Green's Theorem — Question 8

Question 8 – Solution

Strategy. Green’s Theorem turns the circulation condition into a linear equation for $a$.

Step 1: Form the integrand With $P=ay^2$ and $Q=x^2$, $$Q_x-P_y=2x-2ay.$$ Thus $$\begin{align*} \oint_C\mathbf F_a\cdot d\mathbf r &=\int_0^1\int_0^1(2x-2ay)\,dy\,dx\\ &=\int_0^1(2x-a)\,dx=1-a. \end{align*}$$

Step 2: Impose the condition $$1-a=-2 \quad\Longrightarrow\quad \boxed{a=3}.$$

Step 3: Verify For $a=3$, $$\int_0^1\int_0^1(2x-6y)\,dy\,dx =1-3=-2,$$ exactly the specified counterclockwise circulation.

Uniqueness check The circulation $1-a$ has nonzero coefficient $-1$ on $a$, so the equation has exactly one solution.