Pedagogical activities

................................	doc. RNDr. Daniela Velichová, CSc. Institute of Mathematics and Physics, Faculty of Mechanical Engineering Slovak University of Technology in Bratislava Nám. slobody 17, 812 31 Bratislava tel: 02/5729 6115, e-mail: daniela.velichovastuba.sk
	Pedagogical activities Constructive geometry 2^nd year, second term, 2 - 2 weakly (26 hours of lectures, 26 hours of practicals) in Slovak and in English Study materials LECTURES in ENGLISH CURVES SURFACES Mathematics I 1^st year, first term, 4 - 4 weakly (52 hours of lectures, 52 hours of practicals) in Slovak and in English INVITATION TO THE WORLD OF MATHEMATICS Study materials LECTURES in ENGLISH Mathematics II 1^st year, second term, 3 - 3 weakly (39 hours of lectures, 39 hours of practicals) in Slovak Study materials LECTURES in ENGLISH Applied Mathematics 4^th year, first term, 2 - 2 weakly (26 hours of lectures, 26 hours of practicals) in Slovak Study materials LECTURES in ENGLISH


	Constructive geometry 2^nd year, second term, 2 - 2 weakly (26 hours of lectures, 26 hours of practicals) Annotation The subject is based on elements of the Euclidean space geometry. Its aim is to develop spatial abilities and skill in drawing views of elementary solids and surfaces, appearing most frequently on technical drawings. Monge method and orthogonal axonometric method as the most convenient orthographic drawing methods are explained. Basic constructional problems - intersection points of a line and an elementary surface patch (prismatic, pyramidal, cylindrical, conical and spherical), planar cuttings of the elementary surface patches and intersections of surfaces are solved in both projection methods. Basics of the intrinsic geometry of curves and surfaces are introduced and applied on technical curves and surfaces. Basic topics Extended Euclidean space. Transformations of the space: axial affinity, central collineation. Projection of the space, projection methods - Monge method, orthogonal axonometry. Definition, views and problems on elementary solids and surfaces: spherical surface - sphere, prismatic surface - prism, cylindrical surface - cylinder, pyramidal surface - pyramid, conical surface - cone. Geometry of the Creative space, creative and analytic representations of modelled figures. Matrices of transformations and projection methods. Definition, point function and intrinsic geometric propereties of curves, Frenet-Serretov moving trihedron. Technical curves - conic sections, helices - cylindrical, conical, spherical, free-form curves - interpolation and approximation. Definition, point function and intrinsic geometric properties of surfaces, net of iso-parametric curves, tangent plane, twist and normal to the surface. Special types of curves and surfaces used in the technical practise, problems on curves and surfaces. Ruled surfaces, surfaces of revolution, quadratic surfaces, helicoids, envelope surfaces, free-form surfaces - interpolation and approximation. Computer aided geometric modelling. Syllaby Geometry of the Euclidean space and extended Euclidean space Axiomatic system - 5 groups of postulates: incidence, order, continuity, congruence, parallelism Space transformations (mappings) - metric (Euclidean), affine (axial affinity - axis of affinity, direction of affinity), projective (central collineation - centre of collineation, axis of collineation) invariant sets of points and invariant properties Projection of the space - basic notions: projection (image) plane, direction of projection, parallel projection, centre of projection, central projection, projecting line and plane, view of a point, a line, a plane, a trace of a line, a trace of a plane Orthographic projections - ground (horizontal) plane, frontal plane, profile (side) plane, ground (top, plan) view, front view, profile (side) view, fundamental views, multi-view drawings (European and American standard) Projection methods Monge method: related views (ground and front view), reference line Orthogonal axonometry: the Pelz axonometric triangle, axonometric coordinate (principal) axes cross, axonometric view, auxiliary views onto principal planes Problems on superposition of geometric figures (incidence) a point located on a line; a point located in a plane; a line located in a plane; special (principal) lines - level line, frontal line, profile line, slope lines of three frames; two different lines - parallel, intersecting, skew; two different planes - parallel, intersecting; line of intersection (a pierce line); a line and a plane - parallel, intersecting; point of intersection (a pierce point) Metric