Course Information 2DM60 Wiskunde II (Mathematics II, code 2DM60)

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1 Course Information 2DM60 Wiskunde II (Mathematics II, code 2DM60) Responsible lecturer: dr. ir. R. Duits (office: MF 5.071a/Gemini 2.110, tel: 2859/3037) Instructor I: ir. Tom Dela Haije (office: MF , tel: 2261) Instructor II: Ylona Meeuwenberg Instructor III: dr. ir. Sjoerd Rienstra Instructor IV: Christian Vleugels Curriculum: Major course in 2 nd year (1 st quartile) curriculum BMT and WMT (not obligatory) Number of credits: 5 Study points (i.e. 5 x 28 = 140 hours) Course Material: -Elementary Linear Algebra, Bernard Kolman and Divid R.Hill, 9 th edition, Pearson, Prentice Hall. -Calculus a Complete Course, R.A. Adams and C.Essex, 7 th edition, Pearson, Canada. Course Setup: 7 lectures (2 x 45 min) 7 instructions (2 x 45 min) 8 th week is a catch-up week where a test-examination will be done during the lecture hours, and personal questions on this test-examination can be asked during the instruction hours. Required knowledge: Students starting this course are expected to have passed the course Calculus (B-variant, e.g. 2WBB0) in 1 th year curriculum BMT and MWT Recommended follow-up courses (not part of 2DM60): Mechanica (K. Oomes) major course in 2 nd year (2 nd quartile) curriculum BMT and WMT Transport fysica (F. van der Vosse) major course in 2 nd year (3 th quartile) curriculum BMT and WMT Note: These courses include follow up material of much more in-depth multi-variable analysis.

2 Contents: This mathematical course covers 3 topics: Complex numbers: Calculus and solving equations with complex numbers (Euler s formula and polar representations of complex numbers, solving polynomial equations, formula of de Moivre, description of subsets in C, principal roots of complex numbers). (Calculus Adams: Appendix I&II ) Linear Algebra: (Matrices, solving systems of linear equations, vector spaces, inner-product spaces, basis, linear maps, null space and range of a linear map, expressing linear maps into matrices with a given basis on domain and range, basis transformations, determinants, eigenvalues and eigenvectors). (Elementary Linear Algebra, B. Kolman and D. R.Hill 9 th ed, chapters , , , , 4.9, 5.3, 5.4*, 5.6*, 6.1, 6.2, 6.3, 7, and appendix B*) *strongly recommended for self-study but not part of exams Multivariable Analysis: (Functions from R^n to R^m and their derivatives, directional derivatives, optimization on unbounded domain and bounded domain, two-fold integration) (Calculus Adams 7 th ed.: chapters 12.6, 12.7, 13.1, 13.3, 14.1, 14.2, 14.4) Learning Objectives: After completing this course students are able to: 1. Compute with complex numbers, solve basic polynomial equations, apply basic geometry in the complex plane. 2. Compute with matrices and solve linear systems. 3. Know when a linear system has a (unique) solution. 4. Know the difference between a matrix representation of a linear map and the linear map. Set up a matrix representation of a given linear map w.r.t. given basis on domain and given basis on range. 5. Compute the null space and range of a linear map from R^n to R^m. 6. Apply basis transformations. 7. Diagonalize a square matrix A (provided that A can be diagonalized, for example: if A is symmetric).

