\( D \) is the solid cylinder bounded by \( x^{2}+y^{2}=1 \), the \( x y \)-plane, and the plane \( z=2 \). Use the Divergence Theorem to find the flux of the vector field \( \mathbf{F}(x, y, z)=\left\langle 2 x-y z, 9 y+e^{x z}, \cos (7 y)-7 z\right\rangle \) over the surface of \( D \). Consider cylindrical coordinates. Write the exact answer. Do not round. Answer

Step 1 ：Step 1: Compute the divergence: \(\nabla \cdot \mathbf{F}(x, y, z) = \frac{\partial}{\partial x}(2x-yz) + \frac{\partial}{\partial y}(9y + e^{xz}) + \frac{\partial}{\partial z}(\cos(7y) - 7z) = 2 - z + 9 + x e^{xz} - 7\)

Step 2 ：Step 2: Convert to cylindrical coordinates: \(x = r\cos(\theta), y = r\sin(\theta), z = z\), \(\nabla \cdot \mathbf{F}(r, \theta, z) = 2 - z + 9 + r\cos(\theta)e^{r\cos(\theta)z} - 7\)

Step 3 ：Step 3: Compute the flux integral: \(\iint_{S}\vec{F}\cdot d\vec{S} = \iiint_{D}(\nabla \cdot \mathbf{F})dV = \int_{0}^{2}\int_{0}^{1}\int_{0}^{2\pi}(2 - z + 9 + r\cos(\theta)e^{r\cos(\theta)z} - 7)r\,d\theta\,dr\,dz = \int_{0}^{2}\int_{0}^{1}\int_{0}^{2\pi}(2r - rz + 9r - 7r)r\,d\theta\,dr\,dz\)