Answer :
Explanation:
To solve for the circulation and flux of the vector field \( \mathbf{F} = y \mathbf{i} - x \mathbf{j} \) around and across the given curves, we'll use the following principles:
1. **Circulation of \(\mathbf{F}\) around a closed curve \(C\)** is given by:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r}
\]
2. **Flux of \(\mathbf{F}\) across a closed curve \(C\)** is given by:
\[
\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS
\]
where \(S\) is the surface bounded by \(C\), and \(\mathbf{n}\) is the unit normal vector to \(S\).
### Part (a): Circle \( r(t) = \cos(t) \mathbf{i} - \sin(t) \mathbf{j} \), \( 0 \leq t \leq 2\pi \)
**1. Circulation:**
The parameterization of the circle is \( \mathbf{r}(t) = \cos(t) \mathbf{i} - \sin(t) \mathbf{j} \).
Compute \( d\mathbf{r} \):
\[
d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt = (-\sin(t) \mathbf{i} - \cos(t) \mathbf{j}) dt
\]
The vector field \( \mathbf{F} \) in terms of \( t \) is:
\[
\mathbf{F} = y \mathbf{i} - x \mathbf{j} = (-\sin(t)) \mathbf{i} - (\cos(t)) \mathbf{j}
\]
Compute the dot product \( \mathbf{F} \cdot d\mathbf{r} \):
\[
\mathbf{F} \cdot d\mathbf{r} = (-\sin(t) \mathbf{i} - \cos(t) \mathbf{j}) \cdot (-\sin(t) \mathbf{i} - \cos(t) \mathbf{j}) dt = (\sin^2(t) + \cos^2(t)) dt = dt
\]
Integrate over \( 0 \leq t \leq 2\pi \):
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} dt = 2\pi
\]
Thus, the circulation of \( \mathbf{F} \) around the circle is \( 2\pi \).
**2. Flux:**
Compute \( \nabla \times \mathbf{F} \):
\[
\nabla \times \mathbf{F} = \left( \frac{\partial (-x)}{\partial y} - \frac{\partial y}{\partial x} \right) \mathbf{k} = (-1 - 1) \mathbf{k} = -2 \mathbf{k}
\]
The area of the circle \( S \) is:
\[
\text{Area} = \pi \times r^2 = \pi \times 1^2 = \pi
\]
Since \( \mathbf{n} \) (the unit normal vector to the plane) is \( \mathbf{k} \), the flux through \( S \) is:
\[
\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \iint_S -2 \, dS = -2 \times \pi = -2\pi
\]
Thus, the flux of \( \mathbf{F} \) across the circle is \( -2\pi \).
### Summary
1. The circulation of \( \mathbf{F} \) around the circle is \( 2\pi \).
2. The flux of \( \mathbf{F} \) across the circle is \( -2\pi \).