3 8. Understand the geometric meaning of a linear map from a vector space into itself, understand the geometric meaning of an eigenvector, and understand why a matrix becomes diagonal when represented in a basis of eigenvectors. 9. Compute the derivative of a function from R^n to R^m. 10. Compute the directional derivative of a real-valued multivariate function. 11. Optimize a real-valued function of two variables on R^ Optimize a real-valued function of two variables on a subset of R^2 with piecewise smooth boundary, via Euler-Lagrange. 13. Compute two-fold integrals in Cartesian and polar coordinates. Popular Summary: This major course in the second year of the BMT and WMT curriculum provides a mathematical basis for (bio)medical engineering students. After this course students will be equipped with profound linear algebra skills and basic multi-variable analysis skills. Although the course does pay attention to mathematical understanding of formal concepts, it primarily focuses on direct computation skills that can be widely applied in biomedical engineering. Within the Bachelor College design this course consists of 3 parts: 1. Complex numbers (1 week). 2. Linear algebra (4 weeks). 3. Basic Multivariate analysis (2 weeks). Exams and Grading: Instead of instruction at 7 th -8 th hour 25 th of September aud.8 there will be a 2 hour intermediate test (written exam) starting from 15:45h. We follow the standardized Bachelor College rules for examination: Let 0.0 X 10.0 denote the result of this intermediate test at 25 th of September. Let 1.0 Y 10.0 denote the result of the final written examination. (for time, date, and location of final written examination see ) If Y 5.0 then the final grade is obtained by 0.3 X Y rounded off to an integer valued final grade G. If G 6 the student has passed the course 2DM60. If Y < 5.0 and 0.3 X Y 5.5 then the final grade G = 5 and the student has not passed the course 2DM60. If Y < 5.0 and 0.3 X Y < 5.5 then the final grade G 5 is obtained by rounding off 0.3 X Y to an integer value and the student has not passed the course 2DM60.

4 Answers to most of the problems (odd numbers) can be found at the end of the books, practice (intermediate) exams, will gradually appear on study-web. Always check study-web before attending the next instruction. Precise scheduling: see owinfo.tue.nl Go to vakinformatie and enter 2DM60. Instruction Scheme NOTE: regarding below, exercises indicated in blue are recommended to be done (max 15 minutes) on blackboard by instructors. Week 1a: Know how to apply basic calculus with complex numbers, understand and apply Euler s formula for complex numbers, de Moivre, know how to describe sets in complex plane. Book Adams 7 th edition, App I Ex. 1, 3, 5, 7, 11, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 40, 41, 47 Week 1b: Deal with (principle) roots, know how to solve polynomial equations, know what simple mappings from complex plane to complex plane actually do geometrically in complex plane, description of subsets in complex plane Book Adams 7 th edition, App I Ex. 51, 53, 55 Book Adams 7 th edition, App II Ex. 3, 4, 17, 27, 29, 33, 35 Week 2a:

5 Solve linear systems, know the concept of linear combinations, know when a square matrix is invertible and know how to compute the inverse matrix for 2x2 invertible matrices, know how to compute with matrices, know what (anti)symmetric matrices are, know how to write a given matrix as a sum of a symmetric and anti-symmetric matrix, deal with computations with inverse matrices. Book Kolman 9 h edition, Ch: 1.1, Ex. 1, 3, 34, Ch: 1.2, Ex. 9, 19, Ch: 1.3, Ex. 19, 39, Ch: 1.4, Ex. 3 Ch: 1.5, Ex. 29, 33, 35, 53 Week 2b: Know how to solve linear systems by Gaussian elimination ( rijen vegen ), know precisely when a square linear system has a unique solution or no solution, or infinitely many solutions, know how to compute the inverse matrix for invertible nxn matrices. Know how a matrix induces a linear transformation. Book Kolman 9 h edition, Ch: 2.2, Ex. 1, 5a, 7a, 7d, 15 (apply Gaussian elimination, do not use determinants), 17,18,21,23 Ch: 2.3, Ex. 9c. Week 3a: Know the definition of a determinant. Know how to compute a determinant (of at most 3x3 matrices) via co-factor expansion Book Kolman 9 h edition, Ch: 3.1, Ex. 9, 11, 13, 15 Ch:3.2, Ex. 10, Ch:3.3, Ex. 1