problems The true size of a line segment 1, revolution about the vertical (horizontal) axis 2, additional auxiliary views (folding line) 3, lowering to the projection plane Problems: the distance between a point and a line, a point and a plane, two parallel lines, a line parallel to a plane, two parallel planes The true size of an angle between two lines 1, additional view (a point view of a line) 2, revolution of a plane about its trace (principal line) to the true-size view of a plane Problems: the angle between two planes (dihedral angle), a line and a plane; a line perpendicular to a plane; a plane perpendicular to a line The true size of a geometric figure in a plane 1, revolution of a plane 2, additional views (the edge view or a true-size view of a plane) Problems: projection of a circle, a polygon defined by its metric properties, intersection method in axonometry Views of elementary surfaces and solids: sphere, spherical surface - ball right or oblique prism (parallelepiped, cube) - prismatic surface right or oblique cylinder - cylindrical surface right or oblique pyramid (tetrahedron, octahedron) - pyramidal surface right or oblique cone - conical surface Eckhardt intersection method Problems: intersection points of a line and a surface, intersection figure of a plane and a surface Creative geometry - basic notions: Creative space, creative and syntetic representations of geometric figures, vector (point) function Transformation matrices Geometry of curve segments in the extended Euclidean space Definition, vector function, parametric curvilinear coordinate of a curve segment point, double points - cusps, multiple points, regular - irregular points Intrinsic properties of a curve segment at a regular point Frenet-Serret trihedron: unit tangent, normal and binormal vectors, osculating, tangent and normal planes, first (flexion) and second (torsion) curvatures, radius of curvature, centre of curvature, osculating circle Degree of a planar, pierce points of a curve segment and a line Technical curves Circle: vector equation, intrinsic properties, rectifications of a circular arc length Cycloidal curves: synthetic generation, analytic representations - vector equations, types of cycloidal movements and curve segments - orthocycloids, epicycloids (Cardiod, Limacon of Pascal, Nefroid), hypocycloids (Steiner hypocycloid, Asteroid, Archimedean spiral, Logarithmic spiral), pericycloids Conic sections: ellipse, parabola, hyperbola Helices: creative representation, vector equation (radius, pitch, reduced pitch, and slope), intrinsic properties, projection of a helix and a Frenet-Serret trihedron at the given point, rectification of a helix segment length, curvatures - torsion, flexion; cylindrical, conical, spherical helix Free-Form curves: creative and analytic representations, types of interpolations, special curves - cubics: Ferguson, Bezier, B-spline Coons, b-spline (b1 - velocity, b2 - tension), rational splines (weight of a point), NURBS - interpolation of conic sections Intrinsic properties of the interpolation cubic segment at the given point Surface patches in the extended Euclidean space Definition, vector equation, parametric curvilinear coordinates of a surface patch point, net of iso-parametric curves, regular-irregular points, multiple points or curves Intrinsic surface patch properties: tangent vectors to the iso-parametric curves, normal vector and normal to the surface patch, tangent plane, twist vector at the given surface patch point, types of points (elliptic, parabolic, hyperbolic) Classification of surfaces, types of surfaces Mappings of surfaces - isometric, conform; development of surfaces Degree of a surface = number of pierce points of a surface and a line Constructional problems: 1. view of a surface patch (a contoure line and an outline curve) 2. a point on a surface patch (a tangent plane and normal to the surface patch) a curve segment on a surface patch ( a tangent line to the curve segment) 3. an intersection curve of a plane and a surface patch 4. pierce points of a line and a surface patch 5. an intersection curve of 2 surface patches (a tangent line to the intersection) Developable surfaces (torses) Definition, creative law, equations and types of developable surfaces (plane, cylindrical surface, conical surface, surface of a curve segment tangents) Transition (ruled) surfaces defined by two basic curves: construction of a surface patch line (joining points of the basic curves) using a tangent plane to the surface patch (defined by tangent lines to the basic curves in the concerned points) Development of a surface