Real Analysis

Lecturer: Prof. Francis Su
College/University: Harvey Mudd College

1. Constructing the Real Numbers
   
2. Properties of Q (Rational Numbers)
   
3. Construction of the Reals
   
4. The Least Upper Bound Property
   
5. Complex Numbers
   
6. Principle of Induction
   
7. Countable and Uncountable Sets
   
8. Cantor Diagonalization and Metric Spaces
   
9. Limit Points
   
10. The Relationship Between Open and Closed Sets
    
11. Compact Sets
    
12. Relationship of Compact Sets to Closed Sets
    
13. Compactness and the Heine-Borel Theorem
    
14. Connected Sets, Cantor Sets
    
15. Convergence of Sequences
    
16. Subsequences, Cauchy Sequences
    
17. Complete Spaces
    
18. Series
    
19. Series Convergence Tests, Absolute Convergence
    
20. Functions - Limits and Continuity
    
21. Continuous Functions
    
22. Uniform Continuity
    
23. Discontinuous Functions
    
24. The Derivative and the Mean Value Theorem
    
25. Taylor's Theorem, Sequence of Functions
    
26. Ordinal Numbers and Transfinite Induction
    

Probability for Life Science

Lecturer: Prof. Herbert Enderton
College/University: University of California, Los Angeles (UCLA)

1. Events, Sample Space, and Probability Function
   
2. Basic Properties of the Probability Function, Multiplication Principle, Addition Principle, Permutation
   
3. Combinations, Permutation of n Objects Not All Distinct
   
4. Other Properties of the Probabilty Function
   
5. Conditional Probability
   
6. Law of Total Probability, Bayes' Rule, and Independent Events
   
7. Independence of Two or More Events
   
8. Discrete Random Variables, Probability Mass Function, Cumulative Distribution Function, and Expected Value
   
9. Properties of Expected Value, Variance and Standard Deviation of a Random Variable
   
10. Binomial Distribution, Joint Distributions, Independent Random Variables and Other Properties
    
11. Review
    
12. Multinomial Distribution, Geometric Distribution
    
13. Expected Value and Standard Deviation of a Geometric Distribution, Negative Binomial Distribution
    
14. Poisson Distribution
    
15. Expected Value, Variance and Standard Deviation of the Poisson Distribution
    
16. Continuous Random Variable, Probability Density Function
    
17. Mean and Variance of Continuous Random Variable, Exponential Distribution
    
18. Variance and Standard Deviation of an Exponential Distribution, Exponential Distribution with Parameter Lamda as a Function of Another Variable
    
19. Standard Normal Distribution and Its Properties
    
20. Normal Distribution with Mean "Mu" and Standard Deviation "Sigma"
    
21. Central Limit Theorem
    
22. Applications of Central Limit Theorem
    
23. Review
    
24. Confidence Interval (Part I)
    
25. Confidence Interval (Part II)
    
26. Markov's Inequality, Chebyshev's Inequality, Law of Large Numbers
    
27. Determining Sample Size and Confidence Interval
    
28. Review

Multivariable Calculus

Lecturer: Professor Denis Auroux
College/University: Massachusetts Institute of Technology (MIT)

1. Dot Product
   
2. Determinants and Cross Product.
   
3. Matrices and Inverse Matrices
   
4. Square Systems; Equations of Planes
   
5. Parametric Equations for Lines and Curves
   
6. Velocity, Acceleration; Kepler's Second Law
   
7. Review
   
8. Level Curves; Partial Derivatives; Tangent Plane Approximation
   
9. Max-min Problems; Least Squares
   
10. Second Derivative Test; Boundaries and Infinity
    
11. Differentials; Chain Rule
    
12. Gradient; Directional Derivative; Tangent Plane
    
13. Lagrange Multipliers
    
14. Non-independent Variables
    
15. Partial Differential Equations
    
16. Double integrals
    
17. Double integrals in Polar Coordinates; Applications
    
18. Change of Variables
    
19. Vector Fields and Line Integrals in the Plane
    
20. Path Independence and Conservative Fields.
    
21. Gradient Fields and Potential Functions
    
22. Green's Theorem
    
23. Flux; Normal Form of Green's Theorem
    
24. Simply Connected Regions
    
25. Triple Integrals in Rectangular and Cylindrical Coordinates
    
26. Spherical Coordinates; Surface Area
    
27. Vector Fields in 3D; Surface Integrals and Flux
    
28. Divergence Theorem (First Part)
    
29. Divergence theorem (Second Part)
    
30. Line Integrals in Space, Curl, Exactness and Potentials
    
31. Stokes' Theorem (First Part)
    
32. Stokes' theorem (Second Part)
    
33. Topological considerations; Maxwell's equations
    
34. Review (First Part)
    
35. Review (Second Part)

Single Variable Calculus

Lecturer: Professor David Jerison
College/University: Massachusetts Institute of Technology (MIT)

