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CSS Applied Mathematics QUESTION #1326
Question 1
Green's theorem applied to the vector field \(\vec{F}=\left(\dfrac{1}{2}xy^4,\dfrac{1}{2}x^4y\right)\) over the rectangle \([0,3]\times[0,2]\) gives:
  • \(\iint_R 2x^3y-2xy^3\,dA\)✔️
  • \(\iint_R x^3y+xy^3\,dA\)
  • \(\iint_R xy^3-x^3y\,dA\)
  • \(\iint_R 2(x^3+y^3)\,dA\)
Correct Answer Explanation
By Green's theorem: \(\oint_C\vec{F}\cdot d\vec{r}=\iint_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)dA\). With \(P=\dfrac{1}{2}xy^4\) and \(Q=\dfrac{1}{2}x^4y\): the integrand is \(2x^3y-2xy^3\). Integrating over the rectangle gives the circulation.