GATE CE 2015 Set 1 — Question 40
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Engineering Mathematics → Vector Calculus → Directional Derivatives
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Question
The directional derivative of the field
u(x,y,z)=x2−3yz in the direction of the vector
(i^+j^−2k^) at point
(2,−1,4) is
________________.
Correct answer
-5.72 to -5.7
Solution
1.Find the gradient of u(x,y,z): ∇u=∂x∂ui^+∂y∂uj^+∂z∂uk^=2xi^−3zj^−3yk^2.Evaluate the gradient at the point (2,−1,4): ∇u∣(2,−1,4)=2(2)i^−3(4)j^−3(−1)k^=4i^−12j^+3k^3.Find the unit vector in the direction of a=i^+j^−2k^: a^=12+12+(−2)2i^+j^−2k^=6i^+j^−2k^4.The directional derivative is the dot product of the gradient and the unit vector:
Da^u=∇u⋅a^=(4i^−12j^+3k^)⋅6i^+j^−2k^ Da^u=64(1)−12(1)+3(−2)=64−12−6=6−14≈−5.715The value lies in the range -5.72 to -5.70.
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