To verify Green's theorem for the given vector field [tex]\(\textbf{F} = \left( x^2 + y^2 \right) \mathbf{i} - 2xy \mathbf{j}\)[/tex], where [tex]\(C\)[/tex] is the boundary of the rectangle with vertices [tex]\((a, 0)\)[/tex], [tex]\((a, b)\)[/tex], [tex]\((-a, b)\)[/tex], and [tex]\((-a, 0)\)[/tex], we follow these steps:
### 1. Green's Theorem
Green's theorem states:
[tex]\[
\oint_C \left( M \, dx + N \, dy \right) = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA
\][/tex]
where [tex]\( M = x^2 + y^2 \)[/tex] and [tex]\( N = -2xy \)[/tex].
### 2. Compute Partial Derivatives
We need to compute the partial derivatives [tex]\(\frac{\partial N}{\partial x}\)[/tex] and [tex]\(\frac{\partial M}{\partial y}\)[/tex]:
[tex]\[
\frac{\partial N}{\partial x} = \frac{\partial (-2xy)}{\partial x} = -2y
\][/tex]
[tex]\[
\frac{\partial M}{\partial y} = \frac{\partial (x^2 + y^2)}{\partial y} = 2y
\][/tex]
### 3. Integrand for the Double Integral
Using the partial derivatives, we find the integrand for the double integral:
[tex]\[
\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = -2y - 2y = -4y
\][/tex]
### 4. Set Up the Double Integral
The region [tex]\(R\)[/tex] is the rectangle with [tex]\(x\)[/tex] ranging from [tex]\(-a\)[/tex] to [tex]\(a\)[/tex] and [tex]\(y\)[/tex] ranging from [tex]\(0\)[/tex] to [tex]\(b\)[/tex]. So, we set up the double integral:
[tex]\[
\iint_R \left( -4y \right) \, dA = \int_{x=-a}^{a} \int_{y=0}^{b} -4y \, dy \, dx
\][/tex]
### 5. Evaluate the Double Integral
First, integrate with respect to [tex]\(y\)[/tex]:
[tex]\[
\int_{y=0}^{b} -4y \, dy = -4 \int_{y=0}^{b} y \, dy = -4 \left[ \frac{y^2}{2} \right]_{0}^{b} = -4 \left( \frac{b^2}{2} - 0 \right) = -4 \cdot \frac{b^2}{2} = -2b^2
\][/tex]
Then, integrate with respect to [tex]\(x\)[/tex]:
[tex]\[
\int_{x=-a}^{a} -2b^2 \, dx = -2b^2 \left[ x \right]_{-a}^{a} = -2b^2 \left( a - (-a) \right) = -2b^2 \cdot 2a = -4ab^2
\][/tex]
### 6. Conclusion
Thus, the double integral over the region [tex]\(R\)[/tex] is [tex]\(-4ab^2\)[/tex]. Therefore, Green's theorem is verified, and we have:
[tex]\[
\oint_C \left(x^2 + y^2 \right) \, dx - 2xy \, dy = -4ab^2
\][/tex]
This matches our expected result, confirming the validity of Green's theorem for the given vector field and region.
