Exams
Assignments
Problems on these assignments had to be presented in class by a single person in order to earn credit for the problem. About halfway through the course, we added a problem day, which explains the increase in problems.
- Winter 2013 Assignment 1
- Winter 2013 Assignment 2
- Winter 2013 Assignment 3
- Winter 2013 Assignment 4
- Winter 2013 Assignment 5
Notes
These notes are from my iPad using GoodNotes.
- Day 1 -Uniform Continuity; Intermediate Value Theorem
- Day 2 -Sufficient Conditions for the IVT; Partitions; Lower and Upper Sums; Definition of Riemann Integral
- Day 3 -Equivalent Condition for Riemann Integrability; Integrability of Continuous Functions
- Day 4 -Integrating Functions with Finitely Many Discontinuities; Integration Properties
- Day 5 -Lebesgue Measure Zero Sets; α-continuity and uniform α-continuity
- Day 6 -Characterization of Integrability in Terms of Lebesgue Measure, Part 1
- Day 7 -Characterization of Integrability in Terms of Lebesgue Measure, Part 2; Fundamental Theorem of Calculus
- Day 8 -Fundamental Theorem, continued; Integration by Parts; Pointwise and Uniform Convergence of Functions
- Day 9 -Convergence in General Metric Spaces; Uniform Convergence and Riemann Integrability
- Day 10 -Substitution; Weierstrass Approximation Theorem, Part 1
- Day 11 -Weierstrass Approximation Theorem, Part 2
- Day 12 -Weierstrass Approximation Theorem, Part 3; Improper Integration; Logarithms and Exponentials; Uniform Convergence and Differentiation, Part 1
- Day 13 -Uniform Convergence and Differentiation, Part 2, Complete Metric Spaces;Uniformly Cauchy Sequences
- Day 14 -Infinite Series of Functions; Power Series; Weierstrass M-Test
- Day 15 -Power Series: Radius and Interval of Convergence, Uniform Convergence
- Day 16 -Power Series: Continuity and Differentiability
- Day 17 -Uniform Convergence and Differentiation, Part 3
- Day 18 -Taylor and MacLaurin Series; Generalized MVT; Lagrange Remainder Theorem
- Day 19 -A Function Not Equal to its Taylor Series; Generalized Riemann Integral: Definition, Tagged Partitions, Gauges, Gauged Partitions
- Day 20 -Fundamental Theorem of Calculus for Generalized Riemann Integration
- Day 21 -Norms on R^n; Derivatives in R^n; Linear Maps; Matrix Norm
- Day 22 -Properties of Matrix Norm; Invertible Linear Maps; Uniqueness of Derivatives in R^n; Chain Rule
- Day 23 -Partial Derivatives: Relation to Derivative; Convex Sets;
- Day 24 - Contractions; Statement of Inverse Function Theorem
- Day 25 -Inverse Function Theorem: Proof
- Day 26 -Implicit Function Theorem, Part 1
- Day 27 -Implicit Function Theorem, Part 2; Integration in R^n: Definition
- Day 28 -Integration: Extension, Fubini's Theorem, Multilinear Maps (Tensors)
- Day 29 -Examples of Multilinear Maps; Determinants, Alternating Tensors
- Day 30 - Definition of k-Tensor on R^n; Tensor Product: Basic Properties; Dimension of k-Tensors on R^n; Alternating k-Tensors
- Day 31 -Alt: Basic Properties; Wedge Product: Basic Properties
- Day 32 -More Properties of Wedge Product; Dimension of Alternating k-Tensors on R^n; Orientation
- Day 34 -Stokes' Theorem (breakneck pace!)
