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Question

# Find div $\mathbf{F}$ and curl $\mathbf{F}$ for the vector field $\mathbf{F}$ at the given point.$\begin{array}{l}{\mathbf{F}(x, y, z)=\left(e^{-x} \sin y\right) \mathbf{i}+\left(e^{-x} \cos y\right) \mathbf{j}+\mathbf{k} \text { at }} \\ {(1,3,-2)}\end{array}$

Solution

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Here $F(x,y)=( e^{-x}\sin{y})\hat{i}+(e^{-x}\cos{y})\hat{j}+\hat{k}$ So

\begin{align*} div F &= \frac{\partial (e^{-x}\sin{y})}{\partial x}+\frac{\partial(e^{-x}\cos{y})}{\partial y}+\frac{\partial(1)}{\partial z}\\ &=-e^{-x}\sin{y}-e^{-x}\sin{y}\\ &=-2e^{-x}\sin{y}\\ \end{align*}

Hence at the point $(1,3,-2)$ divergence is $-2e^{-1}\sin{3}$.

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