| 1 |
Course Organization and Introduction to Mathematica®, Assignment and Evaluation |
| 2 |
Introduction and Overview of Mathematica®, Assignment and Evaluation, Mathematica® Functions, Operations on Expressions, Lists, Getting Help |
| 3 |
Procedural and Functional Programming, Functions and Rules |
| 4 |
Mathematica®: Symbolic and Numeric Calculations, Roots of Equations, File Input and Output |
| 5 |
Mathematica® Graphics: Plotting Data, Two- and Three-Dimensional Plotting, Graphics Primitives |
| 6 |
Linear Algebra: Matrix Operations, Interpretations of Matrix Operations, Multiplication, Transposes, Index Notation |
| 7 |
Linear Algebra: Solutions to Linear Systems of Equations, Determinants, Matrix Inverses, Linear Transformations and Vector Spaces |
| 8 |
Complex Numbers: Complex Plane, Addition and Multiplication, Complex Conjugates, Polar Form of Complex Numbers, Powers and Roots, Exponentiation, Hyperbolic and Trigonometric Forms |
| 9 |
Matrix Eigenvalues: Eigenvalue/Eigenvector Definitions, Invariants, Principal Directions and Values, Symmetric, Skew-symmetric, and Orthogonal Systems, Orthogonal Transformations |
| 10 |
Hermitian Forms, Similar Matrices, Eigenvalue Basis, Diagonal Forms |
| 11 |
Vector Calculus: Vector Algebra, Inner Products, Cross Products, Determinants as Triple Products, Derivatives of Vectors |
| 12 |
Multi-variable Calculus: Curves and Arc Length, Differentials of Scalar Functions of Vector Arguments, Chain Rules for Several Variables, Change of Variable and Thermodynamic Notation, Gradients and Directional Derivatives |
| 13 |
Vector Differential Operations: Divergence and Its Interpretation, Curl and Its Interpretation |
| 14 |
Path Integration: Integral Over a Curve, Change of Variables, Multidimensional Integrals |
| 15 |
Multidimensional Forms of the Fundamental Theorem of Calculus: Green’s Theorem in the Plane, Surface Representations and Integrals |
| 16 |
Multi-variable Calculus: Triple Integrals and Divergence Theorem, Applications and Interpretation of the Divergence Theorem, Stoke’s Theorem |
| 17 |
Periodic Functions: Fourier Series, Interpretation of Fourier Coefficients, Convergence, Odd and Even Expansions |
| 18 |
Fourier Theory: Complex Form of Fourier Series, Fourier Integrals, Fourier Cosine and Sine Transforms, The Fourier Transforms |
| 19 |
Ordinary Differential Equations: physical interpretations, geometrical interpretations, separable equations |
| 20 |
ODEs: Derivations for Simple Models, Exact Equations and Integrating Factors, The Bernoulli Equation |
| 21 |
Higher Order Differential Equations: Homogeneous Second Order, Initial Value Problems, Second Order with Constant Coefficients, Solution Behavior |
| 22 |
Differential Operators, Damped and Forced Harmonic Oscillators, Nonhomogeneous Equations |
| 23 |
Resonance Phenomena, Higher Order Equations, Beam Theory |
| 24 |
Systems of Differential Equations, Linearization, Stable Points, Classifi-cation of Stable Points |
| 25 |
Linear Differential Equations: Phase Plane Analysis and Visualization |
| 26 |
Solutions to Differential Equations: Legendre’s Equation, Orthogonality of Legendre Polynomials, Bessel’s Equation and Bessel Functions |
| 27 |
Sturm-Louiville Problems: Eigenfunction, Orthogonal Functional Series, Eigenfunction Expansions |