Consider the vector field

\( \mathbf{F}(x, y, z) = x\sin{z} \, \, \mathbf{i} + y\cos{x} \, \, \mathbf{j} + z\tan{y} \, \, \mathbf{k} \).

The curl of this vector field at the point \( ( \frac{7\pi}{6}, \frac{\pi}{6}, \frac{2\pi}{3}) \) can be expressed in the form \( a \, \mathbf{i} + b \, \mathbf{j} + c \, \mathbf{k} \) where \( a \), \( b \) and \( c \) are real numbers. \( a + b + c \) can be expressed in the form \( \frac{d}{e} \pi \) where \( d \) and \( e \) are coprime positive integers. What is \( d + e \)?

If you are unfamiliar with the concept of curl you may wish to read the following webpage.

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