For vec F =left(dfrac 1 2 xy^4,;dfrac 1 2 x^4yright), the quantity dfrac partial Q partial x -dfrac partial P partial y needed for Green's Theorem equals:

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SSC Applied Mathematics QUESTION #1304

Question 1

For \(\vec{F}=\left(\dfrac{1}{2}xy^4,\;\dfrac{1}{2}x^4y\right)\), the quantity \(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\) needed for Green's Theorem equals:

\(2x^3y-2xy^3\)✔️
\(x^3y-xy^3\)
\(2x^3y+2xy^3\)
\(x^3y+xy^3\)

Correct Answer Explanation

Here \(P=\dfrac{1}{2}xy^4\) and \(Q=\dfrac{1}{2}x^4y\). \(\dfrac{\partial Q}{\partial x}=\dfrac{1}{2}\cdot4x^3y=2x^3y\). \(\dfrac{\partial P}{\partial y}=\dfrac{1}{2}\cdot4xy^3=2xy^3\). So \(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}=2x^3y-2xy^3\).

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