MAT-60206 Mathematical Analysis, 5 cr

Additional information

Suitable for postgraduate studies

Person responsible

Janne Kauhanen

Lessons

Implementation 1: MAT-60206 2015-01

Study type P1 P2 P3 P4 Summer

Lectures
Excercises
26 h/per
18 h/per
+16 h/per
+12 h/per

Lecture times and places: Monday 12 - 14 S2/SA203 , Wednesday 12 - 14 SE203

Requirements

Two mid-course exams or final exam. The grade of the course may be improved by bonus points collected from the weekly exercises.
Completion parts must belong to the same implementation

Learning Outcomes

This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Upon successful completion of the course, student will be able to: - Read mathematical texts and proofs, - Use the definitions and apply the basic results that are introduced during this course and - Produce rigorous proofs of results that arise in this course using direct and indirect proof, induction and epsilon-delta technique.

Content

Content Core content Complementary knowledge Specialist knowledge

1. THE REAL NUMBERS Field and ordering properties of the real numbers, supremum and infimum, the Completeness Axiom, open and closed sets, limit points. The Heine-Borel Theorem and the Bolzano-Weierstrass Theorem.

2. THE LIMIT AND CONTINUITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Continuity of sum, product, quotient and composition of continuous functions. Boundedness, the existence of min and max and the Intermediate Value theorem for continuous functions. Monotonic functions and continuity of the inverse function. Uniform continuity.

3. DIFFERENTIABILITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Linear approximation, the derivative of sum, product and quotient of differentiable functions, the Chain Rule, extreme values, the Mean Value Theorem and l'Hospitals Rule. Taylor's Theorem.

4. THE RIEMANN INTEGRAL Riemann sums and upper and lower sums, existence of the integral using upper and lower sums (the Riemann Condition), integrability of monotone and continuous functions. Basic properties of the integral: linearity, comparison of integrals, the Triangle Inequality and additivity on intervals. The Mean Value Theorem, the Fundamental Theorem of Calculus, integration by parts and the change of variable. Local integrability and the improper integral, improper integrals of nonnegative functions, the Comparison Test, absolute and conditional convergence. The set of discontinuities and integrability.

5. SEQUENCES AND SERIES OF FUNCTIONS Pointwise and uniform convergence of sequences and series of functions. Caychy's uniform convergence criterion for sequences and series of functions, Weirstrass's M-Test, the effect of uniform convergence on the limit/sum function with respect to continuity, differentiability and integrability.

Instructions for students on how to achieve the learning outcomes

If the student is mastering the concepts, results and short proofs in concrete examples the evaluation is 3. For the grades 4 and 5 the student should in addition to the previous level be able to independently apply theory to deduce new results.

Assessment scale:

Numerical evaluation scale (1-5) will be used on the course

Partial passing:

Completion parts must belong to the same implementation

Study material

Type Name Author ISBN URL Additional information Examination material

Book Introduction to real analysis (Edition 2.04) William Trench Chapters 1-4 No

Summary of lectures Matemaattinen analyysi Janne Kauhanen Ladattavissa Moodlesta No

Prerequisites

Course Mandatory/Advisable Description

MAT-01160 Matematiikka 1 Mandatory

MAT-01260 Matematiikka 2 Mandatory

MAT-01360 Matematiikka 3 Mandatory

MAT-01460 Matematiikka 4 Mandatory

Additional information about prerequisites
Vaihtoehtoisesti Laajan matematiikan opintojaksot tai hyvin suoritetut Insinöörimatematiikan opintojaksot (18-19 op).

Correspondence of content

Course Corresponds course Description

MAT-60206 Mathematical Analysis, 5 cr MAT-43650 Mathematical Analysis, 6 cr

MAT-60206 Mathematical Analysis, 5 cr MAT-60200 Mathematical Analysis, 5 cr

Last modified 23.03.2015

Study type	P1	P2	P3	P4	Summer
Lectures Excercises	26 h/per 18 h/per	+16 h/per +12 h/per

Content	Core content	Complementary knowledge	Specialist knowledge
1.	THE REAL NUMBERS Field and ordering properties of the real numbers, supremum and infimum, the Completeness Axiom, open and closed sets, limit points.	The Heine-Borel Theorem and the Bolzano-Weierstrass Theorem.
2.	THE LIMIT AND CONTINUITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Continuity of sum, product, quotient and composition of continuous functions. Boundedness, the existence of min and max and the Intermediate Value theorem for continuous functions. Monotonic functions and continuity of the inverse function.	Uniform continuity.
3.	DIFFERENTIABILITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Linear approximation, the derivative of sum, product and quotient of differentiable functions, the Chain Rule, extreme values, the Mean Value Theorem and l'Hospitals Rule.	Taylor's Theorem.
4.	THE RIEMANN INTEGRAL Riemann sums and upper and lower sums, existence of the integral using upper and lower sums (the Riemann Condition), integrability of monotone and continuous functions. Basic properties of the integral: linearity, comparison of integrals, the Triangle Inequality and additivity on intervals. The Mean Value Theorem, the Fundamental Theorem of Calculus, integration by parts and the change of variable. Local integrability and the improper integral, improper integrals of nonnegative functions, the Comparison Test, absolute and conditional convergence.	The set of discontinuities and integrability.
5.	SEQUENCES AND SERIES OF FUNCTIONS Pointwise and uniform convergence of sequences and series of functions.	Caychy's uniform convergence criterion for sequences and series of functions, Weirstrass's M-Test, the effect of uniform convergence on the limit/sum function with respect to continuity, differentiability and integrability.

Type	Name	Author	ISBN	URL	Additional information	Examination material
Book	Introduction to real analysis (Edition 2.04)	William Trench			Chapters 1-4	No
Summary of lectures	Matemaattinen analyysi	Janne Kauhanen			Ladattavissa Moodlesta	No

Course	Mandatory/Advisable	Description
MAT-01160 Matematiikka 1	Mandatory
MAT-01260 Matematiikka 2	Mandatory
MAT-01360 Matematiikka 3	Mandatory
MAT-01460 Matematiikka 4	Mandatory

Course	Corresponds course	Description
MAT-60206 Mathematical Analysis, 5 cr	MAT-43650 Mathematical Analysis, 6 cr
MAT-60206 Mathematical Analysis, 5 cr	MAT-60200 Mathematical Analysis, 5 cr