GATE 2024 BM – Question 50
For $\mathbf F=(x+y)\hat i+(x+y)\hat j$, evaluate $\oint\mathbf F\cdot d\mathbf r$ counterclockwise around the unit square (0,0),(1,0),(1,1),(0,1). Give an integer.
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Show answer and explanation
Correct answer: 0 to 0
Explanation
By Green’s theorem the circulation is the area integral of ∂Q/∂x−∂P/∂y. Both derivatives equal one, so the integrand is zero everywhere. Therefore the closed-path integral is zero. Equivalently F is the gradient of (x+y)²/2.