Course Details
Mathematics 4
Academic Year 2024/25
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.
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
Institute
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.
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.
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)
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)
Offered to foreign students
Not to offer
Course on BUT site
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.