GATE 2016 EC (EC2) – Question 37

Engineering Mathematics · Vector Analysis · 2 marks · Numerical answer

C is the circle $x^2+y^2=1$ oriented anticlockwise. The value of $\oint_C(xy^2\,dx+x^2y\,dy)$ is ________.

Show answer and explanation

Correct answer: -0.03 to 0.03

Explanation

Let $P=xy^2$ and $Q=x^2y$. Green's theorem turns the integral into the disk integral of $\partial Q/\partial x-\partial P/\partial y=2xy-2xy=0$.

Hence the circulation is **0**. Equivalently, the integrand is the exact differential $d(x^2y^2/2)$, whose integral around any closed curve vanishes.

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