# Guide to Essential Math

**A Review for Physics, Chemistry and Engineering Students**

**By**

- Sy Blinder, Wolfram Research, Inc., Chicago, IL, USA and University of Michigan, Ann Arbor, USA

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.

View full description### Audience

Upper-level undergraduates and graduate students in physics, chemistry and engineering

### Book information

- 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**

### Table of 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

1.5 Einstein's Velocity Addition Law

1.6 The Birthday Problem

1.7 p¼ 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

3.6 The Quadratic Formula

3.7 Imagining i

3.8 Factorials, Permutations and Combinations

3.9 The Binomial Theorem

3.10 e is for Euler

4 Trigonometry

4.1 What Use is Trigonometry?

4.2 The Pythagorean Theorem

4.3 ¼ 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

6.1 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

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