## Key Features

- Highly accessible presentation of fundamental mathematical techniques needed in science and engineering courses

- Use of proven pedagogical techniques develolped during the author’s 40 years of teaching experience

- illustrations and links to reference material on World-Wide-Web

- Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables

## Description

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) which 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. By the author's design, no problems are included in the text, to allow the students to focus on their science course assignments.

Readership

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

Guide to Essential Math, 1st Edition

To the Student

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