GATE AG 2011 • Q27 MCQ • 1 Mark

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Divergence and curl

Question Statement:

The divergence of the vector F = sin(xy)i + y cos(z)j + xz cos(z)k (i, j and k represent unit vectors along the three orthogonal axes) at x = y = 0 is

Options:

  • (A) sin(z)
  • (B) cos(z)
  • (C) −sin(z)
  • (D) −cos(z)

Official Answer Key:

B

Step-by-Step Solution & Explanation

div F = ∂/∂x[sin(xy)] + ∂/∂y[y cos z] + ∂/∂z[xz cos z] = y cos(xy) + cos(z) + xz(−sin z) + x cos z·... more carefully: ∂/∂z[xz cos z] = x cos z − xz sin z. So div F = y cos(xy) + cos z + x cos z − xz sin z. At x = y = 0: div F = 0·cos(0) + cos z + 0 − 0 = cos z, matching option (B).

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