Engineering Math

Introduction
1ptSome Prerequisite Topics
Sets And Set Notation
Well Ordering And Induction
The Complex Numbers
Polar Form Of Complex Numbers
Roots Of Complex Numbers
The Quadratic Formula
The Complex Exponential
Dividing Polynomials
The Fundamental Theorem Of Algebra
Exercises
Integrals, Functions of One Variable
Properties of the Integral
Uniform Convergence of Continuous Functions
Uniform Convergence And The Integral
Some Important Improper Integrals
Gamma Function
Laplace Transforms
I 1ptLinear Algebra And Multivariable Calculus
Fundamentals
Fn
Algebra in Rn
Geometric Meaning Of Vector Addition In R3
Lines
Distance in Rn
Geometric Meaning Of Scalar Multiplication In R3
Exercises
Physical Vectors
Exercises
Vector Products
The Dot Product
The Geometric Significance Of The Dot Product
The Angle Between Two Vectors
Work And Projections
The Dot Product And Distance In Cn
Exercises
The Cross Product
The Box Product
Proof of the distributive law
Torque
Center Of Mass
Angular Velocity
Vector Identities And Notation
Planes
Exercises
Systems Of Equations
Systems Of Equations, Algebraic Procedures
Elementary Operations
Gauss Elimination
Balancing Chemical Reactions
Dimensionless Variables
MATLAB And Row Reduced Echelon Form
Exercises
Matrices
Addition And Scalar Multiplication Of Matrices
Multiplication of Matrices
Linear Transformations and Matrices
Multiplication of Matrices
The Transpose
Some Examples of Linear Functions on Rn
Rotations in R2
Projections
Rotations About A Particular Vector
The Inverse of a Matrix
The Identity And Inverses
Finding The Inverse Of A Matrix
MATLAB And Matrix Arithmetic
Exercises
Subspaces Spans and Bases
Subspaces
Exercises
Eigenvalues and Eigenvectors
Definition of Eigenvalues
An Introduction to Determinants
Cofactors and 22 Determinants
The Determinant of a Triangular Matrix
Properties of Determinants
Finding Determinants Using Row Operations
Applications
A Formula For The Inverse
Finding Eigenvalues Using Determinants
Matrices and The Inner Product
Eigenvalues and Eigenvectors
Using Matlab
Distance and Unitary Matrices
Schur's Theorem
Diagonalization
Approximations
Fredholm Alternative
Least Squares
Regression lines
Identifying the Closest Point
Using MATLAB
The Singular Value Decomposition
Exercises
Vector Valued Functions
Vector Valued Functions
Vector Fields
Exercises
Continuous Functions
Sufficient Conditions For Continuity
Limits Of A Function
Properties Of Continuous Functions
Exercises
Open And Closed Sets
Exercises
Some Fundamentals
Combinations Of Continuous Functions
The Nested Interval Lemma
Convergent Sequences, Sequential Compactness
Continuity And The Limit Of A Sequence
The Extreme Value Theorem And Uniform Continuity
Convergence of Functions
Root Test
Convergence of Sums
Connected Sets
Exercises
Vector Valued Functions Of One Variable
Limits Of A Vector Valued Function Of One Real Variable
The Derivative And Integral
Geometric And Physical Significance Of The Derivative
Differentiation Rules
Leibniz's Notation
Exercises
Line Integrals
Arc Length And Orientations
Line Integrals And Work
Another Notation For Line Integrals
Exercises
Independence Of Parametrization
Hard Calculus
Independence Of Parametrization
Motion On A Space Curve
Space Curves
Some Simple Techniques
Geometry Of Space Curves
Exercises
Functions Of Many Variables
Review Of Limits
Exercises
The Directional Derivative And Partial Derivatives
The Directional Derivative
Partial Derivatives
Exercises
Mixed Partial Derivatives
Partial Differential Equations
Exercises
The Derivative Of A Function Of Many Variables
The Derivative Of Functions Of One Variable
Exercises
The Derivative Of Functions Of Many Variables
Exercises
C1 Functions
The Chain Rule
The Chain Rule For Functions Of One Variable
The Chain Rule For Functions Of Many Variables
Exercises
Related Rates Problems
The Derivative Of The Inverse Function
Proof Of The Chain Rule
Exercises
The Gradient
The Gradient And Tangent Planes
Exercises
Optimization
Local Extrema
Exercises
The Second Derivative Test
Exercises
Lagrange Multipliers
Exercises
Proof Of The Second Derivative Test*
The Riemannn Integral On Rn
Methods For Double Integrals
Density And Mass
Exercises
Methods For Triple Integrals
Definition Of The Integral
Iterated Integrals
Exercises
Mass And Density
Exercises
The Integral In Other Coordinates
Polar Coordinates
Exercises
Cylindrical And Spherical Coordinates
Volume and Integrals in Cylindrical Coordinates
Volume And Integrals in Spherical Coordinates
Exercises
The General Procedure
Exercises
The Moment Of Inertia And Center Of Mass
Exercises
The Integral on Two Dimensional Surfaces In R3
The Two Dimensional Area In R3
Surfaces Of The Form z=f( x,y)
MATLAB and Graphing Surfaces
Piecewise Defined Surfaces
Flux Integrals
Exercises
Calculus Of Vector Fields
Divergence And Curl Of A Vector Field
Vector Identities
Vector Potentials
The Weak Maximum Principle
Exercises
The Divergence Theorem
Coordinate Free Concept Of Divergence
Some Applications Of The Divergence Theorem
Hydrostatic Pressure
Archimedes Law Of Buoyancy
Equations Of Heat And Diffusion
Balance Of Mass
Balance Of Momentum
The Reynolds Transport Formula
Frame Indifference
Bernoulli's Principle
The Wave Equation
A Negative Observation
Volumes Of Balls In Rn
Electrostatics
Exercises
Stokes And Green's Theorems
Green's Theorem
Exercises
Stoke's Theorem From Green's Theorem
The Normal and the Orientation
The Mobeus Band
A General Green's Theorem
Conservative Vector Fields
Some Terminology
Moving Coordinate Systems
The Acceleration In Polar Coordinates
Planetary Motion
The Equal Area Rule, Kepler's Second Law
Inverse Square Law, Kepler's First Law
Kepler's Third Law
The Angular Velocity Vector
Angular Velocity Vector on Earth
Coriolis Force and Centripetal Force
Coriolis Force on the Rotating Earth
The Foucault Pendulum
Exercises
Curvilinear Coordinates
Basis Vectors
Exercises
Curvilinear Coordinates
Exercises
Transformation of Coordinates.
