Assignments
September
Rosenlicht
Chapter 3
   ◊ Definition of Metric Space.  Examples
   ◊ Open and Closed Sets
   ◊ Convergent Sequences
   ◊ Completeness
   ◊ Compactness
   ◊ Connectedness
   ◊ Problems


Rudin
Chapter 2 Basic Topology
   ◊ Finite, Countable, and Uncountables Sets
   ◊ Metric Spaces
   ◊ Compact Sets
   ◊ Perfects Sets
   ◊ Connected Sets
   ◊ Exercises
October
Rudin
Chapter 3  Numerical Sequences and Series
   ◊ Convergent Sequences
   ◊ Subsequences
   ◊ Cauchy Sequences
   ◊ Upper and Lower Limits
   ◊ Some Special Sequences
   ◊ Series
   ◊ Series of Nonnegative Terms
   ◊ The Number e
   ◊ The Root and Ratio Tests
   ◊ Power Series
   ◊ Summation by Parts
   ◊ Absolute Convergence
   ◊ Addition and Multiplication of Series
   ◊ Rearrangements
   ◊ Exercises
November
Rosenlicht
Chapter 4   Continuous Functions
   ◊ Definition of Continuity.  Examples
   ◊ Continuity and Limits
   ◊ The Continuity of Rational Operations.  Functions with Values in En
   ◊ Continuous Functions on a Compact Metric Space
   ◊ Continuous Functions on a Connected Metric Space
   ◊ Sequences of Functions
   ◊ Problems


Rudin
Chapter 4  Continuity
   ◊ Limits of Functions
   ◊ Continuity and Compactness
   ◊ Continuity and Connectedness
   ◊ Discontinuities
   ◊ Monotonic Functions
   ◊ Infinite Limits and Limits at Infinity
   ◊ Exercises
December
Rosenlicht
Chapter 5   Differentiation
   ◊ Definition of Derivative
   ◊ Rules of Differentiation
   ◊ The Mean Value Theorem
   ◊ Taylor's Theorem
   ◊ Problems


Protter
Chapter 4  Elementary Theory of Differentiation
   ◊ The Derivative in R1
   ◊ Inverse Functions in R1
   ◊ Exercises


Rudin
Chapter 5   Differentiation
   ◊ The Derivative of a Real Function
   ◊ Mean Value Theorems
   ◊ The Continuity of Derivatives
   ◊ L'Hospital's Rule
   ◊ Derivatives of Higher Order
   ◊ Taylor's Theorem
   ◊ Differentiation of Vector-Valued Functions
   ◊ Exercises


Additional Material
Dini Derivatives
January
Rudin
Chapter 6   Riemann-Stieltjes Integral
    ◊ Definition and Existence of the Integral
    ◊ Properties of the Integral
    ◊ Integration of Vector-valued Functions
    ◊ Rectifiable Curves
    ◊ Exercises
February
Rudin
Chapter 7   Sequences and Series of Functions
    ◊ Discussion of Main Problem
    ◊ Uniform Convergence
    ◊ Unifrom Convergence and Continuity
    ◊ Uniform Convergence and Integration
    ◊ Univofrm Convergence and Differentiation
    ◊ Equicontinuous Families of Functions
    ◊ The Stone-Weierstrass Theorem
    ◊ Exercises
March
Rudin
Chapter 8   Some Special Functions
    ◊ Power Series
    ◊ The Exponential and Logarithmic Functions
    ◊ The Trigonometric Functions
    ◊ The Algebraic Functions
    ◊ Fourier Series
    ◊ The Gamma Function
    ◊ Exercises
April
Rudin
Chapter 9   Functions of Several Variables
   ◊ Linear Transformations
   ◊ Differentiation
   ◊ The Contraction Principle
   ◊ The Inverse Function Theorem
   ◊ The Implicit Function Theorem
   ◊ The Rank Theorem
   ◊ Determinants
   ◊ Derivatives of Higher Order
   ◊ Differentiation of Integrals
   ◊ Exercises
May
Rudin
Chapter 10   Integration of Differential Forms
   ◊ Integration
   ◊ Primitive Mappings
   ◊ Partitions of Variables
   ◊ Differential Forms
   ◊ Simpmlexes and Chains
   ◊ Stokes' Theorem
   ◊ Closed Forms and Exact Forms
   ◊ Vector Analysis
   ◊ Exercises
June
Rudin
Chapter 11   The Lebesgue Theory
   ◊ Set Functions
   ◊ Construction of the Lebesgue Measure
   ◊ Measure Spaces
   ◊ Measurable Functions
   ◊ Simple Functions
   ◊ Integrations
   ◊ Comparison with the Riemann Integral
   ◊ Integration of Complex Functions
   ◊ Functions of Class L2
   ◊ Exercises
July
Rudin
Chapter 11 The Lebesgue Theory
Summer Assignments
Multivariable Calculus & Linear Algebra Summer Assignments
Preliminaries