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GATE Engineering Mathematics: Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem – Previous Year Questions

12 GATE previous year questions on Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem (Engineering Mathematics, Electrical Engineering) with answers and explanations, from every paper.

  1. GATE 2015 EE Q14 (1 mark, Multiple choice) – Consider a function f = 1r2r, where r is the distance from the origin and r is the unit vector in the radial direction. The divergence of this…
  2. GATE 2018 EE Q52 (2 marks, Numerical answer) – As shown in the figure, C is the arc from the point (3,0) to the point (0,3) on the circle x2+y2=9. The value of the integral…
  3. GATE 2019 EE Q28 (1 mark, Numerical answer) – If f=2x3+3y2+4z, the value of line integral Cgradf d r evaluated over contour C formed by the segments (-3,-3,2)(2,-3,2)(2,6,2)(2,6,-1) is .
  4. GATE 2019 EE Q49 (2 marks, Numerical answer) – If A=2x i+3y j+4z k and u=x2+y2+z2, then div(u A) at (1,1,1) is .
  5. GATE 2020 EE Q37 (2 marks, Multiple choice) – The vector function expressed by F= a x(5y-k 1z)+ a y(3z+k 2x)+ a z(k 3y-4x) represents a conservative field, where a x, a y, a z are unit vectors…
  6. GATE 2020 EE Q54 (2 marks, Numerical answer) – Let a x and a y be unit vectors along x and y directions, respectively. A vector function is given by F= a xy- a yx The line integral of the above…
  7. GATE 2021 EE Q15 (1 mark, Multiple choice) – Which one of the following vector functions represents a magnetic field B? ( x, y, and z are unit vectors along x-axis, y-axis, and z-axis,…
  8. GATE 2022 EE Q51 (2 marks, Multiple choice) – Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which one of the…
  9. GATE 2022 EE Q52 (2 marks, Multiple choice) – Let E(x,y,z)=2x+5y+3z k. The value of V( E)\,dV, where V is the volume enclosed by the unit cube defined by 0 x1,\ 0 y1,\ and\ 0 z1, is
  10. GATE 2023 EE Q62 (2 marks, Numerical answer) – The closed curve shown in the figure is described by r=1+, where r=x2+y2; x=r, y=r. The magnitude of the line integral of the vector field F=-y i+x j…
  11. GATE 2025 EE Q51 (2 marks, Multiple select) – Let a R be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of n, the divergence of the radial vector…
  12. GATE 2026 EE Q32 (1 mark, Numerical answer) – Given that F(x,y,z)=(y)\, x+(x)\, y+5\, z, the integral S F(x,y,z) d s over the unit sphere S centered at the origin evaluates to . (Round off to one…