Curly

$$ The curl of $\mathbf{F}$ at the point $\left(\frac{7\pi}{6},\frac{\pi}{6}\,\frac{2\pi}{3}\right)$ is $$ $\begin{aligned} \nabla\times\mathbf{F} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} &\mathbf{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ x\, \sin z & y\,\cos x & z\,\tan y \end{vmatrix}\\ &=\left(\frac{\partial}{\partial y}(z\,\tan y)-\frac{\partial}{\partial z}(y\,\cos x)\right)\mathbf{i}-\left(\frac{\partial}{\partial x}(z\,\tan y)-\frac{\partial}{\partial z}(x\, \sin z)\right)\mathbf{j}+\left(\frac{\partial}{\partial x}(y\,\cos x)-\frac{\partial}{\partial y}(x\, \sin z)\right)\mathbf{k}\\ &=(z\,\sec^2y-0)\,\mathbf{i}-(0-x\,\cos z)\,\mathbf{j}+(-y\,\sin x-0)\,\mathbf{k}\\ &= z\,\sec^2y\;\mathbf{i}+x\,\cos z\;\mathbf{j}-y\,\sin x\;\mathbf{k}\\ &= \frac{8}{9}\pi\;\mathbf{i} - \frac{7}{12}\pi\;\mathbf{j} + \frac{1}{12}\pi\;\mathbf{k} \end{aligned}$ $$ Thus, $a+b+c= \frac{8}{9}\pi- \frac{7}{12}\pi+ \frac{1}{12}\pi=\frac{7}{18}\pi\;$ and $\;d+e=7+18=\boxed{25}$ . $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$

The curl of $\mathbf{F}$ is $$\mathrm{curl} \, \mathbf{F} = \nabla \times \mathbf{F}= \left|\begin{array}{ccc} \mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ x \sin z& y \cos x& z \tan y \end{array} \right|$$ $$=\langle z \sec^2 y, x \cos z, -y \sin x \rangle.$$ At $\left( \frac{7\pi}{6},\frac{\pi}{6},\frac{2\pi}{3} \right)$ , we get $$\langle \frac{8\pi}{9},-\frac{7\pi}{12},\frac{\pi}{12} \rangle.$$ So $a+b+c=\frac{7\pi}{18}$ , and $d+e=\boxed{25}$ .

25