Course detail

Mathematics IV

FAST-HA01Acad. year: 2023/2024

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.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

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.

Rules for evaluation and completion of the course

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Aims

Understanding the basics of the theory of functions of a complex variable.
Understanding the basics of differential geometry of 3D curves and surfaces.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DIRK J. STRUIK. Lectures on classical differential geometry. Dover Publications, 1988 (EN)
ERWIN KREYSZIG: Differential geometry. Akademische Verlagsgesellschaft, Leipzig, 1957. (EN)
S.P.FINIKOV: Diferencialnaja geometrija. Moskva, 1961. (RU)

Recommended reading

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)

Classification of course in study plans

  • Programme N-P-C-GK Master's

    branch GD , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

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 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

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.