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GATE 2026 AE – Question 11

Engineering Mathematics · Calculus: Line, surface and volume integrals, theorems of Stokes, Gauss and Green · 1 mark · Multiple choice

Consider the contour $C$ shown in the figure below. For the vector $\vec F = (x + 2y)\hat e_x + (2x + 4y)\hat e_y$, the integral $\oint_C \vec F \cdot d\vec l = $ ________. Here $d\vec l$ represents an infinitesimal length along the contour $C$.

A closed contour with corners O (0,0), P (1,1), Q (0,2) and R (-1,1), traversed O to P to Q to R and back to O.
  1. 0
  2. 2
  3. 4
  4. 6

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Correct answer: (A) 0

Explanation

By Stokes' theorem the line integral equals the surface integral of the curl. Here $(\nabla \times \vec F)_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 2 - 2 = 0$, so $\vec F$ is conservative and the integral around any closed contour is 0.