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.
Audience
Upper-level undergraduates and graduate students in physics, chemistry and engineering
Contents 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 p? 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 – 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
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