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Example: Use Green's theorem to evaluate the line integral \(\oint_C (x^2\,dx + y^2\,dy)\) where C is the unit square with vertices (0,0), (1,0), (1,1), (0,1) traversed counter‑clockwise. Step 1: Identify P(x,y)=x^2 and Q(x,y)=y^2. Step 2: Compute the partial derivatives: \(\partial Q/\partial x = 0\) and \(\partial P/\partial y = 0\). Step 3: Green's theorem says \(\oint_C (P\,dx + Q\,dy) = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,dA\). Step 4: Substitute the derivatives: integrand = 0 - 0 = 0. Step 5: The double integral over the unit square of 0 is 0. Step 6: Therefore, the line integral around C is 0. Result: \(\oint_C (x^2\,dx + y^2\,dy) = 0\).