GATE ME 2020 Set 2 — Question 36
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Engineering Mathematics → Vector Calculus → Directional Derivatives
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Question
The directional derivative of
f(x,y,z)=xyz at point
(−1,1,3) in the direction of vector
i^−2j^+2k^ is
Correct answer
(C) (7)/(3)
Solution
The directional derivative of
f in the direction of unit vector
u^ is given by
∇f⋅u^.
1.Calculate the gradient of f(x,y,z)=xyz: ∇f=∂x∂fi^+∂y∂fj^+∂z∂fk^=(yz)i^+(xz)j^+(xy)k^2.Evaluate the gradient at point (−1,1,3): ∇f∣(−1,1,3)=(1⋅3)i^+(−1⋅3)j^+(−1⋅1)k^=3i^−3j^−k^3.Find the unit vector in the direction of a=i^−2j^+2k^: u^=∣a∣a=12+(−2)2+22i^−2j^+2k^=1+4+4i^−2j^+2k^=3i^−2j^+2k^4.Calculate the directional derivative:
D.D.=∇f⋅u^=(3i^−3j^−k^)⋅(3i^−2j^+2k^)=33(1)+(−3)(−2)+(−1)(2)=33+6−2=37More questions on Vector Calculus
