GATE 2026 MT – Question 65
Given the scalar field $\phi(x,y,z)=x^2-yz$, the magnitude of a vector in the direction of the most rapid increase of $\phi$ at $P=(3,4,1)$ is ________. (Rounded off to two decimal places.)
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Correct answer: 7.08 to 7.48
Explanation
The direction of the most rapid increase of $\phi$ is the direction of its gradient, and the rate of increase in that direction equals $|\nabla\phi|$.
$\nabla\phi=(2x,-z,-y)$. At $P=(3,4,1)$ this is $(6,-1,-4)$.
$|\nabla\phi|=\sqrt{6^2+(-1)^2+(-4)^2}=\sqrt{36+1+16}=\sqrt{53}=7.28$.
**7.28**