Show that $\overline{\boldsymbol{F}}$ can be expressed as the gradient of a scalar function. Find the work done in moving a particle in this field from (1,2,-4) to (3,3,2) along the straight line joining the points.

$$\overline{F}$ = $\left(x+2y+az\right)i+\left(bx-3y-z\right)j+\left(4x+cy+2z\right)k$$ and $$\overline{r} = x \overline{i}+y\overline{j}+z\overline{k}$$ $$\therefore dr = dx \overline{i}+dy\overline{j}+dz\overline{k}$$

Since $\overline{F}$ is irrotational

Curl $\overline{F}=0$

$\boldsymbol{\therefore }\boldsymbol{\ }\left| \begin{array}{ccc} \overline{i} & \overline{j} & \overline{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ F_x & F_y & F_z \end{array} \right|=0$

$\therefore \ …