patch, Catalano' theorem Conditions of the development (length of curve segments, size of angles between a surface patch line and tangent lines to the basic curve segments in the common points) Rectification of a curve segment - radius of curvature, centre of curvature on a normal line, osculating circle in the osculating plane Surfaces of revolution Definition, creative law, equation and properties, classification (sphere, torus) Views of a surface of revolution, iso-parametric curves on the surface - parallel circles or parallels (equator, crater, neck), meridian sections or meridians (principal meridian) Tangent plane and normal to the surface at the given point, pierce points of a line and a surface, intersection curve of a plane and a surface and its tangent line Special types: sphere, cylindrical surface, conical surface, torus quadratic surfaces: ellipsoids (oblate - flat, prolate - elongate), paraboloid, hyperboloids ( of one or two sheets) Intersections of two surfaces Constructional methods: cutting plane method, sphere method Intersections of elementary surfaces - prismatic, pyramidal, cylindrical, conical Intersections of surfaces of revolution - parallel axes, intersecting axes, skew axes Helical surfaces (helicoids) Definition, creative law, equation and properties, classification, types of helical surfaces, views of a helical surface, curves on the surface - helices (equator, neck), basic curves, a tangent plane and a normal to the surface at the given point, intersection curve of a plane and a surface and its tangent line - normal and meridian cutting planes Envelope surfaces Definition, creative law, equation and properties, classification Characteristic curve - characteristics on an envelope surface, construction and projection of a surface characteristics, a tangent line and a normal to the characteristics Free-form surfaces Definition, creative law, equation and properties, classification, types of interpolation surfaces - Coons, Bezier, B-spline, b-spline, D-spline, rational and NURBS-patches Net of isoparametric curves on a surface, a tangent plane (tangent vectors to isoparametric curves), normal and twist vectors to the surface at the given point Computer aided geometric modelling CAD SYSTEMS - brief introduction to the geometric core Study materials Velichová, D.: Constructive geometry, Vydavateľstvo STU, Bratislava 2012, study book in English Velichová, D.: Constructional geometry I, II, KM SjF STU, Bratislava 1996, lecture notes in English Velichová, D.: Lectures in English, KM SjF STU, 2004/2005, Velichová, D.: Constructive Geometry, electronic book in English, KM SjF STU, 2003 Velichová, D.: Geometry in plane, electronic learning materials, dMath project, 2003/2006, Velichová, D.: Constructive geometry, Publishing House of STU, Bratislava 2003, textbook in Slovak Velichová, D.: Constructive geometry, Publishing House of STU, Bratislava 1996, lecture notes in Slovak Szarková, D., Velichová, D.: Constructive geometry - Spreadsheets and problems, Publishing House SPEKTRUM STU, Bratislava 2012, student book in Slovak Velichová, D.: Constructive geometry, electronic textbook in Slovak, KM SjF STU, 2003 Velichová, D.: Lectures, KM SjF STU, 2004/2005, lecture notes in Slovak, Velichová, D.: Geometry in plane, electronic learing materials in Slovak, KEGA, 2004/2005, Velichová, D.: Geometric modelling - mathematical backgrounds, Publishing House of STU, Bratislava 2006, monography in Slovak Oravec, G., Rybár, J., Zbuňáková, E.: Constructive geometry, Alfa Bratislava, 1994 Medek, V., Zámožík, J.: Constructive geometry for technicians, Alfa Bratislava, 1978 Mathematics I 1^st year, first term, 4 - 4 weakly (52 hours of lectures, 52 hours of practicals) Annotation Subject is an introduction to the mathematics at technical university. Its aim is to develop logical reasoning and relevant knowledge necessary for application of mathematical methods in solving professional problems and tasks that are most frequently appearing in separate mechanical engineering branches. The backgrounds of the subject dwell in mathematical analysis of a function of one real variable (differentiation and integral calculus), and solutions of systems of linear algebraic equations. Basic topics Introduction to mathematical logic, set theory and theory of number systems. Matrices and determinants Systems of linear algebraic eqautions. Sequences, limit of a sequence. Number series, convergence. Real functions of one real variable. Limit and continuity of a function. Derivative of a function. Higher order derivatives and behaviour of a function. Indefinite integral, primitive