Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
To solve the given integral, we will use the vector calculus identity known as the curl of a cross product. Specifically, we will use the following identity:
\[
\mathbf{G} \cdot \text{curl}(\mathbf{F}) - \mathbf{F} \cdot \text{curl}(\mathbf{G}) = \nabla \cdot (\mathbf{F} \times \mathbf{G})
\]
This identity simplifies the integral significantly. Let's proceed step-by-step.

1. **Compute the curl of \(\mathbf{F}\):**
\[
\mathbf{F}(x,y,z) = (\sin(xy), \sin(yz), 0)
\]
The curl of \(\mathbf{F}\) is given by:
\[
\text{curl}(\mathbf{F}) = \nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)
\]
Substituting the components of \(\mathbf{F}\):
\[
\text{curl}(\mathbf{F}) = \left( 0 - \cos(yz), 0 - 0, \cos(yz) - \cos(xy) \right) = (-y \cos(yz), 0, -x \cos(xy))
\]

2. **Compute the curl of \(\mathbf{G}\):**
\[
\mathbf{G}(x,y,z) = (e^{x^2 + y^2 + z^2}, \cos(xz), 0)
\]
The curl of \(\mathbf{G}\) is given by:
\[
\text{curl}(\mathbf{G}) = \nabla \times \mathbf{G} = \left( \frac{\partial G_3}{\partial y} - \frac{\partial G_2}{\partial z}, \frac{\partial G_1}{\partial z} - \frac{\partial G_3}{\partial x}, \frac{\partial G_2}{\partial x} - \frac{\partial G_1}{\partial y} \right)
\]
Substituting the components of \(\mathbf{G}\):
\[
\text{curl}(\mathbf{G}) = \left( 0 - (-xz \sin(xz)), 2z e^{x^2 + y^2 + z^2} - 0, -z \sin(xz) - 0 \right) = (xz \sin(xz), 2z e^{x^2 + y^2 + z^2}, -z \sin(xz))
\]

3. **Compute the divergence of the cross product \(\mathbf{F} \times \mathbf{G}\):**
\[
\mathbf{F} \times \mathbf{G} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \sin(xy) & \sin(yz) & 0 \\ e^{x^2 + y^2 + z^2} & \cos(xz) & 0 \end{matrix} \right|
\]
\[
\mathbf{F} \times \mathbf{G} = \left( 0 - 0, 0 - 0, \sin(xy) \cos(xz) - \sin(yz) e^{x^2 + y^2 + z^2} \right) = (0, 0, \sin(xy) \cos(xz) - \sin(yz) e^{x^2 + y^2 + z^2})
\]

4. **Compute the divergence of \(\mathbf{F} \times \mathbf{G}\):**
\[
\nabla \cdot (\mathbf{F} \times \mathbf{G}) = \frac{\partial}{\partial x}(0) + \frac{\partial}{\partial y}(0) + \frac{\partial}{\partial z}(\sin(xy) \cos(xz) - \sin(yz) e^{x^2 + y^2 + z^2}) = 0
\]

Since the divergence of \(\mathbf{F} \times \mathbf{G}\) is zero, the integral over the region \(W\) is also zero:
\[
\iiint_{W} \nabla \cdot (\mathbf{F} \times \mathbf{G}) \, dV = 0
\]

The final answer is \(\boxed{0}\).