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CSS Applied Mathematics QUESTION #1303
Question 1
Green's Theorem relates the line integral around a closed curve \(C\) to the double integral over the enclosed region \(R\) by:
  • \(\oint_C(P\,dx+Q\,dy)=\iint_R\left(\dfrac{\partial Q}{\partial y}-\dfrac{\partial P}{\partial x}\right)dA\)
  • \(\oint_C(P\,dx+Q\,dy)=\iint_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)dA\)✔️
  • \(\oint_C(P\,dx+Q\,dy)=\iint_R(P+Q)\,dA\)
  • \(\oint_C\vec{F}\cdot d\vec{r}=\iint_R|\vec{F}|\,dA\)
Correct Answer Explanation
Green's Theorem: \(\oint_C(P\,dx+Q\,dy)=\iint_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)dA\), where \(C\) is traversed counter-clockwise. This converts a 1D line integral into a 2D area integral over the region enclosed by \(C\).