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 e-book discusses intimately the innovations and methods in Engineering arithmetic.

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T. (iii) Multiplying (ii) by x and (iii) by y and adding, we have x2 ∂2 f ∂x 2 + 2xy ∂2 f ∂2 f ∂f ∂f + y2 +y =n 2 + x ∂x∂y xy ∂y ∂x F x ∂f + y ∂t I GH ∂x ∂y JK 37 DIFFERENTIAL CALCULUS-I ⇒ x 2 ∂2 f ∂x 2 ∂2 f ∂2 f 2 + 2xy +y ∂x∂y ∂y 2 = n2f − nf = n (n – 1) f Example 1. Verify Euler’s theorem for the function f (x, y) = ax2 + 2hxy + by2. Sol. Here the given function f (x, y) is homogeneous of degree n = 2. (i) ∂y ∂x Now, we are to prove equation (i) as follows: ∂f ∂f = 2 ax + 2 hy, ∂y = 2hx + 2by ∂x ∴ x ∂f ∂f + y ∂y ∂x = 2ax2 + 2hxy + 2hxy + 2by2 = 2(ax2 + 2 hxy + by2) = 2f ⇒ x ∂f ∂f +y ∂x ∂y = 2f, which proves equation (i).

T. y is denoted by ∂y and is defined as ∂z ∂y = Lim δy→0 Notation: The partial derivative b g b g. 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.

H 2K Ans. n is odd, yn (0) = (n – 2)2 (n – 4)2 .... 52 . 32 . 1 n is even, yn (0) = 0. 6 . Find yn (0) when y = sin (m sin h–1 x). a fFGH Ans. n is odd, yn (0) = –1 n−1 2 IJ K {an − 2f 2 + m2 } {an − 4f 2 } + m 2 .... (12 + m2) m · n is even, yn = (0) = 0. LM { N 7 . If y = log x + 1 + x 2 }OPQ 2 , show that yn + 2 (0) = – n2 yn (0) hence find yn (0). a f Ans. n is odd, yn (0), = 0 n is even yn(0) = −1 n− 2 2 (n – 2)2 (n – 4)2 .... 2. PARTIAL DIFFERENTIATION Introduction Real world can be described in mathematical terms using parametric equations and functions such as trigonometric functions which describe cyclic, repetitive activity; exponential, logrithmic and logistic functions which describe growth and decay and polynomial functions which approximate these and most other functions.

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