6 Ch:2.2 Ex.15 (now via determinant) Week 3b: Know that vector space are sets equipped with a scalar multiplication operation and an addition operation satisfying certain rules, know when a subset is a linear subspace, know what a linear span of vectors is, know when vectors are linearly independent, know when a linear span of vectors is a basis for a vector space. Book Kolman 9 h edition, Ch: 4.2, Ex.2, 5, 13, Ch:4.3, Ex. 26, 33, 35, Ch:4.4, Ex. 11,12, Ch:4.5, Ex. 1, 3, 24. Week 4a: Intermediate Test 25 th september (See exams and grading), which will be very similar to exercises so far. Week 4b: -Know about inner product spaces, know properties of inner products. - Given a basis in domain and range, you need to know both how to set up a matrix from a linear transformation, and how to set up a linear transformation from a matrix. Book Kolman 9 h edition Ch:5.3, Ex. 1, 8, 11a, 27a, 11, 27, 33, 32, Ch:6.1, Ex. 7a, 7b, 7c 10, 11, 18. Week 5a: -Given a linear operator f, Know how to determine both the null space (= set of vectors x in the domain of f such that f(x)=0)

7 and the range (= set of vectors y in co-domain that can be written as y = f(x), with vector x in domain of f) and again - Given a basis in domain and range, you need to know both how to set up a matrix from a linear transformation, and how to set up a linear transformation from a matrix. Book Kolman 9 h edition Ch:6.2, Ex. 1, 3, 5, 9, 15, Ch:6.3, Ex. 1, 3, 4. Ch:1.4, Ex. 8 (solve geometrically by the linear map and avoid computations with the matrix) Week 5b: -Know the standard procedure to determine the eigenvalues and eigenvectors of a matrix, summarized on p 449 (Kolman 9 th ed.). -Know that an eigenvector is a vector that is being mapped onto a scalar multiple of itself. Book Kolman 9 h edition Ch: 7.1, Ex. 7a, 7b, 8a, 8b, 8d, 9a, 17a, Week 6a: -Understand the standard procedure to determine the eigenvalues and eigenvectors of a matrix, summarized on p 449 (Kolman 9 th ed.). - Know that an orthogonal matrix maps an orthonormal basis to an orthonormal basis -Know that a symmetric matrix has an orthonormal basis of eigenvectors. -Know how to apply a basis transformation on vectors. See Example 4, p.262 (Kolman 9 th ed.).

8 -Know that a single linear transformation can have multiple matrix representations. Know how to apply a basis transformation of matrices to relate such matrix representation. See Example 2, p.411 (Kolman 9 th ed.). Book Kolman 9 h edition Ch: 4.8, Ex. 15, 16, 17 Ch:7.2, Ex. 2, 10, 13. Ch:7.3, Ex. 7 Week 6b: - Master optimization on unbounded domain R^2. Know how to compute the critical points (maximum, minimum, saddle) of a function from R^2 to R. - Know what a partial derivative is and how to compute it. - Know how to compute a directional derivative of a differentiable function f: R^2 R at a given position (a1,a2) in a given direction (v1,v2) - Know how to compute a Jacobian matrix (derivative) of a function f:r^n R^m at a point (a1,..,a_n) Book Adams 7 th edition Ch: 12.6, Ex. 17, 18, 20 Ch: 12.7, Ex. 1, 3, 7, 10, 11, 17 Week 7a: Learning objective: Integration of basic, real-valued functions of two variables using Cartesian coordinates and polar coordinates. Ch: 14.2, Ex. 1, 3, 14, 15 Ch: 14.4, Ex. 1, 3, 9 Week 7b : Learning objective:

9 Master optimization on bounded domain. Know how to optimize a function f: V R, where V is a closed and connected domain with smooth boundary, via the Method of Euler Lagrange. Book Adams 7 th edition Ch: 13.1, Ex. 1, 7 and Ch: 13.3, Ex. 1, 2, 3, 9, 13, 22 Week 8a: Examination training. Test-exam will be available on studyweb.