Mathematical Analysis

LEARNING OUTCOMES

The student after the successful completion of the course will be able to:

• Appropriately use concepts from Calculus in a mathematical / applied environment
• Implement, with a systematic approach, methods for solving basic mathematical problems from the field of engineering and technological sciences.
• Interpret the results she/he has reached
• Verify the results through critical thinking.

 

General Competences

• Exercise criticism and self-criticism
• Mathematical thinking and analysis
• Mathematical and analytical presentation of geometric concepts
• Search, analyze and synthesize data with the use of the necessary technology
• Autonomous work
• Production of free, creative and inductive thinking

 

SYLLABUS

The course is designed for a set of 13 weeks of lectures. The topics that will be discussed are the following

1. Set of real and complex numbers
   The real number system. Mathematical Induction. The real line. Complex numbers
   and their properties. Polar representation of complex numbers.
2. Real functions of one variable
   Basic Definitions, Algebraic Functions, Trigonometric Functions, Exponential Function
3. Limit of a Function
   Existence and uniqueness of the limit. Algebraic properties of limits. One-sided limits.
   Limit of composite functions
4. Continuity of functions
   Definitions, Continuity of Elementary Functions, Intermediate Value Theorem, Existence of maximum and minimum of continuous functions in closed intervals, Monotonic functions, Continuous and 1-1 functions, Inverse functions, Logarithmic function
5. Differential Calculus of Functions of one variable
   Definition, Differentiation Rules, Derivatives of Elementary Functions, Mean ValueTheorem, Derivative of a Function and Monotonicity, Derivative and Local Extremes, L’Hopital Rule, Curved Functions, Inflection Points,
6. Sequences of real numbers
   Converging sequences. Monotonic sequences. Defining a sequence recursively
   Integrals
7. Series of real numbers
   Series of sequences. Taylor Series.
8. Integral Calculus of functions of one variable
   The fundamental theorem of calculus. Integration techniques (integration by factors, recursively, rational functions, variable change). Definite integral, Integral applications
9. Functions of two variables
10. Sequences in R2, Limits of Functions of Two Variables, Partial Derivative, Taylor
    Theorem, Extremes of Functions of Two Variables, Double Integrals. Green’s Theorem, Stokes Theorem and Gauss Theorem.

 

STUDENT PERFORMANCE EVALUATION

During the semester students will be given problems-exercises which together with the material of the lectures will be an aid for the preparation of the final exams.

 

ATTACHED BIBLIOGRAPHY

1. Wrede, R., Spiegel, M.R. (2013) – Schaum’s Outline of Advanced Calculus, Mcgraw-Hill, 6th Edition
2. Trench W.F. (2013) – Introduction to Real Analysis, Faculty Authored and Edited Books, Trinity University.
3. Thomas, G.B., Finney, R.L., Wier, M.D. (2002) – Thomas’ Calculus, Addison – Wesley, 9th Edition
4. Spivak, M. (2019) – The Hitchhiker’s Guide to Calculus, Vol 57, American Mathematical Society
5. Halidias, N. (2021) – Applied Mathematics for Economists and Engineers, Broken Hill. (in Greek)
6. Rassias, T. (2017) – Mathematics I, Tsotras, 2nd Edition (in Greek).