Fall Semester 2020

Calculus and Elements of Linear Algebra I (JTMS-09)


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This module is the first in a sequence introducing mathematical methods at university level in a form relevant for study and research in the quantitative natural sciences, engineering, Computer Science, and Mathematics. The emphasis in these modules lies in training operational skills and recognizing mathematical structures in a problem context. Mathematical rigor is used where appropriate. However, a full axiomatic treatment of the subject is done in the first-year modules Analysis I and Linear Algebra.

Contact Information:
Instructor:Marcel Oliver
Office hours:  Tu 11:00, We 10:00 in Research I, 107
TA:Denida Blloshmi, d.blloshmi@jacobs-university.de
Bishoy Kaleny, b.kaleny@jacobs-university.de

Time and Place:
Lectures:  Mo, We 14:15-15:30, on Teams

Notes: CLP is a good modern Calculus textbook with relatively comprehensive coverage of all Calculus topics, except that some of the preliminary concepts (factorization of polynomials, fundamental theorem of algebra, inequalities, graphs of equations) are not or insufficiently covered, see reading recommendations in the syllabus below. The book is open source and freely available for browsing on the web and as a printable PDF download. I am trying to base the class on this book to the extent possible.

MW is an extremely clear and well-written classic and definitely one of my favorites. Its biggest drawback is that it follows a "late transcendentals" concept, i.e., trigonometric functions, exponentials, and logarithms are introduced only after integration, which limits the variety of available examples at the time when the key ideas are first introduced. However, for self-study or review, this may not matter so much. This book is also freely and legally available as a scanned PDF download.

The book by Edwards and Penney is a good mainstream Calculus textbook. It used to be the default reading for many years at Jacobs, so there are still quite a few paper copies in the IRC.

RHB is a comprehensive textbook on mathematical methods that covers a great variety of concepts, far beyond first-year Calculus. The introductory chapters are somewhat terse and assume familiarity with high-school Calculus without presenting a systematic introduction from scratch. Yet, the choice of examples and focus on advanced topics make this book a lasting resource.

For the short Linear Algebra part at the end, the class mostly follows RHB except for the details of the process of Gaussian elimination and matrix inversion, which are covered by separate class notes. LADW is good background reading; the book is more advanced than what we are doing, but follows the same style and philosophy. Gaussian elimination is explained in the same way as here (except for a differen sign convention).


Tentative class Schedule (subject to change!)

02/09/2020: Introduction; natural numbers, integers, rational, real numbers (CLP1 0.1, additional reading CLP1 0.2, 0.3)
07/09/2020: Complex numbers (RHB 3.1, 3.2);
polynomials, factorization, fundamental theorem of Algebra (RHB 1.1)
09/09/2020: Long division (see, e.g., CLP2, 1.10.1 or the Wikipedia article on Polynomial Long Division for a detailed explanation in North American notation; I personally prefer the more self-explanatory continental European tradition, see the the Wikipedia article on Long Division for a comparison of notation);
intervals, inequalities (MW R2)
14/09/2020: Equations, graphs, functions (domain, range, inverse); brief review of elementary examples (trigonometric functions, exponential, logarithm) (CLP1 0.4-0.6, MW R4-R6)
16/09/2020: Limits of functions (CLP1 1.3, start with 1.4)
21/09/2020: Asymptotes (parts of CLP1 1.5);
squeeze law and basic trigonometric limit (CLP1 1.4 ctd., also see CLP1 2.8 for the basic trigonometric limit)
23/09/2020: Continuity, types of discontinuities, intermediate value theorem (CLP1 1.6);
continuous functions attain their extreme values on closed, bounded intervals (CLP1 3.5.2)
28/09/2020: Differentiability, derivative, slope, secant, tangent (CLP1 2.1-2.3)
30/09/2020: Rules and theorems for differentiation, power rule, product rule (CLP1 2.4, 2.5 for advanded background reading, 2.6 for examples);
chain rule (CLP1 2.9)
05/10/2020: Derivatives of exp, log and trigonometric functions (parts of CLP1 2.7, 2.8, 2.10);
implicit differentiation and derivative of inverse functions (CLP1 2.11, also MW 2.4);
curvature of a function as an example (cf. RHB 2.1.9, but implicit differentiation makes this much easier)
07/10/2020: Rolle's theorem, mean value theorem, critical point, maximum/minimum of a function (parts of CLP1 2.13)
12/10/2020: First and second derivative tests, inflection points, isoperimetric problems (CLP1 2.13 ctd., 2.14, parts of 3.5)
14/10/2020: Curve sketching (CLP1 3.6)
19/10/2020: Anti-derivative, indefinite integral, integration by substitution, integration by parts (CLP1 4.1, CLP2 1.4, 1.7 - indefinite integral versions only)
21/10/2020: Approximations to area, partitions of intervals, Riemann sum, definite integral, Fundamental Theorem of Calculus (CLP2 1.1-1.3)
26/10/2020: Properties of the integral (CLP2 1.1.2), Area between curves (CLP2 1.5), integration of rational functions by partial fractions (CLP2 1.10)
28/10/2020: Review
02/11/2020: Mock Midterm Exam
04/11/2020: Improper integrals (CLP2 1.12), Applications of integration I: Volume computations (CLP2 1.6)
09/11/2020: Applications of integration II: Work (CLP2 2.1), Separable first-order differential equations (CLP2 2.4)
11/11/2020: Separable first-order differential equations ctd., in particular logistic growth and a brief account of the Volterra-Lotka model (CLP2 2.4, MW 8.5)
16/11/2020: Basic analytic geometry: vectors and scalars in Euclidean space, vector arithmetic, scalar product, cross product (selected topics from RHB Chapter 7)
18/11/2020: Equations for lines and planes (RHB 7.7), vector spaces, linear independence, span, basis, dimension (RHB 8.1, 8.1.1)
23/11/2020: Linear transformations, coordinates, representation of linear transformations by matrices (RHB 8.2, 8.3)
25/11/2020: Matrix arithmetic, matrix multiplication, transpose and Hermitian conjugate (RHB 8.4, 8.6, 8.7), systems of linear equations, Gaussian elimination (handout)
30/11/2020: Structure of the solution set, range, kernel, rank-nullity theorem (RHB 8.18.1), matrix inverse (handout)
02/12/2020: Properties of matrix inverse, appication of matrix inverse: change of basis (RHB 8.15 first half)
07/12/2020: Review for final exam
14/12/2020: Final Exam

Last modified: 2020/12/19
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/fall2020/CO-583/index.html
Marcel Oliver (m.oliver@jacobs-university.de)