Advanced Engineering Mathematics by H.K. Dass

By H.K. Dass

Offers with partial differentiation, a number of integrals, functionality of a posh variable, specified services, laplace transformation, complicated numbers, and statistics.

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U v dx du dv y u u 2 3 2 2 find , . 11. If u = x cos , x  3r  2 s, y  4 r  2 s , z  2r  3s z r s 10. If z = f (x, y) where x = uv, y = u y 4x y 4x y r y u y 6 x s2 y 6 xys y  6 r cos  sin  2 sin ,  2 cos  sin  2 sin r z z z z s z z z z z z (x, y) where x = eu cos v, y = eu sin v. Prove that 2 2 2   f   f    2u   f    e          u  y   v    Ans. 12. If z = f  f     x  2 cos  sin  ,y  and z  f ( x, y ) , then show that u u 13. If x = 2 z  x2  2 z  y2 1 14.

4) Created with Print2PDF. (7) [From (1)] xy x y On adding (6) and (7), we have 2 2 z z z 2  z  y x y 2 2  x  y  x  y = n(n – 1)u + n (n + 1) v + nu – nv x y = nu (n – 1 + 1) + nv (n + 1 – 1) = n2u + n2v = n2 (u + v) = n2z II. Deduction: Prove that x2 2 z  2 xy x2  2u x 2  2 xy Proved.  2u  2u  y 2 2 = g(u) [g(u) – 1] (Nagpur University, Winter 2003) xy y f (u ) g(u) = n f (u ) Proof. By Euler’s deduction formula I u u f (u ) x y n. t. t. ‘y’, we have x  2u  2u u [ g (u )  1] = 2 y x y y Multiplying (2) by x, (3) by y and adding, we get y x2  2u x 2  2 xy  2u  y.

2 ( x 2  y 2  z 2 )2 1 x 1  2 x  = 0 r r r 2 r 12. If x = r cos , y = r sin , prove that Hence deduce that (i) r x  1 x  , r.  . x r x r  2x cos   2u 2  (ii)  2 x 2 13. If v = (x2 – y2). f (xy), prove that   2v x 2 2 y 2  = 0  2v y 2 (c) 2r x 2  2r y 2  2 2  r   1  r        r  x   y   = (x4 – y4) f (xy) u v u v x 2  y 2   14. If ux + vy = 0 and x  y = 1, show that x y y 2  x 2 15. If z = xy + yx, verify that 2 z 2z  x y y x 16.

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