314423 FUNCTIONS OF SEVERAL REAL VARIABLES

1. Course: 314423 FUNCTIONS OF SEVERAL REAL VARIABLES
2. Instructor: Dr.Nimit Nimana
3. Class Meeting Time:
                            Monday                 14.30 – 16.00             Room SC7305
                            Wednesday         14.30 – 16.00             Room SC7305
4. Credit: 3 (3-0-6)
5. Prerequisite:
                            314211 LINEAR ALGEBRA I
                            314221 ADVANCED CALCULUS
                            314321 MATHEMATICAL ANALYSIS I
6. Course Description:
   Basic topology of the Euclidean n-Space, sequences in the Euclidean n-Space, limits and continuity, uniform continuity, differentiation, integration.
7. Topics:
   (1) The Real Line and Euclidean n-Space
                            The Real Number Line 
                            Euclidean n-Space 
   (2) Topology of the Euclidean n-Space 
                            Open Sets
                            Interior of a Set
                            Closed Sets
                            Accumulation Points
                            Closure of a Set
                            Boundary of a Set
                            Sequences Series of the Euclidean n-Space 
   (3) Compact and Connected Sets
                            Compact sets: The Heine-Borel and Bolzano-Weierstrass Theorems
                            Nested Set Property
                            Path-Connected Sets
                            Connected Sets
   (4) Continuous Mappings
                            Continuity
                            Images of Compact and Connected Sets Operations on Continuous Mappings
                            The Boundedness of Continuous Functions of Compact Sets
                            The Intermediate Value Theorem
                            Uniform Continuity
   (5) Uniform Convergence
                            Pointwise and Uniform Convergence
                            The Weierstrass M-Test
                            The Space of Continuous Functions
   (6) Differentiable Mappings
                            Definition of the Derivative
                            Matrix Representation
                            Continuity of Differentiable Mappings; Differentiable Paths
                            Conditions for Differentiability
                            The Chain Rule
                            Product Rule and Gradients
                            The Mean Value Theorem
                            Taylor’s Theorem and Higher Derivatives
                            Maxima and Minima
   (7) The Inverse and Implicit Function Theorems and Related Topics
                            Inverse Function Theorem
                            Implicit Function Theorem
                            Constrained Extrema and Lagrange Multipliers
   (8) Integration Integrable Functions
                            Volume and Sets of Measure Zero
                            Lebesgue’s Theorem
                            Properties of the Integral
   (9) Fubini’s Theorem
                            Fubini’s Theorem
                            Change of Variables Theorem
8. Grading Policy:
                            Problem sets                                                  10 %
                            Presentations                                                25 %
                            Two midterm exams                                   40 % (20/20)
                            Final exam                                                        25 %
9. Textbook:
   Any good book in advanced mathematical analysis and functional analysis should be useful. Our main reference will be:
   📝Marsden, J. E.: (1974). Elementary Classical Analysis. San Francisco: W. H. Freeman and Company.
10. Lecture Notes: