Infinite Series, Power Series / Chapter 1: |
The Geometric Series |
Definitions and Notation |
Applications of Series |
Convergent and Divergent Series |
Convergence Tests |
Convergence Tests for Series of Positive Terms |
Alternating Series |
Conditionally Convergent Series |
Useful Facts about Series |
Power Series; Interval of Convergence |
Theorems about Power Series |
Expanding Functions in Power Series |
Expansion Techniques |
Accuracy of Series Approximations |
Some Uses of Series |
Complex Numbers / Chapter 2: |
Introduction |
Real and Imaginary Parts of a Complex Number |
The Complex Plane |
Terminology and Notation |
Complex Algebra |
Complex Infinite Series |
Complex Power Series; Disk of Convergence |
Elementary Functions of Complex Numbers |
Euler's Formula |
Powers and Roots of Complex Numbers |
The Exponential and Trigonometric Functions |
Hyperbolic Functions |
Logarithms |
Complex Roots and Powers |
Inverse Trigonometric and Hyperbolic Functions |
Some Applications |
Linear Algebra / Chapter 3: |
Matrices; Row Reduction |
Determinants; Cramer's Rule |
Vectors |
Lines and Planes |
Matrix Operations |
Linear Combinations, Functions, Operators |
Linear Dependence and Independence |
Special Matrices and Formulas |
Linear Vector Spaces |
Eigenvalues and Eigenvectors |
Applications of Diagonalization |
A Brief Introduction to Groups |
General Vector Spaces |
Partial Differentiation / Chapter 4: |
Introduction and Notation |
Power Series in Two Variables |
Total Differentials |
Approximations using Differentials |
Chain Rule |
Implicit Differentiation |
More Chain Rule |
Maximum and Minimum Problems |
Constraints; Lagrange Multipliers |
Endpoint or Boundary Point Problems |
Change of Variables |
Differentiation of Integrals |
Multiple Integrals / Chapter 5: |
Double and Triple Integrals |
Applications of Integration |
Change of Variables in Integrals; Jacobians |
Surface Integrals |
Vector Analysis / Chapter 6: |
Applications of Vector Multiplication |
Triple Products |
Differentiation of Vectors |
Fields |
Directional Derivative; Gradient |
Some Other Expressions Involving V. Line Integrals |
Green's Theorems in the Plane |
The Divergence and the Divergence Theorem |
The Curl and Stokes' Theorem |
Fourier Series and Transforms / Chapter 7: |
Simple Harmonic Motion and Wave Motion |
Periodic Functions |
Applications of Fourier Series |
Average Value of a Function |
Fourier Coefficients |
Complex Form of Fourier Series |
Other Intervals |
Even and Odd Functions |
An Application to Sound |
Parseval's Theorem |
Fourier Transforms |
Ordinary Differential Equations / Chapter 8: |
Separable Equations |
Linear First-Order Equations |
Other Methods for First-Order Equations |
Linear Equations (Zero Right-Hand Side) |
Linear Equations (Nonzero Right-Hand Side) |
Other Second-Order Equations |
The Laplace Transform |
Laplace Transform Solutions |
Convolution |
The Dirac Delta Function |
A Brief Introduction to Green's Functions |
Calculus of Variations / Chapter 9: |
The Euler Equation |
Using the Euler Equation |
The Brachistochrone Problem; Cycloids |
Several Dependent Variables |
Lagrange's Equations |
Isoperimetric Problems |
Variational Notation |
Tensor Analysis / Chapter 10: |
Cartesian Tensors |
Tensor Notation and Operations |
Inertia Tensor |
Kronecker Delta and Levi-Civita Symbol |
Pseudovectors and Pseudotensors |
More about Applications |
Curvilinear Coordinates |
Vector Operators |
Non-Cartesian Tensors |
Special Functions / Chapter 11: |
The Factorial Function |
Gamma Function; Recursion Relation |
The Gamma Function of Negative Numbers |
Formulas Involving Gamma Functions |
Beta Functions |
Beta Functions in Terms of Gamma Functions |
The Simple Pendulum |
The Error Function |
Asymptot |
Infinite Series, Power Series / Chapter 1: |
The Geometric Series |
Definitions and Notation |