# Guide to Essential Math: A Review for Physics, Chemistry and Engineering Students (Elsevier Insights)

Language: English

Pages: 275

ISBN: B00BQWUC0Y

Format: PDF / Kindle (mobi) / ePub

This book reminds students in junior, senior and graduate level courses in physics, chemistry and engineering of the math they may have forgotten (or learned imperfectly) that is needed to succeed in science courses. The focus is on math actually used in physics, chemistry, and engineering, and the approach to mathematics begins with 12 examples of increasing complexity, designed to hone the student's ability to think in mathematical terms and to apply quantitative methods to scientific problems. Detailed illustrations and links to reference material online help further comprehension. The second edition features new problems and illustrations and features expanded chapters on matrix algebra and differential equations.

- Use of proven pedagogical techniques developed during the author’s 40 years of teaching experience
- New practice problems and exercises to enhance comprehension
- Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables

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Integrals 7.1 Some Elementary Series 7.2 Power Series 7.3 Convergence of Series 7.4 Taylor Series 7.5 Bernoulli and Euler Numbers 7.6 L´Hôspital’s Rule 7.7 Fourier Series 7.8 Dirac Deltafunction 79 79 81 83 86 88 90 91 93 94 96 99 101 101 102 104 106 108 110 111 116 Contents vii 7.9 Fourier Integrals 7.10 Generalized Fourier Expansions 7.11 Asymptotic Series 119 121 121 8 Differential Equations 8.1 First-Order Differential Equations

formulas; in all cases a constant is to be added to the right-hand side: ex d x = ex , (6.50) ecx , c 1 dx dx = = ln x, x x ecx d x = (6.51) (6.52) sin xd x = − cos x, (6.53) cos xd x = sin x, (6.54) dx = arctan x, (6.55) 1 + x2 dx = arcsin x. (6.56) √ 1 − x2 For all the above integrals, the constant drops out if we put in limits of integration, for example b a dx = ln x x b a = ln b . a (6.57) Calculus 91 You can find many Tables of Integrals which list hundreds of other

a 1 1 1 x(y0 + y1 ) + x(y1 + f y2 ) + x(y2 + y3 ) 2 2 2 1 1 1 x(y3 + y4 ) = x y0 + y1 + y2 + y3 + y4 . + 2 2 2 (6.98) More generally, approximating the integral using n trapezoids, b a f (x)d x ≈ x 1 1 y0 + y1 + y2 + y3 + · · · + yn . 2 2 (6.99) 100 Guide to Essential Math 2e Figure 6.11 Trapezoidal rule for numerical integration. Clearly, the result will become more accurate as n is increased and x is decreased. A somewhat better numerical approximation can be obtained using

= 1 + x + 2x 2 + 3x 3 + 5x 4 + 8x 5 + 13x 6 + 21x 7 + 34x 8 + · · · . (7.5) Therefore f (x) serves as a generating function for the Fibonacci numbers: ∞ Fn x n . f (x) = (7.6) n=0 An infinite geometric series inspired by one of Zeno’s paradoxes is 1 1 1 1 + + + + · · · = 1. 2 4 8 16 (7.7) Zeno’s paradox of motion claims that if you shoot an arrow, it can never reach its target. First it has to travel half way, then half way again—meaning 1/4 of the distance—then continue with an inﬁnite

Cartesian coordinates, the transformation (x, y, z) → (−x, −y, −z). Irreducible representations under this symmetry operation are classified as g (even) or u (odd). Group Theory 179 Figure 10.3 Pictorial representation of the operation S4. 8. Sn : An improper rotation or rotation-reflection axis. Clockwise rotation through an angle of 2!/n radians followed by a reflection in the plane perpendicular to the axis of rotation. Also known as an alternating axis of symmetry. Note that S1 is