Course detail
Mathematics IV
FAST-HA01Acad. year: 2018/2019
Complex-valued functions, limit, continuity and derivative. Cauchy-Riemann conditions, analytic functions. Conformal mappings performed by analytic function.
Curves in space, curvature and torsion. Frenet frame, Frenet formulae.
Explicit, implicit and parametric form of the equation of the surface in the space, first fundamental form of a surface and its applications, second fundamental form of a surface, normal and geodetic curvature of a surface, curvature and asymptotic lines on a surface, mean and total curvature of a surface, elliptic, parabolic, hyperbolic and rembilical points of a surface.
Supervisor
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
Students will achieve the subject's main objectives:
Understanding the basics of the theory of functions of a complex variable.
Understanding the basics of differential geometry of 3D curves and surfaces.
Prerequisites
Basic properties of complex numbers as taught at secondary schools.
Basics of integral calculus of functions of one variable and the basic interpretations.
Basics of calculus. Differentiation.
Basics of calculus of two- and more-functions. Partial differentiation.
Co-requisites
Not applicable.
Recommended optional programme components
Not applicable.
Recommended or required reading
ERWIN KREYSZIG: Differential geometry. Akademische Verlagsgesellschaft, Leipzig, 1957. (EN)
S.P.FINIKOV: Diferencialnaja geometrija. Moskva, 1961. (RU)
DIRK.J.STRUIK: Lectures on classical differential geometry. Addison - Wesley publishing Massachutes USA, 1961. (EN)
DLOUHÝ O., TRYHUK V.: Vybrané části funkce komplexní proměnné a diferenciální geometrie. FAST VUT v Brně, 2010. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
Planned learning activities and teaching methods
Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.
Assesment methods and criteria linked to learning outcomes
Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops.
Language of instruction
Czech
Work placements
Not applicable.
Course curriculum
1. Complex numbers, basic operations, displaying, n-th root. Complex functions.
2. Limit, continuity, derivative of a complex function, Cauchy-Riemann conditions.
3. Analytical functions. Conform mapping implemented by an analytical function.
4. Conform mapping implemented by an analytical function.
5. Planar curves, singular points on a curve.
6. 3D curves, curvature and torsion.
7. Frenet trihedral, Frenet formulas.
8. Explicit, implicit, and parametric equations of a surface.
9. The first basic form of a surface and its use.
10. The second basic form of a surface. Normal and geodetic curvature of a surface. Meusnier's theorem.
11. Asymptotic curves on a surface.
12. Mean and total curvature of a surface.
13. Elliptic, hyperbolic, parabolic and circular points of a surface.
Aims
Understanding the basics of the theory of functions of a complex variable.
Understanding the basics of differential geometry of 3D curves and surfaces.
Specification of controlled education, way of implementation and compensation for absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
1. Complex numbers, basic operations, displaying, n-th root. Complex functions.
2. Limit, continuity, derivative of a complex function, Cauchy-Riemann conditions.
3. Analytical functions. Conform mapping implemented by an analytical function.
4. Conform mapping implemented by an analytical function.
5. Planar curves, singular points on a curve.
6. 3D curves, curvature and torsion.
7. Frenet trihedral, Frenet formulas.
8. Explicit, implicit, and parametric equations of a surface.
9. The first basic form of a surface and its use.
10. The second basic form of a surface. Normal and geodetic curvature of a surface. Meusnier's theorem.
11. Asymptotic curves on a surface.
12. Mean and total curvature of a surface.
13. Elliptic, hyperbolic, parabolic and circular points of a surface.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
1. Complex numbers, basic operations, displaying, n-th root. Complex functions.
2. Limit, continuity, derivative of a complex function, Cauchy-Riemann conditions.
3. Analytical functions. Conform mapping implemented by an analytical function.
4. Conform mapping implemented by an analytical function.
5. Planar curves, singular points on a curve.
6. 3D curves, curvature and torsion.
7. Frenet trihedral, Frenet formulas.
8. Explicit, implicit, and parametric equations of a surface.
9. The first basic form of a surface and its use.
10. The second basic form of a surface. Normal and geodetic curvature of a surface. Meusnier's theorem.
11. Asymptotic curves on a surface.
12. Mean and total curvature of a surface.
13. Elliptic, hyperbolic, parabolic and circular points of a surface. Seminar evaluation.