By Alan Davies; Diane Crann;

Meant for college students of arithmetic in addition to of engineering, actual technology, economics, company experiences, and machine technology, this guide includes important info and formulation for algebra, geometry, calculus, numerical tools, and records. accomplished 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 trouble-free layout.

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Solid cylinder radius r length 2a 9. Cylindrical shell radii R and r length 2a mr 2 4 mr 2 2 m(R2 + r 2 ) 4 m(R2 + r 2 ) 2 Axis mr 2 2 Perpendicular to axis through centroid m Axis m(R2 + r 2 ) 2 Perpendicular to axis through centroid m a2 r 2 + 3 4 a2 R2 + r 2 + 3 4 For entries 2, 6 and 7 the second moment of area J is given by J = IA/m where A is the area. 9 Kinematics and dynamics Kinematics Position vector r = xi + yj + zk (Cartesian coordinates) = rˆ r (Plane polar coordinates) Velocity vector ˆ (Cartesian coordinates) r˙ = xˆ ˙ ı + y˙ˆj + z˙ k ˙ ˆ = rˆ ˙ r + rθθ (Plane polar coordinates) ˆ = s˙ t (Intrinsic coordinates) Acceleration vector ˆ ¨r = x ¨ˆı + y¨ˆj + z¨k (Cartesian coordinates) 1 d ˙ θˆ (Plane polar coordinates) = (¨ r − r θ˙ 2 )ˆ r+ (r 2 θ) r dt s2 = s¨ˆ t+ n ˆ (Intrinsic coordinates) ρ For uniform motion with angular velocity ω in a circle of radius a, speed is v = aω and acceleration is v 2 /a = aω 2 directed towards the centre.

3! 4! n! x3 x5 x7 (−1)n x2n+1 sin x = x − + − + ... + + . . , all x 3! 5! 7! (2n + 1)! x2 x4 x6 (−1)n x2n cos x = 1 − + − + ... + + . . , all x 2! 4! 6! (2n)! x3 2x5 x7 π tan x = x + + + + . . , |x| < 315 2 π 3 −1 15 sin−1 x = − cos x 2 x3 3x5 5x7 (2n)! x2n+1 = x+ + + + . . + 2n + . . )2 (2n + 1) 3 5 2n+1 x x x tan−1 x = x − + + . . + (−1)n+1 + . . , |x| < 1 3 5 (2n + 1) To obtain the series for the corresponding hyperbolic functions, see the note following Osborne’s rule on page 5. Taylor’s series for a function of two variables f (a + h, b + k) = f (a, b) + {hfx (a + b) + kfy (a, b)} 1 + {h2 fxx (a, b) + 2hkfxy (a, b) + k2 fyy (a, b)} + .

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 . 12 Runge-Kutta (fourth order formulae) 1 Yi+1 = Yi + (k1 + 2k2 + 2k3 + k4 ), 6 where Y0 = y 0 k1 = hf (xi , Yi ), k2 = hf (xi + h/2, Yi + k1 /2), k3 = hf (xi + h/2, Yi + k2 /2), k4 = hf (xi + h, Yi + k3 ).

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