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