By H.S. Gangwar, Dr. Prabhakar Gupta

Written for the scholars of BTech I 12 months of UP Technical college, Lucknow and different states, this e-book discusses intimately the thoughts and strategies in Engineering arithmetic.

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

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If V = loge sin x Sol. We have Hence proved. R| π 2x 2 + y 2 + zx 12 U| j |V , find the value of |S e || 2ex 2 + xy + 2yz + z 2 j 13 || T W ∂V ∂V ∂V + y ∂y + z , when x = 0, y = 1, z = 2. (i) 1 1 , e2V = 2 2 π 4 Putting all these values in equation (i), we get ⇒ x ∂V ∂V ∂V +y +z = ∂x ∂y ∂z x ∂V ∂V ∂V +y +z = ∂x ∂y ∂z Example 11. If u = x3y2 sin–1 x 1− 1 2 1 π × × = 1 3 4 2 π 12 12 π × 12 12 . FG y IJ , show that H xK ∂u ∂ 2u ∂u ∂ 2u ∂ 2u 2 + y ∂y = 5u and x2 + 2xy + y = 20u. ∂y2 ∂x ∂x2 ∂x∂y u = x3y2 sin–1 Sol.

If u = 3x cot G J + 16y cos G J then prove that H xK H yK x2 or 2 ∂ 2u ∂ 2u 2∂ u = 2 + 2xy ∂x∂y + y ∂x ∂y 2 4 x –1 2 4 –1 2 ∂ 2u ∂ 2u 2∂ u xy 2 + + y = 12u. ∂x∂y ∂x2 ∂y 2 Sol. The given function is homogeneous function of degree 4. r. r. (iii) 47 DIFFERENTIAL CALCULUS-I Adding (ii) and (iii), we get or F GH I JK x2 2 ∂u ∂u ∂ 2u ∂ 2u 2∂ u +y = 3 x xy y 2 + + 2 2 ∂x ∂y ∂x∂y ∂x ∂y x2 2 ∂ 2u ∂ 2u 2∂ u = 3 × 4u = 12u. Hence proved. 5 Verify Euler’s theorem 1. 2 2 x −y . 1 4 4. x 3 y − 3 tan–1 FG y IJ.

H ∂x ∂y K x + y ∂ F 1 I ∂ F 1 I G J + 2 =2 ∂y GH x + y JK ∂x H x + y K = ∂u ∂u + ∂x ∂y b g j j 2 ... (iii) 2 2 =– 2 b x + yg 2 Example 13. If u = exyz, show that ∂ 3u = (1 + 3xyz + x2y2z2)exyz. ∂x∂y∂z – 2 b x + yg 2 =– 4 bx + yg 2 . Hence proved. xy ∂z u = exyz ∴ Sol. We have = (2xyz + 1) exyz + (x2yz + x) exyz·yz = (1 + 3xyz + x2y2z2) exyz. Hence proved. Example 14. If u = log r, where r2 = (x – a)2 + (y – b)2 + (z – c)2, show that ∂ 2u ∂ 2u ∂ 2u 1 + + = 2. 2 2 2 ∂x ∂z ∂y r Sol. (i) Differentiating partially with respect to x, we get 2r Similarly, Now, ∴ or ∴ or Similarly, ∴ FG x − a IJ .

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