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Question 27.5: Show that the vector field F = ze^x sin yi + ze^x cos yj + e......

Show that the vector field

\mathbf{F}=z\mathbf{e}^{x}\sin y\mathbf{i}+z\mathbf{e}^{x}\cos y\mathbf{j}+\mathbf{e}^{x}\sin y\mathbf{k}

is a conservative field.

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We find curl F:

\nabla \times F = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ ze^{x}\sin y & ze^{x}\cos y & e^{x}\sin y\end{vmatrix}\\ =(\mathrm{e}^{x}\cos y-\mathrm{e}^{x}\cos y)\mathrm{i}-(\mathrm{e}^{x}\sin y-\mathrm{e}^{x}\sin y)\mathrm{j}+(\mathrm{ze}^{x}\cos y-\mathrm{ze}^{x}\cos y)\mathrm{k}

=0

We have shown that ∇ ×F = 0 and so the field is conservative. Note from Section 26.5 that such a field is also said to be irrotational.

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