Mathematical Analysis II

Learning Outcomes

The course’s material includes mathematical definitions, fundamental concepts, results and reasoning methodologies that pertain to basic multivariate calculus and fourier theory objects and models underlying the foundations and applications of computer science. Moreover the relevant connections of the fourier analysis to several more specific branches of computer science are presented. The course directly supports most of the curriculum subjects and lessons: Note that during the lesson, specific examples of application are discussed using new technologies with the help of programs such as Matlab, Octave and R.

Upon successfully completion of the course the students will be in position to:·

• Know, describe and handle basic knowledge of mathematical analysis of many variables (indicatively: Gamma and Beta Function, Laplace Transformation, Multiple Variable Functions (Derivation, Integration), Sequences and Function Sequences, Fourier Series, Fourier Integral, Fourier Integral)·
• Choose the appropriate mathematical concepts and be able to model the particular IT problem that it is called upon to solve. In addition to develop mathematical thinking and being able to analyze and adapt acquired knowledge to applications of computer science. ·
• Combine the essentials of Multivariate Computation and Fourier theory to solve more complex mathematical problems.·
• Know, handle and understand Matlab, Octave and R programs on applications of mathematical analysis of multiple variables and fourier theory in computer science.

Course Contents

• Improper Integrals, Improper Integrals Depending on a Parameter. Gamma and Beta Functions.
• Laplace transformations.
• Vectors in the level and in the space.
• Vector interrelations. Applications: laws of Kepler.
• Functions of Two Variables. Differentiation of Functions of Two Variables. Interrelations of multiple variables (definition, graphic representation, limits, constantly, certain derivative, derivative as for direction, total differential gear, very little – biggest).
• Precise differential equations.
• Double and triple integrals, change of coordinates – applications.
• Sequences and Series of Functions
• Fourier Series and Integrals. Fourier transformations.
• Completion of vector fields (bent – divergence – turn, curve integrals, surface integrals, theorems green, gauss, stokes).

Recommended Readings

• M. Filippakis, Applied Analysis and Fourier Theory, Tsotras Publications, Athens 2017, 2nd Edition.
• Zygmund A. (2003): Trigonometric Series, 3rd Edition, Cambridge University Press.
• Teaching Notes.