How to choose Mathematics courses? Overview

Second semester Engineering and Science Mathematics comes in two flavors that differ only in a small number of topics. 120102 ESM 2A – Linear Algebra,...

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How to choose Mathematics courses? Overview: 1.) Introduction 2.) Basic Courses 2.a) Choice between Regular, Advanced and intermediate variants 2.b) Calculus 2.c) Laboratory courses 3.) Curriculum Requirements 4.) Advanced Courses 5.) Guided Research 6.) A Note on Transdisciplinary Courses 1.) Introduction The main guiding principles for the choice of mathematics courses should be that your mathematical knowledge and abilities are challenged with the courses you take, that the certain basic courses should be included within your mathematics studies at IUB and, last but not least, that you are going to satisfy all the necessary requirements for graduation. Usually, the last point should not be a problem if you keep in mind the first two. Note that for all options presented in the following 'intermediate' choices/combinations are possible due to the flexible structure of the curriculum. Also this document is intended as a guide not as a reference so for exact regulations please consult the corresponding handbook. 2.) Basic Courses As an incoming student you have the following choices, which strongly depend on the previous mathematics education you had at high school or elsewhere. The first option is to start with the so-called 'regular track', which includes the following core courses in the first two semesters: 110101 & 110102 General Mathematics and Computational Science I / II 120101 & 120111 Engineering and Science Mathematics (ESM) 1 & 2 110102 Mathematics Lab Units: Symbolic Software Lab 110112 Mathematics Lab Units: Numerical Software Lab For a more detailed discussion of the options for ESM courses and laboratory units, see sections below. Alternatively, you can follow the 'honors' or 'advanced' version, which includes the following courses: 100211 & 100212 Analysis I & II 100221 & 100222 Linear Algebra I & II 100291 & 100292 Perspectives of Mathematics I & II Mathematics Lab Courses as above Intermediate combinations are always possible, e.g. General Mathematics and

Computational Science I / II and Linear Algebra I / II in your first year. If Analysis and Linear Algebra are not challenging enough for you, then you can also start with even higher level classes. In this case it is essential that you consult with mathematics faculty for advice and details. The main challenge, being presented with the two variants above and all possible intermediate combinations, is to find the combination which suits you best. 2.a) Choice between Regular, Advanced and Intermediate Variants If one or more of the following applies to you, you are likely ready for one of the advanced versions of the curriculum. 1) Have you taken special/advanced mathematics courses at high school level (e.g. for the German Abitur this would be “Leistungskurs Mathematik”)? 2) Have you participated in mathematics competitions for high school students or have you had special training classes for these competitions? 3) Have you attended a high school specialized in mathematics or the natural sciences? 4) Go to the Advanced Placement Exam (this is an exam offered at the beginning of the semester in Fall) for 'Engineering and Science Mathematics I' and see how you do and if you can pass it. If none of the above applies, you should consider the 'regular' track. If you are unsure, keep in mind that it is usually easier to switch from a course like 'Analysis' to 'General Mathematics and Computational Science' than the other way around if the level of difficulty does not suit you – the classes you are taking should be difficult and challenging, but you must be able to cope with the weekly workload. You should use the first weeks before the drop/add deadline wisely to find out what is suitable for you; if in doubt, talk to the respective instructors and to your academic advisor. 2.b) Calculus Another important decision, if you are taking Engineering and Science Mathematics, is which of the two classes offered in the first semester is best. 120101 ESM 1A – Single Variable Calculus 120111 ESM 1B – Multivariable Calculus and ODEs Very roughly speaking, ESM 1A covers similar topics as Analysis I, and ESM 1B similar topics as Analysis II. However, the point of view, the style of presentation, and the learning goals are completely different. Engineering and Science Mathematics will provide you with methods for solving problems, points out certain application domains and provides training to improve your speed and reliability in performing certain calculations. Analysis, on the other hand, builds the theory behind these methods, and provides training in formal and rigorous thinking. Both kind of skills are important for the working mathematician. Experience has shown that people can learn the more technical aspects “on the job” provided their understanding of the fundamental principles is strong. Thus, it is safe to go soft on Engineering and Science Mathematics provided you treat the more advanced courses seriously and you are

not afraid of filling in missing bits and pieces on your own. Conversely, taking the more advanced ESM 1B, possibly in addition to ESM 1A – both courses can be taken in the same semester, is advisable if – –

you are planning on taking advanced classes in Physics, Computational Science, or Electrical Engineering you wish to maximize your exposure to Mathematics in your first year in order to be optimally prepared for the more abstract courses in the second year.

