GATE 2026 BM – Question 40
For $f(x,y)=x^2+y^2$, evaluate $\int_P^Q\nabla f\cdot d\mathbf l$ along the quarter-unit-circle path shown, from P=(1,0) to Q=(0,1).
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Correct answer: (A) 0
Explanation
The integrand is the gradient of a scalar field, so the fundamental theorem for line integrals gives f(Q)−f(P). Both endpoints have x²+y²=1, hence the difference is zero. Equivalently the gradient is radial and the circle’s tangent perpendicular to it. Thus A.