91Ó°ÊÓ

The given force vector is

$F=i-zj-yk$.

Force field is said to be conservative if curl of force becomes zero.

The formula to find $∇×F$ in order to show that field is conservative is mentioned below.

$∇×F=\left|\begin{array}{ccc}i& j& k\\ \frac{∂}{∂x}& \frac{∂}{∂y}& \frac{∂}{∂z}\\ x& y& z\end{array}\right|$

where $F=xi+yj+zk$

In order to find the scalar potential $\mathrm{φ}$,where $F=-∇\mathrm{φ}F=-∇\mathrm{φ}$, the formula for work done is given below.

$$\begin{gathered} W=∫F.dr \\ F=∇W \\ ⇒\mathrm{φ}=-W \end{gathered}$$

Use the formula $∇×F=\left|\begin{array}{ccc}i& j& k\\ \frac{∂}{∂x}& \frac{∂}{∂y}& \frac{∂}{∂z}\\ x& y& z\end{array}\right|$

Put $F=i-zj-yk$

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

This implies that force field is conservative.

Use the formula $W=∫F.dr$

$W=∫dx-zdy-ydz$

Take path from (0,0,0) to (x,y,z)

(i) From (0,0,0) to (x,0,0)

y = 0

z = 0

dy = 0

dz = 0

$$\begin{gathered} W_1=\underset{0}{\overset{x}{∫}}dx \\ {W}_1=x \end{gathered}$$

(ii) From (x,0,0) to (x,0,z)

y = 0

dy = 0

dx = 0

$$\begin{gathered} W_2=\underset{0}{\overset{z}{∫}}0-z\left(0\right)-0\left(dz\right) \\ {W}_2=0 \end{gathered}$$

(iii) From (x,0,z) to (x,y,z)

dx = 0

dz = 0

$$\begin{gathered} W_3=\underset{0}{\overset{y}{∫}}0-zdy-y\left(0\right) \\ {W}_3=-zy \end{gathered}$$

Total work done is given below.

$$\begin{gathered} W=W_1+W_2+W_3 \\ W=x+0-zy \\ W=x-zy \end{gathered}$$

Use the formula $F=∇W$

Also, use the given relation between force and scalar potential which is given below.

$F=-∇\mathrm{φ}$

From both the above formulas we get that $\mathrm{φ}=-W$

$$\begin{gathered} ⇒\mathrm{φ}=-\left(x-yz\right) \\ ⇒\mathrm{φ}=-x+yz \end{gathered}$$

The force field is conservative and the scalar potential $\mathrm{φ}=-x+yz$