Second semester Engineering and Science Mathematics comes in two flavors that differ only in a small number of topics. 120102 ESM 2A – Linear Algebra, Probability, Statistics 120112 ESM 2B – Linear Algebra, Fourier, Probability For Mathematics majors, both options are good. It more or less depends on your other courses, which choice is better. In general one could say that if you are more interested or take courses in life sciences, then ESM 2A is the better choice. If you are taking electrical engineering courses, you will definitely benefit from ESM 2B. Otherwise it does not really matter too much. 2.c) Mathematics Laboratory Courses In addition to taking the two Mathematics Lab Units (Numerical Software and Symbolic Software), you will benefit from taking the Computer Science Lab Units at some point during your studies at IUB (i.e. not necessarily during the first year). These courses are introductions to programming in C and C++, respectively. Generally it is not a bad idea for a mathematics major to have some exposure to programming although you can also get along without it. All other Laboratory Courses are not directly related to your major and you should usually take those, which are associated to the electives from the School of Engineering of Science (SES). For example if you take as electives General Physics and General Computer Science in addition to General Mathematics and Computational Science then you normally also take the associated labs to these “General”-courses. The Mathematics Lab Units may be replaced by other lab units offered in the SES. Please consult with Mathematics faculty or with your academic advisor whether this makes sense for you. 3. Curriculum Requirements Beyond the initial core of the curriculum you also have to think about the requirements for the Bachelor of Science in Mathematics. Each course you take carries a certain number of credits, which are accessible with the course catalog / course description or via CampusNet. In general these include as minimum requirements: – –

120 credits overall 96 credits in the School of Engineering and Science (SES)



24 credits from transdisciplinary courses including * 3-5 University Study Courses * 3-5 courses in the School of Humanities and Social Sciences (SHSS)

The 96 credits from the SES must include at least 57 credits of mathematics classes and and least – – –

15 credits at first-year level or above 18 credits at second-year level or above 24 credits at third-year level or above

Note that higher-level (or graduate-level) classes may be used to satisfy lower level credit requirements. The level of the courses is currently indicated also in the course number which has 6 digits, The first three for mathematics, mathematics service or computational science classes are 100, 110 or 120. The fourth (!) digit carries the information on the level: • • • •

xyz1.. = first year level xyz2.. = second year level xyz3.. = third year level xyz4.. = graduate level course.

The last two digits characterize the topic of the course. The important case you should be aware of is that courses from Computational Sciences (120...) count only toward the graduation requirements in mathematics provided the second to last digit of the course number is less than or equal to 5. 4. Advanced Courses If you have not taken Linear Algebra and/or Analysis during your first year, you should now take these classes during your second year and if you are interested also Perspectives of Mathematics. If time permits you are encouraged to take also classes described below as more advanced courses. For the advanced courses this document is restricted to a list with prerequisites for each course. Regarding the prerequisites angular brackets “[...]” will denote classes which are not absolutely essential but helpful, so you might be able to take the course without this knowledge, but this might entail additional work. This table should help you to make some long-term planning if you are interested in specific topics, but are not sure which path to take there. Note that some advanced courses are not offered every year, but these courses are often offered in alternating years. Look at the course catalogs of previous years for more information. “[x]” for x=1,2,3,4,5 will mark the level of the course. The [5] denotes specialized topics classes for graduate students, which are only suitable in very rare occasions for undergraduates. For prerequisites the table works as a tree, e.g. a course A depends on B & C and D depends on A, then D also depends on B & C in general. It might be possible to take a course without some prerequisites, but in this case you MUST talk to the instructor of

