm{φ}=-z-xy\end{array}$

Hence the scalar potential is $-xy-z$