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## Higher Analysis

 Module name (EN): Higher Analysis Degree programme: Mechatronics and Sensor Technology, Master, ASPO 01.10.2011 Module code: MST.HAN Hours per semester week / Teaching method: 2V+1U+1F (4 hours per week) ECTS credits: 5 Semester: according to optional course list Mandatory course: no Language of instruction: German Assessment: Written exam/paper [updated 01.10.2020] Applicability / Curricular relevance: MTM.HAN Mechatronics, Master, ASPO 01.04.2020, optional course, technical MST.HAN Mechatronics and Sensor Technology, Master, ASPO 01.04.2016, optional course, technical MST.HAN Mechatronics and Sensor Technology, Master, ASPO 01.10.2011, optional course, technical Workload: 60 class hours (= 45 clock hours) over a 15-week period.The total student study time is 150 hours (equivalent to 5 ECTS credits).There are therefore 105 hours available for class preparation and follow-up work and exam preparation. Recommended prerequisites (modules): None. Recommended as prerequisite for: Module coordinator: Prof. Dr. Barbara Grabowski Lecturer: Prof. Dr. Barbara Grabowski [updated 07.10.2014] Learning outcomes: After successfully completing this course, students will be able to investigate force and velocity fields and other fields of physics and electrical engineering using vector analysis methods.In addition, they will be able to describe curves and curved surfaces in R2 and R3 parametrically using curvilinear coordinate systems and calculate properties such as lengths, curvatures, areas, volumes and centers of gravity, solve complex applied extreme value problems for functions in several variables with and without constraints, and handle eigenvalues, eigenvectors, and quadrics in practical applications. [updated 01.10.2020] Module content: 1.        Curves as vector-valued functions in a variable1.1 Definition of vector functions and their geometrical meaning 1.2 Differentiation and integration of curves, Jordan curves1.3 Tangent vectors and orientation of a curve1.4 Case studies: Applications2.        Real-value functions in several variables2.1 Definition, surfaces of revolution and planes2.2 The directional derivatives, partial derivatives and their properties2.4 The gradient, tangent plane and total differential2.5 Extreme value search methods with and without auxiliary conditions2.6 Case studies: practical applications3.        Coordinate transformation _ Curvilinear coordinates3.1 The Jacobi matrix and its determinants3.2 Coordinate lines and bases in curvilinear coordinate systems3.3 Spherical, cylindrical and polar coordinates3.4 Multiple integrals and integral transform3.5. Case studies: practical applications4.        Scalar and vector fields4.1 Definitions4.2 Gradient of a scalar field, rotation and divergence of vector fields and their meaning4.3 Potential fields and potential function 4.4 Del and Laplace operator and useful equations - Maxwell´s equations4.5  Line, surface and volume integrals over scalar and vector fields and their physical meaning 4.6. Theorems of Gauss and Stokes4.7 Case studies: applications5.        Eigenvalues and eigenvectors, quadrics5.1 Scalar products and orthogonality 5.2. Orthogonal matrices, orthogonal bases, change between orthogonal bases 5.3. Eigenvalues and eigenvectors, eigenvalue estimation 5.4. EWe and EVe of symmetrical matrices, principal axis transformation (diagonalizability of a matrix) 5.5 Square shapes 5.6 Positive/negative (semi) definite matrices 5.7. Quadrics, normal form in R^2 and R^3 5.8 Case studies: practical applications [updated 01.10.2020] Teaching methods/Media: Projector, smart notebook, lecture notesPC lab: AMSeL [updated 01.10.2020] Recommended or required reading: Papula: Mathematik für Ingenieure und Naturwissenschaftler, Band 1-3, Vieweg 2000.MARSHDEN, TROMBA: Vektoranalysis, Spektrum, 1995.Bourne, Kendall: Vektoranalysis, Teubner, 1966.J.Stoer, R. Bulirsch "Einführung in die Numerische Mathematik I und II", Springer; Auflage: 5. Aufl. 2005 Springer; Auflage: 10., neu bearb. 2007. D. Wille "Repetitorium der Linearen Algebra, Teil 1" Binomi 1997.  D. Wille, M. Holz "Repetitorium der Linearen Algebra, Teil 2"  Binomi Verlag; Auflage 2, 2006. G.Merziger, T. Wirth "Repetitorium der höheren Mathematik" Binomi; Auflage 5, 2006. B.Griese "Übungsbuch zur Linearen Algebra: Aufgaben und Lösungen" Vieweg+Teubner Verlag; Auflage 7, überarb. u. erw. Aufl. 2011. K.Jänich "Lineare Algebra" Springer; 11. Aufl. 2008. 2., korr. Nachdruck 2013.  [updated 01.10.2020]
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