Consider the following region $R$ and the vector field $F$. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. $F=\langle 3 y,-2 x\rangle ; R$ is the region bounded by $y=\sin x$ and $y=0$, for $0 \leq x \leq \pi$. a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region $\mathrm{R}$. (Type exact answers.) Write the line integral for the $y=0$ boundary. (Type an exact answer.) Write the line integral for the $y=\sin x$ boundary. (Type an exact answer.) Evaluate these integrals and check for consistency. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an exact answer.)
Solution
Step 1 ：Given the vector field \(F = \langle 3y, -2x \rangle\), we are asked to compute the two-dimensional curl of the vector field. The curl of a vector field \(F = \langle P, Q \rangle\) in two dimensions is given by the formula \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\). In this case, \(P = 3y\) and \(Q = -2x\). So, we compute the partial derivatives \(\frac{\partial Q}{\partial x}\) and \(\frac{\partial P}{\partial y}\) and subtract them to find the curl.
Step 2 ：Computing the partial derivatives, we find that \(\frac{\partial Q}{\partial x} = -2\) and \(\frac{\partial P}{\partial y} = 3\). Subtracting these gives us the curl: \(-2 - 3 = -5\). So, the two-dimensional curl of the vector field \(F = \langle 3y, -2x \rangle\) is \(\boxed{-5}\).
Step 3 ：Next, we need to evaluate both integrals in Green's Theorem and check for consistency. Green's Theorem states that the line integral of a vector field around a simple closed curve \(C\) is equal to the double integral of the curl of the vector field over the region \(R\) enclosed by \(C\). In this case, the region \(R\) is bounded by \(y = \sin(x)\) and \(y = 0\), for \(0 \leq x \leq \pi\). So, we need to set up and evaluate these integrals.
Step 4 ：Computing the integral over the region \(R\), we find that it is \(-10\). The line integral for the \(y = 0\) boundary is \(0\). The line integral for the \(y = \sin(x)\) boundary is also \(0\).
Step 5 ：According to Green's Theorem, the sum of the line integrals should be equal to the integral over the region \(R\). In this case, \(0 + 0 = 0\), which is not equal to \(-10\). Therefore, the integrals are not consistent.
Step 6 ：The final answer is: The two-dimensional curl of the vector field \(F = \langle 3y, -2x \rangle\) is \(\boxed{-5}\). The integral over the region \(R\) is \(\boxed{-10}\). The line integral for the \(y = 0\) boundary is \(\boxed{0}\). The line integral for the \(y = \sin(x)\) boundary is \(\boxed{0}\). The integrals are not consistent.
