By Alan Davies; Diane Crann;
Meant for college students of arithmetic in addition to of engineering, actual technology, economics, company reviews, and machine technology, this instruction manual comprises important details and formulation for algebra, geometry, calculus, numerical tools, and information. finished tables of normal derivatives and integrals, including the tables of Laplace, Fourier, and Z transforms are incorporated. A spiral binding that permits the instruction manual to put flat for simple reference complements the ordinary layout.
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35 Complex Fourier series f (x) is defined for −l < x < l. e. interval (−l, l). F = ∂F1 ∂F2 ∂F3 + + ∂x ∂y ∂z ˆj ˆ ˆı k ∂/∂x ∂/∂y ∂/∂z F1 F2 F3 curlF ≡ ∇ × F = =( ∂F3 ∂F2 ∂F1 ∂F3 ˆ ∂F2 ∂F1 ˆ − )ˆı + ( − )j + ( − )k ∂y ∂z ∂z ∂x ∂x ∂y ∂φ n ˆ ∂n where n ˆ is the unit vector normal to the surface φ = constant. grad φ = Directional derivative. 1 The Laplace transform L[f (t)] = ∞ 0 e−st f (t)dt or L[x(t)] = ∞ e−st x(t)dt 0 = F (s) = x ¯(s) f (t) = L−1 [F (s)] x(t) = L−1 [¯ x(s)] Table of Laplace transforms 1 f (t) L −1 [F (s)] 1 t tn L[f (t)] F (s) 1 s 1 s2 n!
X→a g(x) x→a g (x) If f (a) = g(a) = 0, then lim Taylor’s theorem f (x) = f (a) + (x − a)f ′ (a) + where ǫn = (x − a)n (n) (x − a)2 ′′ f (a) + . . + f (a) + ǫn 2! n! (x − a)n+1 (n+1) f (c) for some c ∈ (a, x). (n + 1)! Taylor’s series for a function of one variable f (x) = f (a) + (x − a)f ′ (a) + (x − a)n (n) (x − a)2 ′′ f (a) + . . + f (a), 2! n! or, writing x = a + h, f (a + h) = f (a) + hf ′ (a) + h2 ′′ hn (n) f (a) + . . + f (a) + . . 2! n! Maclaurin series (put a = 0, h = x in Taylor’s series) f (x) = f (0) + xf ′ (0) + x2 ′′ xn (n) f (0) + .
5 Numerical solution of ordinary differential equations First order equations y ′ = f (x, y) with y(x0 ) = y0 The equation is solved successively at discrete points x1 , x2 , x3 , . . where xi+1 − xi = h. yn is the exact value y(xn ). Yn is the approximation to yn , used in the recurrence relation. Single-step methods Euler’s method Yi+1 = Yi + hf (xi , Yi ), The error is Y0 = y 0 h2 ′′ y (ξ) where xi < ξ < xi+1 . 2 Modified Euler method Yi+1 = Yi + h P f (xi , Yi ) + f (xi+1 , Yi+1 ) , 2 Y0 = y 0 P = Y + hf (x , Y ) where Yi+1 i i i h3 ′′′ The error is − y (ξ) where xi < ξ < xi+1 .