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Preface.
1 Precalculus.
Functions.
· Composition of Functions.
· Inverse Functions.
· Graphs in the x-y Plane.
Analytic Geometry.
· Lines.
· Parabolas.
· Circles.
· Ellipses.
· Hyperbolas.
Polynomial Equations.
· The Division Algorithm, Remainder Theorem, and Factor Theorem.
· The Fundamental Theorem of Algebra and Roots of Polynomial Equations.
· Sum and Product of the Roots.
Logarithms.
Trigonometry.
· Trig Functions of Acute Angles.
· Trig Functions of Arbitrary Angles.
· Trig Functions of Real Numbers.
· Trig Identities and Formulas.
· Periodicity of the Trig Functions.
· Graphs of the Trig Functions.
· The Inverse Trig Functions.
Chapter 1 Review Questions.
2 Calculus i.
Limits of Sequences.
Limits of Functions.
· Limits of Functions as x - -+inf.
Continuous Functions.
· Theorems Concerning Continuous Functions The Derivative.
The Derivative.
· Linear Approximations Using Differentials.
· Implicit Differentiation.
· Higher-Order Derivatives.
Curve Sketching.
Properties of the First Derivative.
Properties of the Second Derivative.
Theorems Concerning Differentiable Functions.
Maxi Min Problems.
Related Rates.
Indefinite Integration (Antidifferentiation).
· Techniques of Integration.
Integration by Substitution.
Integration by Part.
Trig Substitutions.
The Method of Partial Fractions.
Definite Integration.
The Fundamental Theorem of Calculus.
· The Average Value of a Function.
· Finding the Area Between Two Curves.
Polar Coordinates.
Volumes of Solids of Revolution.
Arc Length.
The Natural Exponential and Logarithm Functions.
L'Hopital's Rule.
Improper Integrals.
Infinite Series.
· Alternating Series.
Power Series.
· Functions Defined by Power Series.
· Taylor Series.
· Taylor Polynomials.
Chapter 2 Review Questions.
3 Calculus ii.
Analytic Geometry of R^3.
· The Dot product.
· The Cross product.
· The Triple Scalar Product.
· Lines in -Space.
· Planes in -Space.
· Cylinders.
· Surfaces of Revolution.
· Level Curves and Level Surfaces.
· Cylindrical Coordinates.
· Spherical Coordinates.
Partial Derivatives.
· Geometric Interpretation of f_x and f_y.
Higher-Order Partial Derivatives.
· The Tangent Plane to a Surface.
· linear Approximations.
· The Chain Rule for Partial Derivatives.
Directional Derivatives and the Gradient.
Max/Min Problems.
· Max/Min Problems with a Constraint.
· The Lagrange Multiplier Method.
Line Integrals.
· Line Integrals with Respect to Arc Length.
· The Line Integral of a Vector Field.
· The Fundamental Theorem of Calculus for line Integrals.
Double Integrals.
· Double Integrals in Polar Coordinates.
Green's Theorem.
· Path Independence and Gradient Fields.
Chapter 3 Review Questions.
4 Differential equations.
Separable Equations.
Homogeneous Equations.
Exact Equations.
Nonexact Equations and Integrating Factors.
First-Order Linear Equations.
Higher-Order Linear Equations with Constant Coefficients.
Chapter 4 Review Questions.
5 Linear algebra.
Solutions of Linear Systems.
Matrices and Matrix Algebra.
· Matrix Operations.
· Identity Matrices and Inverses.
Gaussian Elimination.
Solving Matrix Equations Using A-i.
Vector Spaces.
· The Nullspace.
· Linear Combinations.
· The Rank, Column Space, and Row Space of a Matrix.
· Other Vector Spaces.
Determinants.
· Laplace Expansions.
· The Adjugate Matrix.
· Cramer's Rule.
Linear Transformations.
· Standard Matrix Representative.
· The Rank plus Nullity Theorem.
· A note on Inverses and Compositions.
Eigenvalues and Eigenvectors.
· Eigenspaces.
· The Cayley-Hamilton Theorem.
Chapter 5 Review Questions.
6 Number theory and abstract algebra.
Part A: number Theory.
Divisibility.
· The Division Algorithm.
· Primes.
· The Greatest Common Divisor and the Least Common Multiple.
· The Euclidean Algorithm.
· The Diophantine Equation ax + by = c.
Congruences.
The Congruence Equation ax = b (mod n).
Part B: abstract Algebra.
Binary Structures and the Definition of a Group.
· Examples of Groups.
· Cyclic Groups.
Subgroups.
· Cyclic Subgroups.
· Generators and Relations.
· Some Theorems Concerning Subgroups.
The Concept of Isomorphism.
The Classification of Finite Abelian Groups.
Group Homomorphisms.
Rings.
· Ring Homomorphisms.
· Integral Domains.
Fields.
Chapter 6 Review Questions.
7 Additional topics.
Set Theory.
· Subsets and Complements.
· Union and Intersection.
· Venn Diagrams.
· Cordinality.
Combinatorics.
· Permutations and Combinations.
· The Pigeonhole Principle.
Probability and Statistics.
· Probability Spaces.
· Bernoulli Trials.
· Random Variables.
Point-Set Topology.
· The Subspace Topology.
· The Interior, Exterior, Boundary, limit Points, and Closure of a Set.
· Basis for a Topology.
· The Product Topology.
· Connectedness.
· Compactness.
· Metric Spaces.
· Continuous Functions.
Open Maps and Homeomorphisms.
Real Analysis.
· The Completeness of the Real Numbers.
· Lebesgue Measure.
· Lebesgue Measurable Functions.
· Lebesgue Integrable Functions.
Complex Variables.
· The Polar Form.
· The Exponential form.
Complex Roots.
· Complex Logarithms.
· Complex Powers.
· The Trigonometric Functions.
· The Hyperbolic Functions.
· The Derivative of a function of a Complex Variable.
· The Cauchy-Riemann Equations.
· Analytic functions.
· Complex Line Integrals.
· Theorems Concerning Analytic functions.
· Taylor Series for Functions of a Complex Variable.
· Singularities, Poles, and Laurent Series.
· The Residue Theorem.
8 Solutions to the chapter review questions.
9 Practice test.
10 Practice test answers and explanations.
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