Differential and Fractal Geometry

Major: Earth Sciences
Code of subject: 8.103.00.M.026
Credits: 3.00
Department: Cartography and Geospatial Modelling
Lecturer: Ph.D., Associate Professor Brydun Andrii
Semester: 4 семестр
Mode of study: денна
Мета вивчення дисципліни: The purpose of teaching the discipline is for graduate students to master the most important questions of the differential and fractal geometry course, which have theoretical and practical application in further scientific research.
Завдання: As a result of studying an academic discipline, a graduate student should be able to demonstrate the following learning outcomes: • find the length of the curve, • carry out natural parameterization, • find curvature and twist, special points of a curve, • calculate the evolute and the involute, • find the first and second quadratic forms, • find curves and asymptotic lines, • find geodesic lines, • use and compose equations to construct fractals. The study of an academic discipline involves the formation and development of competencies in graduate students: general: • In-depth knowledge in the field of geodesy, space monitoring of the Earth, cartography and geotechnical engineering. • Basic knowledge and understanding of the philosophical methodology of cognition, the key principles of professional ethics, the system of moral and cultural values. • Ability to initiate and conduct original scientific research, identify relevant scientific problems, search for and critically analyze information, produce innovative constructive ideas, and apply non-standard approaches to solving complex and atypical tasks. • The ability to demonstrate oratory and rhetorical skills when presenting the results of scientific research, to conduct a professional scientific conversation and debate with the wider scientific community and the public in Ukrainian, to form scientific texts in written form, to organize and conduct training sessions, to use progressive information and communication tools. • The ability to present and discuss the obtained results of scientific research in a foreign language in oral and written form, to freely read and fully understand foreign scientific texts. • The ability to be purposeful and persistent, to self-improve throughout life, to be aware of social and moral responsibility for the obtained scientific results. • The ability to justify and manage current scientific projects of an innovative nature, to conduct scientific research independently, to interact in a team and to show leadership abilities in the implementation of scientific projects. professional: • In-depth knowledge of classical and modern scientific trends in the study of natural phenomena, processes in various fields of Earth sciences. • Ability to apply and integrate knowledge and understanding of other engineering disciplines. • Ability to apply professional knowledge and practical skills to solve scientific problems of the specialty, as well as to choose technical means for their implementation. • The ability to argue the choice of methods for solving specialized problems, critically evaluate the obtained results and defend the decisions made.
Learning outcomes: The learning outcomes of this discipline detail the following program learning outcomes: • Ability to demonstrate in-depth knowledge and understanding of the scientific and mathematical principles underlying the Earth sciences. • Ability to demonstrate knowledge of the current state of affairs and the latest technologies, and skills in conducting experiments, data collection, modeling and analysis of the obtained results in the Earth sciences. • Ability to demonstrate in-depth knowledge of domestic and foreign research and theoretical and applied principles in at least one of the areas of Earth sciences: geodesy, cartography, geophysics, geodynamics, meteorology and climatology. • The ability to choose and apply the methodology and tools of scientific research when conducting theoretical and empirical research in the field of Earth sciences. • Ability to demonstrate in-depth knowledge of professionally oriented disciplines of the specialty. • Carry out geospatial modeling of objects, processes and phenomena. • To think systematically and apply creative abilities to the formation of fundamentally new ideas. • Ability to perform geospatial modeling of objects, processes and phenomena. • The ability to formulate and improve an important research problem, collect the necessary information for its solution, and formulate conclusions that can be defended in a scientific context. • Conduct scientific research and implement scientific projects based on the identification of current scientific problems, definition of goals and objectives, formation and critical analysis of the information base, substantiation and commercialization of research results, formulation of author's conclusions and proposals. • Ability to formulate own author's conclusions, proposals and recommendations.
Required prior and related subjects: • Higher Mathematics. • Web mapping.
Summary of the subject: Differential geometry is a section of mathematics that studies the general properties of curves and surfaces by methods of analysis of infinitesimally small, that is, properties of arbitrarily small pieces of curves and surfaces. Fractal geometry is also widely used in computer graphics to construct images of natural objects such as trees, shrubs, mountain landscapes, seas.
Опис: Topic 1. Theory of curves. • Curves in Rn. Tangent vector. Tangent. • Curve length, natural parameterization. Lengths of curves in different coordinate systems. • Serre-Frene base. Frenet's formulas. • Curve and twist. Lines given by general equations. Special points. • Touching curves. Circumstantial Evolute and involute. Topic 2. Theory of surfaces. • Surfaces. Tangent plane and normal vector. • The first quadratic form. Isometric surfaces. • Second quadratic form, normal curvature. • Main curves. Dupin's indicatrix. Gaussian and mean curves. • Classification of points on the surface. • Weingarten's derivative equations. Christophel symbols. • Gauss and Peterson-Kodazzi formulas. Bonnet's theorem. • Curve lines and asymptotic lines. Geodetic lines. Topic 3. Fractals. • Fractals. • Classification of fractals (algebraic, geometric and stochastic fractals). • Using the universal equation to construct fractals. • Use in natural sciences for image compression, etc.
Assessment methods and criteria: • Written reports on practical work, oral examination (40%) • Final control (exam): written, verbal form (60%)
Критерії оцінювання результатів навчання: • Work in practical classes (max 20 points). • Completion of individual homework (max. 20 points). • Exam: written and oral form (max 60 points).
Recommended books: 1. Борисенко О.А. Диференціальна геометрія і топологія / Борисенко О.А. – Х., 1995. 2. Куратовский К. Топология Т. 1. / Куратовский К. – M., 1966. 3. Мищенко А.С. Курс дифференциальной геометрии и топологи / Мищенко А.С., Фоменко А.Т. – M., 1980. 4. James Gleick, Chaos – Making a New Science, Viking, New York, 1987. 5. Б. Мандельброт (Не)послушные рынки. Фрактальная революция в финансах / Б. Мандельброт и Ричард Л. Хадсон – Вильямс, 2006. ISBN 5-8459-0922-8.