GATE 2019 EE – Question 28

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 1 mark · Numerical answer

If $f=2x^3+3y^2+4z$, the value of line integral $\int_C\text{grad}f\cdot d\mathbf r$ evaluated over contour $C$ formed by the segments $(-3,-3,2)\to(2,-3,2)\to(2,6,2)\to(2,6,-1)$ is ________.

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Correct answer: 139

Explanation

The field is conservative, so the integral depends only on the end points: $f(2,6,-1)-f(-3,-3,2)=(16+108-4)-(-54+27+8)=120-(-19)=139$.