1. Derivative: Geometric Interpretation
   
2. Derivative as Rate of Change, Differentiability, and Continuity of Functions
   
3. Derivatives of Products, Quotients, and Trigonometric Functions
   
4. Chain Rule and Higher Derivatives
   
5. Implicit Differentiation and Inverses
   
6. Exponetial and Logarithmic Differentiation; Hyperbolic Functions
   
7. Hyperbolic functions (cont.)
   
8. Linear and quadratic approximations
   
9. Curve Sketching
   
10. Max-min Problems
    
11. Related Rates
    
12. Newton's method and other applications
    
13. Mean Value Theorem; Inequalities
    
14. Differentials, antiderivatives
    
15. Differential equations, separation of variables
    

Differential Equations

Lecturer: Professor Arthur Mattuck
College/University: Massachusetts Institute of Technology (MIT)

1. The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
   
2. Euler's Numerical Method for y'=f(x,y) and its Generalizations
   
3. Solving First-order Linear ODE's; Steady-state and Transient Solutions
   
4. First-order Substitution Methods: Bernouilli and Homogeneous ODE's
   
5. First-order Autonomous ODE's: Qualitative Methods, Applications
   
6. Complex Numbers and Complex Exponentials
   
7. First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods
   
8. Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models
   
9. Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases
   
10. Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations
    
11. Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
    
12. Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's
    
13. Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Exponentials
    
14. Interpretation of the Exceptional Case: Resonance
    
15. Introduction to Fourier Series; Basic Formulas for Period 2(pi)
    
16. Continuation: More General Periods; Even and Odd Functions; Periodic Extension
    
17. Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds
    
18. Introduction to the Laplace Transform; Basic Formulas
    
19. Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
    
20. Using Laplace Transform to Solve ODE's with Discontinuous Inputs
    
21. Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
    
22. Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System
    
23. Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
    
24. Matrix Exponentials; Application to Solving Systems
    
25. Decoupling Linear Systems with Constant Coefficients
    
26. Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum
    
27. Limit Cycles: Existence and Non-existence Criteria
    

Probability and Random Variables

Lecturer: Professor M. Chakraborty
College/University: Indian Institute of Technology Kharagpur

1. Introduction to the Theory of Probability
   
2. Axioms of Probability (First Part)
   
3. Axioms of Probability (Second Part)
   
4. Introduction to Random Variables
   
5. Probability Distributions and Density Functions
   
6. Conditional Distribution and Density Functions
   
7. Function of a Random Variable (First Part)
   
8. Function of a Random Variable (Second Part)
   
9. Mean and Variance of a Random Variable
   
10. Moments
    
11. Characteristic Function
    
12. Two Random Variables
    
13. Function of Two Random Variables (First Part)
    
14. Function of Two Random Variables (Second Part)
    
15. Covariance and Correlation
    
16. Vector Space of Random Variables
    
17. Joint Moments
    
18. Joint Characteristic Functions
    
19. Joint Conditional Densities (First Part)
    
20. Joint Conditional Densities (Second Part)
    
21. Sequences of Random Variables (First Part)
    
22. Sequences of Random Variables (Second Part)
    
23. Correlation Matrices and their Properties (First Part)
    
24. Correlation Matrices and their Properties (Second Part)
    
25. Conditional Densities of Random Vectors
    
26. Characteristic Functions and Normality
    
27. Chebychev's Inequality and Estimation
    
28. Central Limit Theorem
    
29. Introduction to Stochastic Process
    
30. Stationary Processes
    
31. Cyclostationary Processes
    
32. System with Random Process at Input
    
33. Ergodic Processes
    
34. Spectral Analysis (First Part)
    
35. Spectral Analysis (Second Part)
    
36. Spectrum Estimation - Non Parametric Methods
    
37. Spectrum Estimation - Parametric Methods
    
38. Autoregressive Modeling and Linear Prediction
    
39. Linear Mean Square Estimation - Wiener (FIR)
    
40. Adaptive Filtering - LMS Algorithm
    

Calculus

Lecturer: Prof. S. K. Ray
College/University: Indian Institute of Technology Kanpur

1. Real Number
   
2. Sequences (First Part)
   
3. Sequences (Second Part)
   
4. Sequences (Third Part)
   
5. Continuous Function
   
6. Properties of Continuous Fuction
   
7. Uniform Continuity
   
8. Differentiable Function
   
9. Mean Value Theorems
   
10. Maxima and Minima
    
11. Taylor's Theorem
    
12. Curve Sketching
    
13. Infinite Series (First Part)
    
14. Infinite Series (Second Part)
    
15. Tests of Convergence
    
16. Power Series
    
17. Riemann Integral
    
18. Riemann Integrable Functions
    
19. Applications of Riemann Integral
    
20. Length of a Curve
    
21. Line Integrals
    
22. Functions of Several Variables
    
23. Differentiation
    
24. Derivatives
    
25. Mean Value Theorem
    
26. Maxima and Minima
    
27. Methods of Lagrange Multipliers
    
28. Multiple Integrals
    
29. Surface Integrals
    
30. Green's Theorem
    
31. Stokes Theorem
    
32. Gauss Divergence Theorem