GATE 2021 CE (CE2) – Question 13

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple choice

Find the unit normal to $x^2+y^2+z^2-48=0$ at (4,4,4).

$(1,1,1)/\sqrt2$
$(1,1,1)/\sqrt3$
$(2,2,2)/\sqrt2$
$(1,1,1)/\sqrt5$

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Correct answer: (B) $(1,1,1)/\sqrt3$

Explanation

The gradient is $(2x,2y,2z)$, giving (8,8,8) at the point. Its magnitude is 8√3, so normalising gives **(1,1,1)/√3 (B)**. A surface also has the opposite unit normal, but the listed outward normal is B.