Differentiation and Christoffel Symbols
Gradients and Divergence
Exercises
Curl and Cross Products
Implicit Function Theorem*
More Continuous Partial Derivatives
The Method Of Lagrange Multipliers
The Local Structure Of C1 Mappings
II Linear Algebra and Differential Equations
Determinants
Basic Techniques And Properties
Cofactors And 22 Determinants
The Determinant Of A Triangular Matrix
Properties Of Determinants
Finding Determinants Using Row Operations
Applications
A Formula For The Inverse
Finding Eigenvalues Using Determinants
Cramer's Rule
MATLAB And Determinants
The Cayley Hamilton Theorem
Exercises
The Mathematical Theory Of Determinants
The Function sgn
The Determinant
The Definition
Permuting Rows Or Columns
A Symmetric Definition
The Alternating Property Of The Determinant
Linear Combinations And Determinants
The Determinant Of A Product
Cofactor Expansions
Formula For The Inverse
Cramer's Rule
Methods and Recipes for First Order ODE
First Order Linear Equations
Bernouli Equations
Separable Differential Equations, Stability
Homogeneous Equations
Exact Equations
The Integrating Factor
The Case Where M,N Are Affine Linear
Linear and Nonlinear Differential Equations
Computer Algebra Methods
Maple
Mathematica
MATLAB
Scientific Notebook
Exercises
First Order Linear Systems
The Theory of the Fundamental Matrix
Laplace Transform Methods
Using a Computer Algebra System
Using Computer Algebra to Find Fundamental Matrix
Homogeneous Particular and General Solutions
Picard Iteration
Higher Order Scalar Linear Equations
Numerical Solutions For Systems
A Few Numerical Methods
Using MATLAB to Find Solutions
Stability of Equilibrium Points
Periodic Orbits, Poincare Bendixon Theorem
Exercises
Solutions Near a Regular Singular Point
The Euler Equations
Some Simple Observations on Power Series
Regular Singular Points
Abel's Formula
Finding the Solution
Method of Frobenius
The Bessel Equations
The Case where 0=x"0117=0
The Case of 0=x"0117 Not an Integer
Case Where 0=x"0117 is an Integer
Other Properties of Bessel Functions
Exercises
Boundary Value Problems, Fourier Series
Boundary Value Problems
Eigenvalue Problems
Fourier Series
Mean Square Approximation
Pointwise Convergence of Fourier Series
Explanation of Pointwise Convergence Theorem
Mean Square Convergence
Integrating and Differentiating Fourier Series
Odd and Even Extensions
Exercises
Some Partial Differential Equations
Laplacian in Orthogonal CurvilinearCoordinates
Heat and Wave Equations
Heat Equation
The Wave Equation
Nonhomogeneous Problems
Laplace Equation
Rectangles
Circular Disks
Exercises
III Fundamentals of Complex Analysis
Analytic Functions
Cauchy Riemann Equations
The Cauchy Riemann Equations
Contour Integrals
Cauchy Integral Theorem
Primitives and Cauchy Goursat Theorem
Functions Differentiable on a Disk, Zeros
Liouville's Theorem
Riemann Sphere
Exercises
Isolated Singularities and Analytic Functions
Open Mapping Theorem for Complex Valued Functions
Functions Analytic on an Annulus
Isolated Singularities
Meromorphic Functions
The Residue Theorem
Evaluation of Improper Integrals
Exercises
Some Fundamental Functions and Transforms
Gamma Function
Laplace Transform
Fourier Transform
The Inversion of Laplace Transforms
The Bromwich Integral
Exercises
IV Probability and Statistics
Probability
Improper Integrals
Combinations
The Binomial Theorem
Exercises
Counting and Basic Probability
Exercises
General Considerations Probability
Moment Generating Functions
Independence and Conditional Probability
Statistical Tests
The Distribution of nS2/0=x"011B2
Confidence Intervals for Variance
The T and F Distributions
The T Distribution
Confidence Intervals for the Mean
Testing For Two Different Means
The F Distribution
Confidence Intervals for the Ratio of Two Variances
Maximum Likelihood Estimates
Quadratic Forms
Linear Regression
Goodness of Fit
Contingency Tables
The Theory Of The Riemannn Integral
An Important Warning
Basic Definition
Basic Properties
Which Functions Are Integrable?
Iterated Integrals
The Change Of Variables Formula
Some Observations
A Rigid Body Rotating About a Point
Lagrangian Mechanics
The Spinning Top and the Euler Angles