By H.S. Gangwar, Dr. Prabhakar Gupta

Written for the scholars of BTech I yr of UP Technical collage, Lucknow and different states, this e-book discusses intimately the recommendations and methods in Engineering arithmetic.

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Extra info for A Textbook of Engineering Mathematics-I, 2nd Edition

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If u = tan G x J , show that x H K ∂x2 22. If F x +y I GH x + y JK . e x+ y 2 1 3 1 3 2 2 tanu ∂ 2u ∂ 2u 2 ∂ u = + 2xy + y 2 2 12 ∂x∂y ∂x ∂y 2 –1 2 ∂f ∂f log x − log y 1 + , show that x + y ∂y + 2f (x, y) = 0. 2 2 ∂x xy x x +y ∂u ∂u 2 5 . If u = sec–1 {(x3 + y3) / (x + y)}, show that x + y ∂y = 2 cot u. ∂x 2 4 . (i) where x = φ (t) and y = ψ (t), then z can be expressed as a function of t alone by substituting the values of x and y in terms of t from the last two equations in equation (i). And we can find the ordinary differential coefficient dz , which is called total differential dt coefficient of z with respect to t.

2 = b f ″ (x – by) + b φ′′ (x + by) = b ∂y ∂x 2 ′ and Example 17. If u (x, y, z) = log (tan x + tan y + tan z). Prove that sin 2x ∂u ∂u ∂u + sin 2y ∂y + sin 2z = 2. ∂x ∂z Sol. , 2006) 33 DIFFERENTIAL CALCULUS-I ∂u ∂y = sec 2 y tan x + tan y + tan z sec 2 z ∂u = tan x + tan y + tan z ∂z ∂u ∂u ∂u ∴ sin 2x + sin 2y ∂y + sin 2z ∂x ∂z = = sin 2x sec 2 x + sin 2y sec 2 y + sin 2z sec 2 z tan x + tan y + tan z b 2 tan x + tan y + tan z btan x + tan y + tan zg g = 2. Hence proved. 4 ∂ 3u 2 2 2 1. Find ∂x∂y∂z if u = ex +y +z .

If u (x, y, z) = log (tan x + tan y + tan z). Prove that sin 2x ∂u ∂u ∂u + sin 2y ∂y + sin 2z = 2. ∂x ∂z Sol. , 2006) 33 DIFFERENTIAL CALCULUS-I ∂u ∂y = sec 2 y tan x + tan y + tan z sec 2 z ∂u = tan x + tan y + tan z ∂z ∂u ∂u ∂u ∴ sin 2x + sin 2y ∂y + sin 2z ∂x ∂z = = sin 2x sec 2 x + sin 2y sec 2 y + sin 2z sec 2 z tan x + tan y + tan z b 2 tan x + tan y + tan z btan x + tan y + tan zg g = 2. Hence proved. 4 ∂ 3u 2 2 2 1. Find ∂x∂y∂z if u = ex +y +z . Ans. 8xyzu 2. Find the first order derivatives of LMAns.

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