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GATE 2022 EC – Question 11

Engineering Mathematics · Vector Analysis · 1 mark · Multiple choice

Consider the two-dimensional vector field $\vec F(x,y)=x\,\hat\imath+y\,\hat\jmath$, where $\hat\imath$ and $\hat\jmath$ denote the unit vectors along the $x$-axis and the $y$-axis, respectively. A contour $C$ in the $x$-$y$ plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral

$$\oint_C \vec F(x,y)\cdot(dx\,\hat\imath+dy\,\hat\jmath)$$

is ________.

Diagram for GATE 2022 EC question 11
  1. 0
  2. 1
  3. $8+2\pi$
  4. $-1$

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Show answer and explanation

Correct answer: (A) 0

Explanation

$\vec F=\nabla\left(\frac{x^2+y^2}{2}\right)$ is a gradient field, so it is conservative and its line integral around any closed path is zero. Equivalently, $\nabla\times\vec F=0$ and Stokes' theorem gives $0$.