To evaluate the surface integral $\int_{S} \mathbf{F} \cdot d\mathbf{A}$, we will use the Divergence Theorem. The Divergence Theorem relates a surface integral over a closed surface to a volume integral over the region enclosed by the surface. It states that:

$\int_{S} \mathbf{F} \cdot d\mathbf{A} = \int_{V} (\nabla \cdot \mathbf{F}) \, dV$

where $\mathbf{F} = \left( \frac{bx}{a} \right) \mathbf{i} + \left( \frac{ay}{b} \right) \mathbf{j}$ and $V$ is the volume enclosed by the surface $S$.

First, we need to compute the divergence of $\mathbf{F}$:

$\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} \left( \frac{bx}{a} \right) + \frac{\partial}{\partial y} \left( \frac{ay}{b} \right)$

Calculate the partial derivatives:

$\frac{\partial}{\partial x} \left( \frac{bx}{a} \right) = \frac{b}{a}$

$\frac{\partial}{\partial y} \left( \frac{ay}{b} \right) = \frac{a}{b}$

So, the divergence of $\mathbf{F}$ is:

$\nabla \cdot \mathbf{F} = \frac{b}{a} + \frac{a}{b}$

Next, we need to integrate this divergence over the volume $V$ of the elliptic cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and $|z| \leq c$.

The volume element in cylindrical coordinates for this elliptic cylinder can be written as $dV = dA \, dz = \left( \pi ab \right) dz$, where $\pi ab$ is the area of the elliptic cross-section.

Thus, the volume integral becomes:

$\int_{V} (\nabla \cdot \mathbf{F}) \, dV = \int_{-c}^{c} \int_{\text{elliptic cross-section}} \left( \frac{b}{a} + \frac{a}{b} \right) \, dA \, dz$

The area of the elliptic cross-section is $\pi ab$, so:

$\int_{\text{elliptic cross-section}} \, dA = \pi ab$

Hence, the volume integral simplifies to:

$\int_{V} (\nabla \cdot \mathbf{F}) \, dV = \left( \frac{b}{a} + \frac{a}{b} \right) \int_{-c}^{c} (\pi ab) \, dz$

The integral with respect to $z$ over $[-c, c]$ is:

$\int_{-c}^{c} dz = 2c$

So, the volume integral is:

$\int_{V} (\nabla \cdot \mathbf{F}) \, dV = \left( \frac{b}{a} + \frac{a}{b} \right) (\pi ab) (2c)$

Simplifying further:

$\int_{V} (\nabla \cdot \mathbf{F}) \, dV = \left( \frac{b}{a} + \frac{a}{b} \right) (2\pi abc)$

Therefore, the value of the surface integral $\int_{S} \mathbf{F} \cdot d\mathbf{A}$ is:

$\int_{S} \mathbf{F} \cdot d\mathbf{A} = 2\pi abc \left( \frac{b}{a} + \frac{a}{b} \right)$

Is there anything specific you would like further details on or any questions?

Relative Questions:

1. What is the Divergence Theorem and how is it used in vector calculus?
2. How do you compute the divergence of a vector field?
3. What are cylindrical coordinates and how do they simplify volume integrals?
4. How is the area of an elliptic cross-section calculated?
5. Why do we integrate with respect to $z$ from $-c$ to $c$?
6. What are some applications of the Divergence Theorem in physics?
7. How would the solution change if the orientation of the surface was towards the z-axis?
8. Can the Divergence Theorem be applied to non-closed surfaces?

Tip:

When applying the Divergence Theorem, always ensure that the surface integral you are evaluating is over a closed surface to correctly use the relationship between the surface integral and the volume integral.