Let's address each of the problems presented in the image one by one:

Problem A1:

Given that $\mathbf{F}(x, y, z) = (\cos z + xy^2) \mathbf{i} + (xe^{-z}) \mathbf{j} + (\sin y + x^2z) \mathbf{k}$ and $S$ is the surface of the solid bounded by $z = x^2 + y^2$ and $z = 4$, calculate the flux of $\mathbf{F}(x, y, z)$ across $S$.

Solution: To find the flux, we need to apply the Divergence Theorem, which states:

$\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV$

where $V$ is the volume enclosed by the surface $S$. The divergence $\nabla \cdot \mathbf{F}$ is:

$\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (\cos z + xy^2) + \frac{\partial}{\partial y} (xe^{-z}) + \frac{\partial}{\partial z} (\sin y + x^2z)$

Let's calculate $\nabla \cdot \mathbf{F}$:

$\nabla \cdot \mathbf{F} = y^2 + 0 + x^2 = y^2 + x^2$

Now, integrate $\nabla \cdot \mathbf{F}$ over the volume bounded by $z = x^2 + y^2$ and $z = 4$:

$\iiint_V (x^2 + y^2) \, dV$

We convert this to cylindrical coordinates, where $x = r\cos\theta$, $y = r\sin\theta$, and $dV = r\, dz\, dr\, d\theta$. The limits for $r$ range from 0 to 2, $\theta$ from 0 to $2\pi$, and $z$ from $r^2$ to 4.

Further calculation would proceed from here.

Problem A2:

Find the volume of the region enclosed by the plane $z = 4$ and the surface $z = (2x - y)^2 + (x + y - 1)^2$.

Solution: The volume can be found by setting up a double integral over the projection of the region onto the $xy$-plane:

$\text{Volume} = \iint_D \left(4 - [(2x - y)^2 + (x + y - 1)^2]\right) \, dx \, dy$

Here, $D$ is the domain in the $xy$-plane where $z$ values are valid. Further simplification would involve determining the exact bounds of $D$.

Problem A3:

Evaluate:

$\oint_C \frac{e^z dz}{z^3 - z^2 - 5z - 3},$

where $z$ is complex.

Solution: To evaluate this contour integral, we must find the poles of the integrand within the contour $C$. Factor the denominator to find the poles. We apply the Residue Theorem to evaluate the integral.

Problem A4:

For $\mathbf{F}(x, y, z) = (x + y^2)\mathbf{i} + (y + z^2)\mathbf{j} + (z + x^2)\mathbf{k}$ and $C$ is the triangle with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$, verify Stokes' Theorem.

Solution: Stokes' Theorem states:

$\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_C \mathbf{F} \cdot d\mathbf{r}$

where $\nabla \times \mathbf{F}$ is the curl of $\mathbf{F}$, and $S$ is the surface bounded by the curve $C$. We need to compute $\nabla \times \mathbf{F}$ and evaluate the surface integral, and then compare it to the line integral around $C$.

Would you like me to proceed with detailed calculations for any of these problems?

Related Questions:

1. How do you calculate surface integrals using parameterization?
2. What is the method for converting Cartesian coordinates to cylindrical coordinates?
3. Can the Divergence Theorem be applied to non-closed surfaces?
4. What is the importance of finding residues in complex integration?
5. How can we determine the bounds of integration for non-standard shapes?

Tip: When applying the Divergence or Stokes' Theorems, always ensure the vector field and the surface (or volume) meet the necessary conditions (e.g., differentiability, bounded regions).