Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007)

Table of Contents

See also Table of Figures

Tutorial

A tutorial for using this document. A must for the newcomer to this courseware.

Introduction

• A general introduction with important information for the user.
• Useful links and hints
• Author, about the author of this courseware.
• Graphing calculator, an excellent tool for graphic display.
• Grapher application, a sophisticated tool for two and three dimensional graphic display

Chapter 1 - Differentiation

1. Real Numbers

1. Irrational Numbers
• Rational numbers
• Countability
• Irrational numbers
• Exercises
2. Operations
• Basic operations on real numbers
• Integer power
• Rational power
• Binomial expansion
• Exercises
3. Sequences
• Convergence
• Divergence and Infinity
• Real-number power
• Practical rules
• Examples
• Exercises
4. Presentation
• Decimal form
• Significant figures
• Order of magnitude
• Units
• Physical dimensions
• Tips
• Exercises



2. Real Functions

1. Basics
• Definition of a function
• Graphical presentation
• Limits and Continuity
• Examples
• Exercises
2. Advanced Graphics
• Classification by shapes
• Inverse functions
• Exercises
3. Trigonometric Functions
• Definitions
• Inverse trigonometric functions
• Exercises
4. Exponent and Log
• Exponential function
• Logarithmic function
• Hyperbolic functions
• Inverse hyperbolic functions
• Exercises
5. Functions and Geometry
• Implicit function
• Parametric presentation
• Geometry and translation
• Straight line
• Parabola
• Polar coordinates and rotation
• Ellipse
• Hyperbola
• Exercises



3. Derivatives

1. Definition and Rules
• Definition of a derivative
• Rules for differentiation and derivative of x^p
• Parity of the derivative
• Exercises
2. Properties
• Sign of a derivative
• High order derivatives
• Stationary points
• Inflection points
• Mini-Maxi applications
• Infinities
• Exercises
3. Elementary Functions
• Implicit functions
• Trigonometric functions
• Log and exponent
• Hyperbolic functions
• Parametric functions
• Exercises
4. Applications
• Rule of L'Hôpital
• Combined L'Hôpital's rule
• Asymptotes
• Exercises
5. Taylor Series
• Approximations
• Series
• Operations with power series
• Binomial series
• Exercises

Chapter 2 - Integration

1. Indefinite Integrals

1. Definitions and Basics
• Differential
• Indefinite integral
• Elementary rules
• Parity
• Exercises
2. Methods
• Substitution
• By Parts
• Parametric functions
• Exercises
3. Special Methods
• Rational Functions
• Trigonometric Substitution
• Table of Integrals
• Exercises
4. Simple Differential Equations
• Introduction
• Derivative as function of the independent variable:   $\frac{{\text{d}}^{n}y}{\text{d}{x}^n}=f_n\left(x\right)$
• Separation of variables:   $\frac{\text{d}y}{\text{d}x}=f_y\left(y\right)f_x\left(x\right)$
• Non-explicit independent variable (x):   $F_1\left(y,\frac{\text{d}y}{\text{d}x},\mathrm{....},\frac{{\text{d}}^{n}y}{\text{d}{x}^n}\right)=0$
• Non-explicit dependent variable (y):   $F_2\left(x,\frac{\text{d}y}{\text{d}x},\mathrm{....},\frac{{\text{d}}^{n}y}{\text{d}{x}^n}\right)=0$
• Exercises



2. Definite Integrals

1. Fundamentals
• Area under a curve
• The definite integral
• Relation with indefinite integrals
• Few facts about definite integrals
• Learning by examples
• Exercises
2. Applications
• Enclosed area
• Mean value
• Energy conservation
• Exercises
3. Beyond Riemann
• Improper integral of the first kind
• Improper integral of the second kind
• Cauchy principal value
• Mean value and improper integrals
• Polar coordinates
• Exercises
4. Geometrical Applications
• Length of an arc
• Volume of revolution around x
• Volume of revolution around y
• Surface of revolution
• Exercises

Chapter 3 - Many Variables

1. Differentiation

1. Function of Many Variables
• Basic notions
• Domain of a function
• Level lines
• Limit and continuity
• 3D coordinate system
• Displaying functions of two variables
• Curves in space
• Physical dimensions
• Exercises
2. Partial Derivatives
• Definition
• Total differential
• Function of functions
• Implicit functions
• First approximation
• Exercises
3. Derivatives of Higher Order
• Differentiation of derivatives
• Notation
• Second approximation
• Taylor series
• Differentiation under the integral sign
• Exercises
4. Change of Variables
• Two variables
• Matrix notation
• Transformations in two dimensions
• Transformations in three dimensions
• Physical dimensions
• Legendre transformation
• Exercises
5. Stationary Points
• Overview
• Sufficient conditions
• Applications
• More classes
• Exercises
6. Constraints
• Definition
• Lagrange multipliers
• Exercises
7. Jacobian
• Two variables
• Features
• Determinants
• Many variables
• Exercises



2. Integration

1. Double Integral
• Definition
• Evaluation
• Intepretations
• Centre of mass
• Exercises
2. Multiple Integral
• Triple integral
• Applications
• Moment of inertia
• Higher multiplicity
• Exercises
3. Two Dimensional Transformation
• Substitution
• Jacobian as magnifier
• Polar coordinates
• Exercises
4. Multidimensional Transformation
• Generalization from two dimensions
• Cylindrical coordinates
• Spherical coordinates
• Simple surface integrals
• Exercises

Appendices

1. Natural Base
• Convergence of the sequence
• Consistency with the series