GATE Calculus: Vector Calculus – Previous Year Questions

10 GATE previous year questions on Vector Calculus (Calculus, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q22 (1 mark, Multiple choice) – A twice continuously differentiable f:ℝ³→ℝ satisfies div(∇f)=6. On the unit sphere S let n be the unit outward normal. Find ∫ S∇f·n dS.
2. GATE 2021 MA Q26 (1 mark, Numerical answer) – Γ is the counterclockwise boundary of the square with vertices (0,0),(2,0),(2,2),(0,2). Evaluate ∮ Γ(1−y²)dx+xdy.
3. GATE 2022 MA Q45 (2 marks, Multiple choice) – Find work of F=(x+y)i−(x²+y²)j along the upper unit semicircle from (1,0) to (−1,0).
4. GATE 2022 MA Q59 (2 marks, Numerical answer) – Find the surface area of z=x²+y² between planes z=0 and z=1, to one decimal.
5. GATE 2023 MA Q31 (1 mark, Numerical answer) – The portion of z=2x+2y−100 inside x²+y²=1 has surface area απ. Find α.
6. GATE 2023 MA Q55 (2 marks, Numerical answer) – C is x²+y²=4,z=2 oriented counterclockwise viewed from above. If C( x+ex)dx+4x\,dy+ez2z\,dz =, find α.
7. GATE 2024 MA Q35 (1 mark, Numerical answer) – Outward flux of F=(x³,y³,z³) through the unit sphere is απ. Find α to two decimals.
8. GATE 2024 MA Q61 (2 marks, Numerical answer) – A C¹ path runs from (0,2) to (3,0). For F(x,y)=(1−2y,1−2x), find ∫F·dr to two decimals.
9. GATE 2025 MA Q32 (1 mark, Numerical answer) – F=(y−z)i+(z−x)j+(x−y)k. S is the outward upper hemisphere x²+y²+(z−1)²=9, 1≤z≤4. Find -1 S( F) n\,dS (integer).
10. GATE 2026 MA Q56 (2 marks, Numerical answer) – = S F n\,dS, with F=(2x+3z) i+(xz-y) j+(y2+2z) k. S is the radius-9 sphere centered at (3,−1,2), with outward normal. Find α/(36π) (integer).