Evaluate ∫SF⃗ ⋅dA⃗
, where F⃗ =(bx/a)i⃗ +(ay/b)j⃗
and S
is the elliptic cylinder oriented away from the z
-axis, and given by x^2/a^2+y^2/b^2=1
, |z|≤c
, where a
, b
, c
are positive constants.

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.

Learn how to evaluate the surface integral ∫SF⃗ ⋅dA⃗ over an elliptic cylinder oriented away from the z-axis using vector calculus and the Divergence Theorem. Detailed step-by-step solution with computations of divergence, volume integration in cylindrical coordinates, and practical applications. Suitable for advanced undergraduate students and professionals in mathematics and physics.

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