Guide to Essential Math : A Review for Physics, Chemistry and Engineering Students /
Sy M. Blinder.
- Second edition
- 1 online resource (285 pages)
Intro -- Half Title -- Title Page -- Copyright -- Contents -- To the Reader -- Preface to Second Edition -- Mathematical Thinking -- 1.1 The NCAA March Madness Problem -- 1.2 Gauss and the Arithmetic Series -- 1.3 The Pythagorean Theorem -- 1.4 Torus Area and Volume -- 1.5 Einstein's Velocity Addition Law -- 1.6 The Birthday Problem -- 1.7 Fibonacci Numbers and the Golden Ratio -- 1.8 sqrtpi in the Gaussian Integral -- 1.9 Function Equal to Its Derivative -- 1.10 Stirling's Approximation for N! -- 1.11 Potential and Kinetic Energies -- 1.12 Riemann Zeta Function and Prime Numbers -- 1.13 How to Solve It -- 1.13.1 Understanding the Problem -- 1.13.2 Devising a Plan -- 1.13.3 Carrying Out the Plan -- 1.13.4 Looking Back -- 1.14 A Note on Mathematical Rigor -- Numbers -- 2.1 Integers -- 2.2 Primes -- 2.3 Divisibility -- 2.4 Rational Numbers -- 2.5 Exponential Notation -- 2.6 Powers of 10 -- 2.7 Binary Number System -- 2.8 Infinity -- Algebra -- 3.1 Symbolic Variables -- 3.2 Legal and Illegal Algebraic Manipulations -- 3.3 Factor-Label Method -- 3.4 Powers and Roots -- 3.5 Logarithms -- 3.6 The Quadratic Formula -- 3.7 Imagining i -- 3.8 Factorials, Permutations and Combinations -- 3.9 The Binomial Theorem -- 3.10 e is for Euler -- Trigonometry -- 4.1 What Use is Trigonometry? -- 4.2 Geometry of Triangles -- 4.3 The Pythagorean Theorem -- 4.4 π in the Sky -- 4.5 Sine and Cosine -- 4.6 Tangent and Secant -- 4.7 Trigonometry in the Complex Plane -- 4.8 de Moivre's Theorem -- 4.9 Euler's Theorem -- 4.10 Hyperbolic Functions -- Analytic Geometry -- 5.1 Functions and Graphs -- 5.2 Linear Functions -- 5.3 Conic Sections -- 5.4 Conic Sections in Polar Coordinates -- Calculus -- 6.1 A Little Road Trip -- 6.2 A Speedboat Ride -- 6.3 Differential and Integral Calculus -- 6.4 Basic Formulas of Differential Calculus -- 6.5 More on Derivatives. 6.6 Indefinite Integrals -- 6.7 Techniques of Integration -- 6.8 Curvature, Maxima and Minima -- 6.9 The Gamma Function -- 6.10 Gaussian and Error Functions -- 6.11 Numerical Integration -- Series and 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'Hôpital's Rule -- 7.7 Fourier Series -- 7.8 Dirac Deltafunction -- 7.9 Fourier Integrals -- 7.10 Generalized Fourier Expansions -- 7.11 Asymptotic Series -- Differential Equations -- 8.1 First-Order Differential Equations -- 8.2 Numerical Solutions -- 8.3 AC Circuits -- 8.4 Second-Order Differential Equations -- 8.5 Some Examples from Physics -- 8.6 Boundary Conditions -- 8.7 Series Solutions -- 8.8 Bessel Functions -- 8.9 Second Solution -- 8.10 Eigenvalue Problems -- Matrix Algebra -- 9.1 Matrix Multiplication -- 9.2 Further Properties of Matrices -- 9.3 Determinants -- 9.4 Matrix Inverse -- 9.5 Wronskian Determinant -- 9.6 Special Matrices -- 9.7 Similarity Transformations -- 9.8 Matrix Eigenvalue Problems -- 9.9 Diagonalization of Matrices -- 9.10 Four-Vectors and Minkowski Spacetime -- Group Theory -- 10.1 Introduction -- 10.2 Symmetry Operations -- 10.3 Mathematical Theory of Groups -- 10.4 Representations of Groups -- 10.5 Group Characters -- 10.6 Group Theory in Quantum Mechanics -- 10.7 Molecular Symmetry Operations -- Multivariable Calculus -- 11.1 Partial Derivatives -- 11.2 Multiple Integration -- 11.3 Polar Coordinates -- 11.4 Cylindrical Coordinates -- 11.5 Spherical Polar Coordinates -- 11.6 Differential Expressions -- 11.7 Line Integrals -- 11.8 Green's Theorem -- Vector Analysis -- 12.1 Scalars and Vectors -- 12.2 Scalar or Dot Product -- 12.3 Vector or Cross Product -- 12.4 Triple Products of Vectors -- 12.5 Vector Velocity and Acceleration -- 12.6 Circular Motion. 12.7 Angular Momentum -- 12.8 Gradient of a Scalar Field -- 12.9 Divergence of a Vector Field -- 12.10 Curl of a Vector Field -- 12.11 Maxwell's Equations -- 12.12 Covariant Electrodynamics -- 12.13 Curvilinear Coordinates -- 12.14 Vector Identities -- Partial Differential Equations and Special Functions -- 13.1 Partial Differential Equations -- 13.2 Separation of Variables -- 13.3 Special Functions -- 13.4 Leibniz's Formula -- 13.5 Vibration of a Circular Membrane -- 13.6 Bessel Functions -- 13.7 Laplace's Equation in Spherical Coordinates -- 13.8 Legendre Polynomials -- 13.9 Spherical Harmonics -- 13.10 Spherical Bessel Functions -- 13.11 Hermite Polynomials -- 13.12 Laguerre Polynomials -- 13.13 Hypergeometric Functions -- Complex Variables -- 14.1 Analytic Functions -- 14.2 Derivative of an Analytic Function -- 14.3 Contour Integrals -- 14.4 Cauchy's Theorem -- 14.5 Cauchy's Integral Formula -- 14.6 Taylor Series -- 14.7 Laurent Expansions -- 14.8 Calculus of Residues -- 14.9 Multivalued Functions -- 14.10 Integral Representations for Special Functions -- About the Author.
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.