Course Details

Mathematics 4

Academic Year 2026/27

NAA026 course is part of 1 study plan

​ NPC-GK Winter Semester 1st year

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.

Credits

5 credits

Language of instruction

Czech

Semester

winter

Course Guarantor

prof. RNDr. Josef Diblík, DrSc.

Institute

Institute of Mathematics and Descriptive Geometry

Forms and criteria of assessment

course-unit credit and examination

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.

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.

Basic Literature

Dlouhý Oldřich, Tryhyk Václav. Matematika IV, Vybrané části funkce komplexní proměnné a diferenciální geometrie, Brno, VUT, FAST, Studijní opora, 2009 (cs)

Recommended Reading

ERWIN KREYSZIG. Differential geometry. Dover Publications, 1991 (en)
DIRK J. STRUIK. Lectures on classical differential geometry. Dover Publications, 1988 (en)
P. FINNIKOV. Differencialnaja geometrija. Moskva, 1961. (ru)
Sushil Shukla, Shikha Tiwari. Functions of Complex Variable: A Textbook of Complex Analysis, ‎ LAP LAMBERT Academic Publishing, 2020 (en)

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.

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/303631

Lecture

13 weeks, 2 hours/week, elective

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

13 weeks, 2 hours/week, compulsory

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.