Question 27.7: Show that the vector field F = ze^k sin yi + ze^x cos yj + ......
Show that the vector field
\mathbf{F}=z \mathrm{e}^x \sin y \mathbf{i}+z \mathrm{e}^x \cos y \mathbf{j}+\mathrm{e}^x \sin y \mathbf{k}is a conservative field.
We find curl F:
\begin{aligned} \nabla \times \mathbf{F} & =\left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ z \mathrm{e}^x \sin y & z \mathrm{e}^x \cos y & \mathrm{e}^x \sin y \end{array}\right| \end{aligned}
\begin{aligned} =\left(\mathrm{e}^x \cos y-\mathrm{e}^x \cos y\right) \mathbf{i}-\left(\mathrm{e}^x \sin y-\mathrm{e}^x \sin y\right) \mathbf{j}+\left(z \mathrm{e}^x \cos y-z \mathrm{e}^x \cos y\right) \mathbf{k} \end{aligned}
\begin{aligned} =\mathbf{0} \end{aligned}
We have shown that \nabla \times \mathbf{F}=\mathbf{0} and so the field is conservative. Note from Section 26.5 that such a field is also said to be irrotational.