91Ó°ÊÓ

We have to determine whether or not each of the vector fields in the given exercises is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y,z)=\mathbf{i}+2\mathbf{j}−3\mathbf{k}$

A vector field $\mathbf{F}(x,y,z)=\left(F_1\right(x,y,z),F_2(x,y,z),F_3(x,y,z\left)\right)$is conservative if and only if localid="1650559372319" $\frac{∂F_3}{∂y}=\frac{∂F_2}{∂z},\frac{∂F_1}{∂z}=\frac{∂F_3}{∂x},\frac{∂F_2}{∂x}=\frac{∂F_1}{∂y}$

$$\begin{gathered} \frac{∂F_3}{∂y}=\frac{∂}{∂y}3\frac{∂F_2}{∂z}=\frac{∂}{∂z}2 \\ \frac{∂F_3}{∂y}=0\frac{∂F_2}{∂z}=0 \end{gathered}$$

Hence,$\frac{∂F_3}{∂y}=\frac{∂F_2}{∂z}$

Since, $\mathbf{F}(x,y,z)=\mathbf{i}+2\mathbf{j}−3\mathbf{k}$

$$\begin{gathered} f(x,y,z)=∫1dx+B+C \\ f(x,y,z)=x+\mathrm{Î\pm }+B+C \end{gathered}$$

where $\mathrm{Î\pm }$ is an arbitrary constant and B is the integral with respect to y of the terms in $F_2(x,y,z)$ in which the factorx does not appear.

$$\begin{gathered} B=∫2dy \\ B=2y+\mathrm{β} \end{gathered}$$

where β is an arbitrary constant.

$$\begin{gathered} C=-∫3dz \\ C=-3z+\mathrm{γ} \end{gathered}$$

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

We have,

$f(x,y,z)=x+2y-3z$