Question

Determine whether the following force field

Fis conservative:

F=xi-yj+zk

Answer 1

The conservative field $\bf F$ must satisfy condition

$$\rm rot\:{\bf F}=0 .$$

We have

$${\rm rot}\:(x\hat i-y\hat j+z\hat k)=\begin{vmatrix} \hat i & \hat j & \hat k \\ x & -y & z\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y}& \frac{\partial}{\partial z} \end{vmatrix}$$

$$=\hat i\left(\frac{\partial (-y)}{\partial z}-\frac{\partial z}{\partial y})\right)-\hat j\left(\frac{\partial x}{\partial z}-\frac{\partial z}{\partial x})\right)+$$

$$+\hat k\left(\frac{\partial x}{\partial y}-\frac{\partial (-y)}{\partial x})\right)=0\hat i+0\hat j+0\hat k=\bf 0.$$

Hence, the field $\bf F$ is conservative.