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Let's go through the problems one by one.

### A1. Flux Calculation
You are asked to calculate the flux of the vector field **F(x, y, z)** across the surface **S**. The vector field is:
\[
\mathbf{F}(x, y, z) = (\cos z + xy^2) \mathbf{i} + (xe^{-z}) \mathbf{j} + (\sin y + x^2z) \mathbf{k}
\]
and the surface **S** is bounded by \(z = x^2 + y^2\) and \(z = 4\). 

To calculate the flux of a vector field through a closed surface, you use the **Divergence Theorem**:
\[
\text{Flux} = \iint_S \mathbf{F} \cdot d\mathbf{A} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV
\]
First, compute the divergence of **F(x, y, z)**:
\[
\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (\cos z + xy^2) + \frac{\partial}{\partial y} (xe^{-z}) + \frac{\partial}{\partial z} (\sin y + x^2z)
\]
Simplify each term:
- \(\frac{\partial}{\partial x} (\cos z + xy^2) = y^2\)
- \(\frac{\partial}{\partial y} (xe^{-z}) = 0\)
- \(\frac{\partial}{\partial z} (\sin y + x^2z) = x^2\)

Thus:
\[
\nabla \cdot \mathbf{F} = y^2 + x^2
\]
Now, we need to integrate this over the volume between the surfaces \(z = x^2 + y^2\) and \(z = 4\). Using cylindrical coordinates (\(x = r \cos \theta\), \(y = r \sin \theta\), \(z = z\)), the volume element \(dV = r \, dz \, dr \, d\theta\).

The divergence in cylindrical coordinates becomes:
\[
\nabla \cdot \mathbf{F} = r^2
\]
The flux integral becomes:
\[
\text{Flux} = \int_0^{2\pi} \int_0^2 \int_{r^2}^4 r^2 r \, dz \, dr \, d\theta
\]

### A2. Volume of Region
The region is enclosed by the plane \(z = 4\) and the surface \(z = (2x - y)^2 + (x + y - 1)^2\).

The volume can be computed using a triple integral:
\[
V = \iiint_V dz \, dx \, dy
\]
The boundaries are more complex, so transforming coordinates may simplify the calculation.

### A3. Complex Integral
You are asked to evaluate the integral:
\[
\oint_C \frac{e^z}{z^3 - z^2 - 5z - 3} dz
\]
This is a complex contour integral. To solve it, we need to find the poles of the integrand by solving \(z^3 - z^2 - 5z - 3 = 0\) and applying the **Residue Theorem**.

### A4. Verify Stokes Theorem
For this problem, you are asked to verify **Stokes' Theorem** for a given vector field and triangular surface. Stokes' Theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary curve:
\[
\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{A} = \oint_C \mathbf{F} \cdot d\mathbf{r}
\]

Would you like further details or any clarification on any of the steps?

### 5 Related Questions:
1. What is the physical meaning of flux in the context of vector fields?
2. How can the Divergence Theorem simplify flux calculations?
3. What are some common techniques for solving complex contour integrals?
4. In what cases is it beneficial to use cylindrical or spherical coordinates for volume integrals?
5. How does Stokes' Theorem generalize Green's Theorem in three dimensions?

#### Tip:
When solving integrals over complex surfaces, identifying symmetries or singularities can significantly simplify the process.

A1. Given that F(x, y, z) = (cos z + xy^2)i + (xe^{-z})j + (sin y + x^2z)k and S is the surface of the solid bounded by z = x^2 + y^2 and z = 4, calculate the flux of F(x, y, z) across S.

Learn how to calculate the flux of a vector field F(x, y, z) = (cos z + xy^2)i + (xe^{-z})j + (sin y + x^2z)k across a surface S bounded by z = x^2 + y^2 and z = 4. This solution uses the Divergence Theorem and covers the essential concepts of vector calculus.

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