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GATE 2026 ME – Question 12

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 1 mark · Multiple choice

Let $\varphi$ be a scalar function. Then, $\nabla\varphi$ is

  1. always perpendicular to the surface of constant $\varphi$
  2. always parallel to the surface of constant $\varphi$
  3. the minimum rate of change of scalar $\varphi$
  4. always zero

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Show answer and explanation

Correct answer: (A) always perpendicular to the surface of constant $\varphi$

Explanation

The gradient points in the direction in which $\varphi$ changes fastest, which is normal to the surface where $\varphi$ is constant. It gives the maximum rate of change, not the minimum, and it is not zero in general.