record about this possibility. Note also that some advanced Computational Science courses might be of interest for mathematics majors as well (see list below). There is the possibility to take further Engineering and Science Mathematics courses beyond ESM I / II. In general this is not advisable for mathematics majors due to overlapping content with other advanced classes, but if you are interested in Physics or Applied Mathematics, these courses might be an option for you. The table below shows that some advanced courses might be more essential than others, but still you have the free choice, although some knowledge taught in a course might be required if you go to graduate school. These courses include, but are not limited to: Introductory Algebra, Integration and Manifolds, Introductory Complex Analysis, a course in differential equations (ODE or PDE) and Introductory Topology. It is selfevident that you should know all the material of Analysis I / II and Linear Algebra I / II very well anyway if you proceed mathematics studies anywhere. Course #

Name

Prerequisites

100321 Introductory Algebra

Linear Algebra I

100341 Introductory Topology

Linear Algebra I, Analysis I

100351 Introductory Geometry

Integration and Manifolds, Introductory Algebra

100331 Introductory Number Theory

Analysis I, Introductory Algebra

100311 Integration and Manifolds

Analysis I / II

100312 Introductory Complex Analysis

Analysis I / II

100362 Partial Differential Equations

Analysis I / II

100361 ODE / Dynamical Systems

Analysis I / II, Linear Algebra I

Discrete Mathematics

Linear Algebra I, [Introductory Algebra]

100381 Probability

Analysis I / II

100411 Real Analysis

Integration and Manifolds

100471 Functional Analysis

Real Analysis

100421 Algebra

Introductory Algebra

100442 Algebraic Topology

Introductory Algebra, Introductory Topology, [Algebra]

100422 Advanced Algebra

Introductory Algebra, [Algebra]

100431 Lie Groups

Introductory Algebra, Integration and Manifolds

100451 Differential Geometry

Integration and Manifolds, Introductory Geometry

100455 Algebraic Geometry

Algebra, Introductory Geometry

100462 Ergodic Theory

Real Analysis, ODE/Dynamical Systems

100431 Number Theory

Introductory Number Theory

100461 Dynamical Systems

ODE / Dynamical Systems

100412 Complex Analysis

Introductory Complex Analysis

100491 Graduate Research Seminar

permission of instructor

100511 Topics in Complex Analysis

permission of instructor

Course #

Name

Prerequisites

100512 Topics in Analysis & Dynamics

permission of instructor

100541 Homology & Dynamics

permission of instructor

Note that the above table is not exhaustive and subject to change, i.e. there might be other courses offered not included in the above. For this case pay attention to the published course catalog and/or the IUB Intranet [CampusNet]. Also some Computational Science Courses [as described above] qualify as Mathematics courses, so make sure that you also take a look at the similar list for CPS as well. 5. Guided Research During the last year of undergraduate studies students usually take two semesters of Guided Research. This project oriented work will ultimately lead to their Bachelor's thesis. The Guided Research courses count as usual third-year mathematics classes towards your degree. The course is structured in such a way that you have a faculty adviser for the topic you are interested in and who plans with you all the details. Note that you are should take advantage of the fact that IUB's mathematics faculty although not excessively large covers a very large number of research areas and you should simply talk to a faculty member if you are interested. Also the choice of guided research adviser is not limited to mathematics faculty only, but some associated IUB faculty e.g. from Physics or Computer Science might supervise your Guided Research. In any case you MUST talk to the instructor of record of Guided Research for the choice of topics and/or advisers. 6. A note on transdisciplinary courses You are strongly encouraged to broaden you horizon and simply take those University Study Courses and electives from the SHSS that you find interesting. Since you might not directly be able to judge from the course title or a topic list, which transdisciplinary course might be suitable it might be advisable to attend the first lectures of several courses to decide on a course and drop the others before the Drop/Add deadline Although not many courses offered in the SHSS have relations to the mathematical component of your undergraduate studies, some should be mentioned as they might turn out to be useful. First it is self-evident that if you are interested in Mathematical Finance then you should consider taking some of the Economics courses offered by the SHSS. If you are interested in the application of mathematics in social sciences, the most prominent classes to attend would be Statistics I and Statistics II.