function, improper integrals. Definite integrals and applications. Power series, radius and interval of convergence. Syllaby Introduction to linear algebra - Matrices Systems of linear equations 1 Systems of linear equations 2 Determinants Number sequences, limit of a sequence Number series, sums, convergence criteria Real function of real variable Elementary functions Limit of a function Continuity of a function, asymptotes to the graph Derivative of a function, rules for derivation Derivatives of elementary functions Derivatives of higher orders, Taylor polynomial, L´Hospital rule Basic theorems of calculus Monotonicity of a function, function local and global extrema Convexity and concavity Investigation of function behaviour Antiderivatives, indefinite integrals Methods of integration for definite integrals Integration of special functions Definite integrals, Newton-Leibniz formula Geometric and physical applications of definite integrals Improper integrals Functional series, power series, radius and interval of convergence. Study materials Velichová, D.: Mathematics I, Vydavateľstvo STU, Bratislava 2014, study book in English Velichová, D.: Mathematics II,vydavateľstvo STU, Bratislava 2016, study book in English Velichová, D.: Lectures, KM SjF STU, 2012/2013 Velichová, D.: Problems, KM SjF STU, 2012/2013, European Virtual Laboratory of Mathematics EVLM database of e-learning modules Kováčová, M., Velichová, D.: Calculus - Differentiation and Integration, e-learning modules, XMath project, 2004 Velichová, D.: Geometry in Plane, elektronic learning materials, KEGA, 2004/2005, in Slovak Dicsöová, A., Pekárková, R., Poláková, V.: Mathematics I, Publishing House of STU, Bratislava 2002, lecture notes in Slovak Ivan, J.: Mathematics I, Publishing House Alfa, Bratislava 1996, book in Slovak Velichová, D.: Mathematics I, KM SjF STU, 2005, electronic textbook in Slovak http://km.sjf.stuba.sk/~velichov/KNIHA/ Velichová, D.: Mathematics II, KM SjF STU, 2006, electronic textbook in Slovak http://sjf.stuba.sk/~velichov/Matematika2/Kniha/kniha.html Velichová, D.: Lecture notes I, KM SjF STU, 2004/2005, in Slovak http://sjf.stuba.sk/~velichov/PREDNASKY/Prednasky.htm Velichová, D.: Exercises I, KM SjF STU, 2004/2005, in Slovak http://sjf.stuba.sk/~velichov/CVICENIA/Cvicenia.htm To view correctly mathematical formulas in the electronic learning materials 8 - 11 it is necessary to install software MathPlayer available free on Internet. It is recommended to install special mathematical fonts enabling correct view of mathematical symbols and special denotations, which are available free on server Mozilla, by downloading the file "font installer" and following the instructions while running it. Mathematics II 1^st year, second term, 3 - 3 weakly (39 hours of lectures, 39 hours of practicals) Annotation Subject is continuation to introduction to the mathematics at technical university. Its aim is to develop logical reasoning and relevant knowledge necessary for application of mathematical methods in solving professional problems and tasks that are most frequently appearing in separate mechanical engineering branches. The backgrounds of the subject dwell in solutions of ordinary differential equations of special types, mathematical analysis of a functions of more real variables (differentiation and integral calculus of function of two variables), and in solutions of application problems using multiple integrals. Basic topics Expansion of functions to Taylor and McLaurin series. Ordinary differential equations - basic notions and concepts. ODE with separated and separable variables. Linear differential equation of the first order. Linear differential equation of the second order. Vectors and operations on vectors. Basics of coordinate geometry in space,linear figures. Euclidean space - quadric figures. Real function of more variables. Limit a continuity of function of two variables. Partial derivatives od function of two variables. Local, global and constrained extremes of function of two variables. Double and triple integrals. Usage of double and triple integrals in solutions of application problems. Syllaby Functional series, power series, radius and interval of convergence. Expansion of function to Taylor and McLaurin series. Ordinary differential equations - basic concepts ODE with separated and separable variables Ordinary linear differential equations of order 1 Ordinary linear differential equations of order 2 - homogeneous Non-homogeneous ordinary linear differential equations of order 2 with special right-hand members and method of variation of constants Summary of linear difrerential