By H.S. Gangwar, Dr. Prabhakar Gupta

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**Extra resources for A Textbook of Engineering Mathematics: v. 1**

**Sample text**

B x + yg . 1 4 1 + y4 I Fx JK GH 1 5 1 I JK + y5 . xy 3. cos–1 6. 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.

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. If u = sin–1 {(x2 + y2)/(x + y)}, show that x 14. 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 .

Ii) x2 + y2 Thus, from (i) and (ii), we get ∂2 f ∂2 f = . Hence proved. ∂x∂y ∂y∂x Example 6. If V = (x2 + y2 + z2)–1/2, show that ∂ 2V ∂ 2V ∂ 2V + + = 0. 2 ∂y ∂x 2 ∂z 2 Sol. Given V = (x2 + y2 + z2)–1/2. (i) 1 (x2 + y2 + z2)–3/2 2x = – x (x2 + y2 + z2)–3/2 2 LM RS 3 ex NT 2 = – x – 2 + y2 + z2 j –5/2 UV e W ⋅ 2x + x 2 + y 2 + z 2 = 3x2 (x2 + y2 + z2)–5/2 – (x2 + y2 + z2)–3/2 = (x2 + y2 + z2)–5/2 [3x2 – (x2 + y2 + z2)] or and ∂ 2V = (x2 + y2 + z2)–5/2 (2x2 – y2 – z2) ∂x 2 Similarly from (i), we can find j –3/2.