The value of the directional derivative of the function \phi(x, y, z)=x y^{2}+y z^{2}+z x^{2} at the point (2,-1,1) in the direction of the vector p=i+2 j+2 k is

GATE Electrical Engineering (EE) 2018 Solved Paper

Question : 15 of 65

Marks: +1, -0

The value of the directional derivative of the function

φ(x, y, z)=x y^{2}+y z^{2}+z x^{2}

at the point

(2,-1,1)

in the direction of the vector

p=i+2 j+2 k

Solution:

\∅=x y^{2}+y z^{2}+z x^{2}

∇ ∅=\;{∂\∅}/{∂ x} {ı}↖{∧}+\;{∂\∅}/{∂ y} {j}↖{∧}+\;{∂\∅}/{∂z} {k}↖{∧}=(y^{2}+2 x z) {ı}↖{∧}+(2 x y+z^{2}) {j}↖{∧}+(x^{2}+2 y z) {k}↖{∧}

∇\∅_{(2,-1,1)}=5 {ı}↖{∧}-3 {j}↖{∧}+2 {k}↖{∧}

The directional derivative of

φ(x, y, z)

at in

(2,-1,1)

the direction of

\ov{P}

∇ϕ_{at,p}{\ov{P}}/{|P↖{-}|}

=(5 \ov{i}-3 \ov{j}+2 \ov{k}) ⋅(\;{\ov{i}+2 \ov{j}+2 \ov{k}}/{3})

=\;{5-6+4}/{3}=1

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