We are tasked with calculating the circulation of the vector field $\vec{F}(x, y) = (4x - 3y)\vec{i} + 4x\vec{j}$ around the given sector $C$, using Green's Theorem. Let's break this down step by step:

Green's Theorem

Green's Theorem states: $\oint_C \vec{F} \cdot d\vec{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA,$ where:

• $\vec{F}(x, y) = P(x, y)\vec{i} + Q(x, y)\vec{j}$,
• $P(x, y) = 4x - 3y$,
• $Q(x, y) = 4x$,
• $R$ is the region enclosed by the curve $C$.

Step 1: Compute $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$

• $\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(4x) = 4$,
• $\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(-3y) = -3$.

Thus, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 4 - (-3) = 7$.

Step 2: Set up the area of the region $R$

The region $R$ is a sector of a circle of radius 4 with a central angle of $\pi/4$. The area of a sector of a circle is given by: $\text{Area of Sector} = \frac{1}{2} r^2 \theta,$ where $r = 4$ and $\theta = \pi/4$. Thus: $\text{Area of Sector} = \frac{1}{2} (4^2) \frac{\pi}{4} = \frac{16 \pi}{8} = 2\pi.$

Step 3: Evaluate the double integral

The circulation is given by: $\oint_C \vec{F} \cdot d\vec{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA.$ Substituting $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 7$: $\iint_R 7 \, dA = 7 \cdot \text{Area of Sector} = 7 \cdot 2\pi = 14\pi.$

Final Answer

The circulation of $\vec{F}$ around $C$ is: $\boxed{14\pi}.$

Would you like further explanation or to explore more about Green's Theorem? Here are some related questions for further study:

Related Questions:

1. How does Green's Theorem relate to the Divergence Theorem?
2. What happens if the vector field is not continuous in the region?
3. How do you compute circulation if the curve $C$ is oriented clockwise?
4. Can Green's Theorem be applied to regions with holes?
5. How does the formula change for scalar fields instead of vector fields?

Tip:

Always check the orientation of the curve (counterclockwise in this case) when applying Green's Theorem. Reversing the orientation reverses the sign of the circulation.