# Guide to Essential Math

## A Review for Physics, Chemistry and Engineering Students

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.

Audience

Hardbound, 320 Pages

Published: February 2013

Imprint: Elsevier

ISBN: 978-0-12-407163-6

## Reviews

• "Blinder throws a life saver to upper-level and early graduate students of physics, chemistry, and engineering who passed the prerequisite freshman and sophomore mathematics courses but are now discovering that they did not really learn very much. All the information is still in their heads, he says, it just needs to be found, dusted off, and loosened up with some exercise."--Reference & Research Book News, October 2013

## Contents

• 1 Mathematical Thinking
1.1 The NCAA Problem
1.2 Gauss and the Arithmetic Series
1.3 The Pythagorean Theorem
1.4 Torus Area and Volume
1.6 The Birthday Problem
1.7 pÂ¼ in the Gaussian Integral
1.8 Function Equal to its Derivative
1.9 Log of N Factorial for Large N
1.10 Potential and Kinetic Energies
1.11 Lagrangian Mechanics
1.12 Riemann Zeta Function and Prime Numbers 1.13 How to Solve It
1.14 A Note on Mathematical Rigor

2. Numbers
2.1 Integers
2.2 Primes
2.3 Divisibility
2.4 Fibonacci Numbers
2.5 Rational Numbers
2.6 Exponential Notation
2.7 Powers of 10
2.8 Binary Number System
2.9 Infinity

3 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
3.7 Imagining i
3.8 Factorials, Permutations and Combinations
3.9 The Binomial Theorem
3.10 e is for Euler

4 Trigonometry
4.1 What Use is Trigonometry?
4.2 The Pythagorean Theorem
4.3 Â¼ in the Sky
4.4 Sine and Cosine
4.5 Tangent and Secant
4.6 Trigonometry in the Complex Plane
4.7 De Moivre's Theorem
4.8 Euler's Theorem
4.9 Hyperbolic Functions

5 Analytic Geometry
5.1 Functions and Graphs
5.2 Linear Functions
5.3 Conic Sections
5.4 Conic Sections in Polar Coordinates

6 Calculus
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

7 Series and Integrals
7.1 Some Elementary Series
7.2 Power Series
7.3 Convergence of Series
7.4 Taylor Series
7.5 L'H'opital's Rule
7.6 Fourier Series
7.7 Dirac Deltafunction
7.8 Fourier Integrals
7.9 Generalized Fourier Expansions
7.10 Asymptotic Series

8 Differential Equations
8.1 First-Order Differential Equations
8.2 AC Circuits
8.3 Second-Order Differential Equations
8.4 Some Examples from Physics
8.5 Boundary Conditions
8.6 Series Solutions
8.7 Bessel Functions
8.8 Second Solution

9 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 Eigenvalue Problems
9.9 Group Theory
9.10 Minkowski Spacetime

10 Multivariable Calculus
10.1 Partial Derivatives
10.2 Multiple Integration
10.3 Polar Coordinates
10.4 Cylindrical Coordinates
10.5 Spherical Polar Coordinates
10.6 Differential Expressions
10.7 Line Integrals
10.8 Green's Theorem

11 Vector Analysis
11.1 Scalars and Vectors
11.2 Scalar or Dot Product
11.3 Vector or Cross Product
11.4 Triple Products of Vectors
11.5 Vector Velocity and Acceleration
11.6 Circular Motion
11.7 Angular Momentum
11.8 Gradient of a Scalar Field
11.9 Divergence of a Vector Field
11.10 Curl of a Vector Field
11.11 Maxwell's Equations
11.12 Covariant Electrodynamics
11.13 Curvilinear Coordinates
11.14 Vector Identities

12 Special Functions
12.1 Partial Differential Equations
12.2 Separation of Variables
12.3 Special Functions
12.4 Leibniz's Formula
12.5 Vibration of a Circular Membrane
12.6 Bessel Functions
12.7 Laplace Equation in Spherical Coordinates
12.8 Legendre Polynomials
12.9 Spherical Harmonics
12.10 Spherical Bessel Functions
12.11 Hermite Polynomials
12.12 Laguerre Polynomials

13 Complex Variables
13.1 Analytic Functions
13.2 Derivative of an Analytic Function
13.3 Contour Integrals
13.4 Cauchy's Theorem
13.5 Cauchy's Integral Formula
13.6 Taylor Series
13.7 Laurent Expansions
13.8 Calculus of Residues
13.9 Multivalued Functions
13.10 Integral Representations for Special Functions