equations Introduction to vector algebra - operations with vectors Euclidean n-dimensional space Linear figures in the Euclidean space Non-linear figures in the space, quadratic surfaces Function of more variables - definition, domain of definition, graph Limit and continuity of functions of more variables Partial derivatives of function of two variables, tangent plane to the function graph, total differential Local extrema of function of two variables Constrained and global extrema of function of two variables Multiple integrals of functions of more variables Double and triple integrals on normal regions Transformations of coordinate systems Geometric and physical applications of double and triple integrals Study materials Velichová, D.: Mathematics I, Vydavateľstvo STU, Bratislava 2014, study book in English Velichová, D.: Mathematics II,vydavateľstvo STU, Bratislava 2016, study book in English Velichová, D.: Lectures, KM SjF STU, 2012/2013 Velichová, D.: Problems, KM SjF STU, 2012/2013, European Virtual Laboratory of Mathematics EVLM database of e-learning modules Harman, B., Dobrakovová, J., Šamajová, H.: Mathematics II, Publishing House of STU, Bratislava 2003, lecture notes in Slovak Ivan, J.: Mathematika I, Alfa, Bratislava 1983, textbook in Slovak Ivan, J.: Mathematika II, Alfa, Bratislava 1985, textbook in Slovak Velichová, D.: Mathematics I, KM SjF STU, 2005, electronic textbook in Slovak http://km.sjf.stuba.sk/~velichov/KNIHA/ Velichová, D.: Mathematics II, KM SjF STU, 2006, electronic textbook in Slovak http://sjf.stuba.sk/~velichov/Matematika2/Kniha/kniha.html Velichová, D.: Lecture notes II, KM SjF STU, 2006, in Slovak http://sjf.stuba.sk/~velichov/Matematika2/Prednasky/prednasky.htm Velichová, D.: Exercices II, KM SjF STU, 2006, in Slovak http://sjf.stuba.sk/~velichov/Matematika2/Cvicenia/cvicenia.htm To view correctly mathematical formulas in the electronic learning materials 7 - 10 it is necessary to install software MathPlayer available free on Internet. It is recommended to install special mathematical fonts enabling correct view of mathematical symbols and special denotations, which are available free on server Mozilla, by downloading the file "font installer" and following the instructions while running it. Applied Mathematics 4^th year, winter term, 2 - 2 weakly (26 hours of lectures, 26 hours of practicals) Annotation Subject represents an introduction to the study of vector analysis, theory of vector fields, differential geometry and geometric modelling. Presented are selected chapters from applied mathematics, as basics from theory of series with variable terms, curvilinear and surface integrals and from geometric modelling. The aim of subject is to provide information on some domains of mathematics, which are frequently appearing in applications to mechanical engineering, applied mechanics and in mechatronics. Main objective is to present backgrounds of vector analysis of functions of more real variables (differential and integral calculus), elements of differential geometry of curves and surfaces, basics of vector fields theory (gradient, curl and Laplacian of a vector function), and Taylor and Fourrier series. Basic topics Introduction to vector algebra. Vector functions and their properties, operations on vector functions. Differential and integral calculus of vector functions. Differential geometry of curves and surfaces. Basics of vector fields theory. Curvilinear and surface integrals. Series with variable terms - functional series. Elements of tenzor calculus. Introduction to geometric modelling. Modelling of curves and surfaces. Minkowski set operations. Sylaby Vector space, operations on vectors Vector functions Properties of vector functions Differential calculus of vector functions Integral calculus of vector functions Operations with vector functions Elements of differential geometry of curves Elements of differential geometry of surfaces Basics of vector fields theory Curvilinear integrals Surface integrals Series with variable terms Fourier series Minkowski set operations Study materials Velichová, D.: Lectures AM, KM SjF STU, 2012, in Slovak Velichová, D.: Problems AM, KM SjF STU, 2012, in Slovak Velichová, D.: Constructive geometry, Vydavateľstvo STU, Bratislava 2012, study book in English Ivan, J.: Matematika I, Vydavateľstvo Alfa, Bratislava 1983, učebnica Ivan, J.: Matematika II, Vydavateľstvo Alfa, Bratislava 1985, učebnica Velichová, D.: Konštrukčná geometria, Vydavateľstvo STU, Bratislava 2003, učebnica Velichová, D.: Geometrické modelovanie - matematické základy, Vydavateľstvo STU, Bratislava 2006, monografia