New PDF release: A Textbook of Engineering Mathematics-I, 2nd Edition

By H.S. Gangwar, Dr. Prabhakar Gupta

ISBN-10: 8122424759

ISBN-13: 9788122424751

Written for the scholars of BTech I 12 months of UP Technical collage, Lucknow and different states, this booklet discusses intimately the ideas and methods in Engineering arithmetic.

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

Example text

If u = sin–1 {(x + y)/( x + ∂u ∂u + y ∂y = 1. , 2008) ∂u ∂u + y ∂y = tan u. ∂x y )}. Show that x ∂u 1 ∂u + y ∂y = tan u. 2 ∂x ∂u 1 ∂u + y ∂y = sin 2u. 2 ∂x 16. If u is a homogeneous function of degree n, show that 15. If u = tan–1 [(x2 + y2)/(x + y)], then prove that x ∂ 2u ∂ 2u ∂u . (a) x 2 + y ∂x∂y = (n – 1) ∂x ∂x ∂ 2u ∂u ∂ 2u . (b) y 2 + x ∂x∂y = (n – 1) ∂y ∂y 48 A TEXTBOOK OF ENGINEERING MATHEMATICS—I FG y IJ , show that x ∂u + y H xK ∂x LM x + y OP , prove that x 1 8 . If u = sin M MN x + y PPQ 17.

Ii) = – bf ′ (x – by) + bφ′ (x + by) ∂2z ∂ 2z 2 2 2 from (ii). Hence proved. 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.

FG y IJ , verify that ∂ f H xK ∂y∂x F yI f = tan G J H xK 2 Example 2. If f = tan–1 Sol. We have = ∂2 f . (iv) 2 Differentiating (iii) partially with respect to x, we get F GH ∂2 f x ∂ = 2 ∂x∂y ∂x x + y 2 = Example 3. If u(x + y) = x Sol. (v) F ∂u ∂u I + y , prove that G − J H ∂x ∂y K F1 − ∂u − ∂u I . GH ∂x ∂y JK 2 =4 x2 + y2 x+y bx + yga2xf − ex + y ja1f = x + 2xy − y b x + yg bx + y g bx + ygb2yg − ex + y ja1f = y + 2xy − x bx + yg bx + yg 2 = ∂u ∂y = ∂u ∂u + ∂x ∂y = b x + yg = 1− = ex 2 2 2 2 2 2 = 2 2 2 2 2 2 2 2 4xy 2 4xy ( x + y) 2 = (x − y) 2 (x + y) 2 j e bx + y g 2 ex − y j 2 bx − yg = bx + yg bx + y g 2 + 2xy − y 2 − y 2 + 2xy − x 2 2 2 ∴ 3 ∂2 f ∂2 f = .

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A Textbook of Engineering Mathematics-I, 2nd Edition by H.S. Gangwar, Dr. Prabhakar Gupta


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