QUIZ – 3, IIT HYDERABAD
ME5010 – Mathematical Methods for Engineers
Date: 10/09/2014
Maximum Marks – 30
Time duration: 1 hours
Instructions:
i)
ii)
1.
Mobile is not allowed in the exam.
Only calculator is allowed.
a) If a vector 𝑃⃑ is defined over a volume V enclosed by an area A which surround by a boundary l , briefly define
Stokes theorem, Divergence theorem, and Helmholtz theorem.
.
b) Evaluate the integral 𝐼 = ∫𝑆(∇ × 𝐮)𝑇 . 𝐧 𝑑𝑆, using Stroke’s theorem, where, u = 𝑥 3 𝐣 − (𝑧 + 1) 𝐤 and S is
the surface 𝑧 = 4 − 4𝑥 2 − 𝑦 2 for 𝑧 ≥ 0.
2.
Marks: (2 + 5 + 10) = 7
Answer the following
(a) For a given functional 𝐹(𝑥, 𝑢(𝑥), 𝑢′(𝑥)), find the expression for 𝑑(𝐹) and 𝛿(𝐹).
(b) Find the total differentiation 𝑑(𝐹) and first variation 𝛿(𝐹) of the following functions
i) 𝐹(𝑢, 𝑢′ , 𝑢′′ ) = 𝑎 𝑢2 + 𝑏 𝑢′2 + 𝑐𝑢′′2 + 𝑑 𝑢 𝑢′
ii) 𝐹(𝑢, 𝑣, 𝑢′ , 𝑣′) = 𝑎 𝑢2 + 𝑏 𝑣 2 + 𝑐𝑢′2 + 𝑑 𝑢′𝑣′
(c) Find the minima of the following function and functional
i) 𝑓(𝑥, 𝑦) = 𝑥 2 + 𝑦 2
𝑥
2
𝑖𝑖) 𝐼 = ∫𝑥 2 √1 + 𝑢′2 𝑑𝑥, where 𝑢(𝑥1 ) = 𝑢1 , 𝑢(𝑥2 ) = 𝑢2
1
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1(a)
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