Question

For \( a>0, b>0 \), let \[ \mathbf{F} = \frac{xj - yk}{b x^2 + a y^2}. \] Let \[ C = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = a^2 + b^2\}. \] Then the line integral \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \]

• A. \( \frac{2\pi}{ab} \)
• B. \( 2\pi \)
• C. \( 2nab \)
• D. 0

Hint:

When evaluating line integrals over closed curves, parametrization and Green's Theorem are often helpful tools.

Answer 1

The Correct Option is A

Step 1: Parametrizing the curve.
The curve \( C \) is a circle, and we parametrize it as: \[ x = (a + b)\cos t, \quad y = (a + b)\sin t, \quad t \in [0, 2\pi]. \]
Step 2: Line integral computation.
The vector field \( \mathbf{F} \) has the form \( \frac{xj - yk}{b x^2 + a y^2} \), and the line integral over the closed curve \( C \) involves applying Green's Theorem or directly computing the integral using the parametrization. The result is \( \frac{2\pi}{ab} \).
Step 3: Conclusion.
Thus, the correct answer is (A).