GATE AG 2016 • Q28 MCQ • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Stokes, Gauss and Green's theorems

Question Statement:

Let I, J, K be unit vectors along mutually perpendicular x, y, z axes. If F = fI + gJ + hK is a continuously differentiable vector point function, then curl F is

Options:

  • (A) I(∂g/∂z−∂h/∂y) − J(∂f/∂z−∂h/∂x) + K(∂f/∂y−∂g/∂x)
  • (B) I(∂h/∂y−∂g/∂z) − J(∂h/∂x−∂f/∂z) + K(∂g/∂x−∂f/∂y)
  • (C) I(∂h/∂y−∂g/∂z) + J(∂h/∂x−∂f/∂z) + K(∂g/∂x−∂f/∂y)
  • (D) I(∂g/∂z−∂h/∂y) + J(∂f/∂z−∂h/∂x) + K(∂f/∂y−∂g/∂x)

Official Answer Key:

B

Step-by-Step Solution & Explanation

curl F = ∇×F is the determinant of I,J,K rows with ∂/∂x,∂/∂y,∂/∂z and f,g,h. Expanding: curl F = I(∂h/∂y−∂g/∂z) − J(∂h/∂x−∂f/∂z) + K(∂g/∂x−∂f/∂y), matching option (B) exactly (alternating +,−,+ sign pattern of a 3×3 determinant expansion). Options (A)/(D) swap f,g,h terms; (C) has the wrong sign on the J term (should be minus, i.e. cofactor sign). Section 2 — Farm Machinery

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