Vector Calculus MCQ [Free PDF] – 5 Objective Questions with Answers

For A = x i + y j + z k, each component depends only on its corresponding coordinate. All cross-partial derivatives in ∇×A are zero, so the field is curl-free (irrotational).

The surface z=0 with 30°<φ<60° is a sector-shaped region of the xy-plane between two rays. Because neither φ=0° nor φ=90° is included, it contains neither the positive x-axis nor positive y-axis, while the radial extent is not bounded by the angular condition. The statement combination is option D.

A solenoidal vector field is divergence-free: ∇·A = 0. Physically, it has no net source or sink within the region.

Use Gauss' divergence theorem. For F = 3x i + 2y j, ∇·F = 3 + 2 = 5. The sphere x²+y²+z²=9 has radius 3 and volume 36π, so the outward flux is 5×36π = 180π.

The three basic vector differential operators have distinct outputs: gradient maps scalar→vector, divergence maps vector→scalar, and curl maps vector→vector. Remembering input/output type is often enough to eliminate distractors. Key relation: grad φ=∇φ; div A=∇·A; curl A=∇×A; ∮_S A·dS = ∫_V (∇·A)dV.

Exam focus: Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector. Common trap: Calling gradient a scalar because the original field is scalar.