GATE 2016 ME (ME2) – Question 36
For $\nabla\varphi=yz\hat i+xz\hat j+xy\hat k$, evaluate $\int_C\nabla\varphi\cdot d\mathbf r$, where $\mathbf r=x\hat i+y\hat j+z\hat k$ and C has $x=t,y=t^2,z=3t^2$, $1\le t\le3$.
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Correct answer: 725.9 to 726.1
Explanation
The given gradient corresponds to potential φ=xyz+C. Along the path, xyz=t·t²·3t²=3t⁵. By the gradient theorem the integral is endpoint potential difference3·3⁵−3·1⁵=729−3=**726**. Direct integration of the gradient along the curve gives the same value.