 
Abbreviation: B23A02  Load: 30(L)
+ 0(P)
+ 0(FE)
+ 0(LE)
+ 0(S)
+ 0(PEE)
+ 30(E)
+ 0(DE)

Lecturers in charge:  doc. dr. sc. Jelka BebanBrkić 
Lecturers:  v. pred. mr. sc. Željka Tutek
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Exercises
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Course description:
To recognize the mathematical and numerical skills acquired within the theory of curves and surfaces in the field of study. To use the mathematical and numerical skills acquired within the theory of curves and surfaces for solving problems in the field of study.
Learning outcomes at the level of the programme to which the course contributes
 Understand mathematical methods and physical laws applied in geodesy and geoinformatics.
 Apply knowledge of mathematics and physics for the purpose of recognizing, formulating and solving of problems in the field of geodesy and geoinformatics.
 Use information technology in solving geodetic and geoinformation tasks
 Exercise appropriate judgements on the basis of performed calculation processing and interpretation of data obtained by means of surveying and its results.
 Take responsibility for continuing academic development in the field of geodesy and geoinformatics, or related disciplines, and for the development of interest in lifelong learning and further professional education.
Learning outcomes expected at the level of the course
 identify various forms of curve equations, calculate arc length, curvature and determine the associated vector fields; Identify and differentiate between types of second order surfaces;
 analyze the second order surfaces with emphasis on the sphere and the ellipsoid of revolution: determine the parameter curves, the tangent plane and the normal vector to the surface;
 determine the first fundamental form of the surface and use it to calculate arc length, surface area and angle between two curves on a surface;
 determine the second fundamental form of the surface and use it for classifying points on the surface, calculating the normal, principal, Gaussian and mean curvature of the surface;
 detect some special curves on surfaces (lines of curvature, asymptotic lines);
 define the concept of the geodesic curvature along a curve on a surfaces and the term geodesic; calculate the geodesic curvature of parameter curves in order to identify whether it is a matter of geodesic coordinates;
 pronounce the Theorema Egregium of Gauss;
 distinguish and name types of mappings of surfaces according to the mapping invariants;
 use a variety of tools for visualizing and solving problems related to the theory of curves and surfaces.
Course content broken down in detail by weekly class schedule (syllabus)
 Basic concepts of vector algebra and vector analysis. 1h
 Representations of curves. 1h
 Arc length and reparameterisation of a curve. 2h
 Moving trihedron. Curvature and torsion. FrenetSerret formulas. 2h
 Concept of a surface: definition, parametric representation, coordinate patches, parameter curves. 2h
 Concept of a surface: the tangent plane and the normal vector to the surface. 2h
 Review of previous work. 1h
 1st preliminary exam 1h
 First fundamental form and its applications (arc length, surface area and angle between two curves on a surface). 2h
 Second fundamental form and its applications (normal curvature, types of points on the surface). 2h
 Asymptotic and principal directions and lines. Principal, Gaussian and mean curvature of the surface. 2h
 Euler's theorem and Dupin's indicatrix. 1h
 GaussWeingarten equations and Christoffel symbols. The fundamental theorem of surfaces. 2h
 Review of previous work. 1h
 2nd preliminary exam 1h
 Geodesics (geodesic curvature, geodesic coordinates, arcs of minimum length) 3h
 Mappings of surfaces (Stereographic projection, Mercator projection) 2h
 Review of previous work. 1h
 The final exam. 1h
Screening student work
 Class attendance  Requirement for the signature
 independent assignments  15%
 interactive tasks  5%
 Tests  80%
 Oral exam  optional
 Written exam  100%

Compulsory literature: 
1.  Beban Brkić, J.: Diferencijalna geometrija, Interna skripta, dostupna na eučenju 
2.  ŽarinacFrančula, B.: Diferencijalna geometrija, Zbirka zadataka i repetitorij. Školska knjiga, Zagreb 1990. 
Recommended literature: 
3.  Lipschutz, M. M.: Differential Geometry, Schaum's Outline Series, McGrawHill Book Company, N. Y. 1969.

4.  Gray, A.: Modern Differential Geometry of Curver and Surfaces With Mathematica, CRS Press, Boston, London, 1998. 
 