Calculus of One and Many Variables

I Functions of One Variable
Fundamental Concepts
Numbers and Simple Algebra
Exercises
Set Notation
Order
Exercises
The Binomial Theorem
Well Ordering Principle, Math Induction
Exercises
Completeness of R
Existence of Roots
Completing the Square
Dividing Polynomials
The Complex Numbers
Polar Form of Complex Numbers
Roots of Complex Numbers
Exercises
Functions
General Considerations
Graphs of Functions and Relations
Circular Functions
Reference Angles and Other Identities
The sin( x) /x Inequality
The Area of a Circular Sector
Exercises
Exponential and Logarithmic Functions
The Function bx
Applications
Interest Compounded Continuously
Exponential Growth and Decay
The Logistic Equation
Using MATLAB to Graph
Exercises
Sequences and Compactness
Sequences
Exercises
The Limit of a Sequence
The Nested Interval Lemma
Exercises
Compactness
Sequential Compactness
Closed and Open Sets
Cauchy Sequences
Exercises
Continuous Functions and Limits of Functions
Equivalent Formulations of Continuity
Exercises
The Extreme Values Theorem
The Intermediate Value Theorem
Continuity of the Inverse
Exercises
Uniform Continuity
Examples of Continuous Functions
Sequences of Functions
Exercises
Limit of a Function
Exercises
The Derivative
The Definition of the Derivative
Finding the Derivative
Derivatives of Inverse Functions
Circular Functions and Inverses
Exponential Functions and Logarithms
The Complex Exponential
Related Rates and Implicit Differentiation
Exercises
Local Extreme Points
Exercises
Mean Value Theorem
Exercises
First and Second Derivative Tests
Exercises
Taylor Series Approximations
Exercises
L'Hôpital's Rule
Interest Compounded Continuously
Exercises
Infinite Series
Basic Considerations
Absolute Convergence
Ratio and Root Tests
Exercises
Convergence Because of Cancellation
Double Series
Exercises
Series of Functions
Weierstrass Approximation Theorem*
Exercises
The Integral
The Integral of 1700's
The Integral of 1800's
Properties of the Integral
Uniform Convergence and the Integral
Exercises
Methods for Finding Anti-derivatives
The Method of Substitution
Exercises
Integration by Parts
Exercises
Trig. Substitutions
Exercises
Partial Fractions
Rational Functions of Trig. Functions
Using MATLAB
Exercises
A Few Standard Applications
Lengths of Curves and Areas of Surfaces of Revolution
Lengths
Surfaces of Revolution
Exercises
Force on a Dam and Work
Force on a Dam
Work
Using MATLAB
Exercises
Improper Integrals and Stirling's Formula
Stirling's Formula
The Gamma Function
Laplace Transforms
Exercises
Power Series
Functions Defined in Terms of Series
Operations on Power Series
Power Series for Some Known Functions
The Binomial Theorem
Exercises
Multiplication of Power Series
Exercises
Some Other Theorems
Some Historical Observations
Polar Coordinates
Graphs in Polar Coordinates
The Area in Polar Coordinates
The Acceleration in Polar Coordinates
The Fundamental Theorem of Algebra
Polar Graphing in MATLAB
Exercises
Algebra and Geometry of Rp
Rp
Algebra in Rp
Geometric Meaning Of Vector Addition In R3
Lines
Distance in Rp
Geometric Meaning of Scalar Multiplication in R3
Exercises
Vector Products
The Dot Product
Geometric Significance of the Dot Product
The Angle Between Two Vectors
Work and Projections
Exercises
The Cross Product
The Box Product
Proof of the Distributive Law
Torque
Center of Mass
Angular Velocity
Vector Identities and Notation
Planes
Exercises
Sequences, Compactness, and Continuity
Sequences of Vectors
Open and Closed Sets
Cartesian Products
Sequential Compactness
Vector Valued Functions
Continuous Functions
Sufficient Conditions for Continuity
Limits of a Function of Many Variables
Vector Fields
MATLAB and Vector Fields
Exercises
The Extreme Value Theorem and Uniform Continuity
Convergence of Functions
Fundamental Theorem of Algebra
Exercises
Space Curves
Using MATLAB to Graph Space Curves
The Derivative and Integral
Geometric and Physical Significance of the Derivative
Differentiation Rules
Leibniz's Notation
Arc Length and Orientations
Arc Length and Parametrizations
Hard Calculus
Independence of Parametrization
Exercises
Motion on Space Curves
Some Simple Techniques
Geometry of Space Curves
Exercises
Some Physical Applications
Spherical and Cylindrical Coordinates
Exercises
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
II Functions of Many Variables
Linear Functions
The Matrix of a Linear Transformation
Row Operations and Linear Equations
Using MATLAB
Uniqueness
The Inverse
MATLAB and Matrix Arithmetic
Exercises
Subspaces Spans and Bases
Linear Independence
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
MATLAB and Determinants
Applications
A Formula for the Inverse
Finding Eigenvalues Using Determinants
MATLAB and Eigenvalues
Matrices and the Dot Product
Distance and Orthogonal Matrices
Diagonalization of Symmetric Matrices
Exercises
Functions of Many Variables
Graphs
Review of Limits
Exercises
Directional 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
The Derivative
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
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
Line Integrals
Line Integrals and Work
Conservative Fields and Notation
Exercises
The Riemannn Integral on Rp
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
Applications of the Divergence Theorem
Hydrostatic Pressure
Archimedes Law of Buoyancy
Equations of Heat and Diffusion
Balance of Mass
Balance of Momentum
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
Exercises
Curvilinear Coordinates
Basis Vectors
Exercises
Curvilinear Coordinates
Exercises
Transformation of Coordinates.
Differentiation and Christoffel Symbols
Gradients and Divergence
Exercises
Measures and Integrals
Countable Sets
Simple Functions, 0=x"011B Algebras, Measurability
Measures and Outer Measures
Measures from Outer Measures
Riemann Integrals for Decreasing Functions
Lebesgue Integrals of Nonnegative Functions
Nonnegative Simple Functions
The Monotone Convergence Theorem
The Integral's Righteous Algebraic Desires
Integrals of Real Valued Functions
Dynkin's Lemma
Product Measures
Exercises
The Lebesgue Measure and Integral in Rp
An Outer Measure on P( R)
One Dimensional Lebesgue Measure
The Lebesgue Integral and Riemann Integral
p Dimensional Lebesgue Measure and Integrals
Iterated Integrals
p Dimensional Lebesgue Measure and Integrals
Lebesgue Measure and Linear Maps
Change of Variables for Nonlinear Maps
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
Row, Column, and Determinant Rank
Formula for the Inverse
The Cayley Hamilton Theorem
Cramer's Rule
p Dimensional Parallelepipeds
Implicit Function Theorem*
More Continuous Partial Derivatives
The Method of Lagrange Multipliers