GATE ME 2026 Set 1 — Question 12
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Engineering Mathematics → Vector Calculus → Gradient, Divergence & Curl
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Question
Let be a scalar function. Then, is
Correct answer
(A) always perpendicular to the surface of constant φ
Solution
The gradient of a scalar function , denoted by , is a vector field. A fundamental property of the gradient is that at any point, the vector is normal (perpendicular) to the level surface (surface of constant ) that passes through that point. This is because the directional derivative of along any tangent vector to the level surface is zero, implying the gradient is orthogonal to all such tangent vectors.
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