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

By H.S. Gangwar, Dr. Prabhakar Gupta

Written for the scholars of BTech I yr of UP Technical collage, Lucknow and different states, this publication discusses intimately the innovations and strategies in Engineering arithmetic.

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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. If u = f –1 1 4 1 4 1 6 1 6 2 2 0 . Verify Euler’s theorem for f = 2 1 . If u = y2 ∂ 2u 1 ∂ 2u ∂ 2u 2 2 + 2xy + y 2 = 144 tan u [tan u – 11]. 2 ∂ ∂ x y ∂y ∂x 5 ∂u ∂u +y = tan u if u = sin–1 2 ∂x ∂y 1 9 .

Cos–1 F xI GH y JK F xI GH y JK + cot–1 + cot–1 FG y IJ. H xK FG y IJ . H xK F x + y I , show that x ∂u + y ∂u = 3. , 2000) GH x + y JK ∂y ∂x ∂u 9 ∂u I F I F 8. U. (AG), 2005] 20 ∂x F xI F xI ∂z ∂z 9. If z = x y sin G y J + log x – log y, show that x +y = 6x y sin G y J . H K H K ∂x ∂y 4 4 7. U. , 2003] ∂ ∂ ∂ u u u 10. If u = x3 + y3 + z3 + 3xyz; show that x +y +z = 3u. ∂x ∂y ∂z 11. If u = cos–1 LM x − y OP , prove that x Nx + y Q ∂u ∂u + y ∂y = 0. ∂x 12. If u = log [(x2 + y2)/(x + y)], prove that x 13.

F x, y + δy − f x, y δy ∂z ∂z ∂f is also denoted by or fx similarly ∂y is denoted ∂x ∂x ∂f by ∂y or fy. The partial derivatives for higher order are calculated by successive differentiation. Thus, ∂2 f ∂ 2z = ∂x 2 ∂x 2 = fxx, ∂2z ∂2 f = = fyy ∂y 2 ∂y 2 ∂2z ∂2 f = ∂x∂y ∂x∂y = fxy, ∂2 f ∂2z = = fyx and so on. ∂y∂x ∂y∂x ∂z ∂z and : ∂ y ∂x Let z = f(x, y) represents the equation of a surface in xyzcoordinate system. Suppose APB is the curve which a plane through any point P on the surface O to the xz-plane, cuts.

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