Admission requirements
Prior knowledge of Analyse 2 (NA) en Lineare Algebra 1 (NA)
Description
This course introduces you to topics from mathematical analysis that you will need to succeed as a physicist. The material taught will come up repeatedly in future courses in the undergraduate Physics and/or Astronomy programmes. The theory and a few examples will be covered in the lectures, but it is up to you to work on the weekly practice problems to cement your understanding of the covered topics.
We’ll start by covering the basics of ODEs, with a focus on linear ODEs. In learning about their solutions, we’ll also discover how functions can be viewed as vectors and why this is useful for determining how many solutions exist.
Following this topic, we’ll move on to PDEs. At a first glance, they seem like a natural extension of our discussion on ODEs, but after seeing some elementary techniques of solving them in ‘easy’ cases, our limitations will force us to digress into Fourier series as a way of solving PDEs. This in turn will lead us to discover the all-important Fourier transform. We’ll spend quite some time on its properties and solving PDEs using it before using it to study particularly important cases such as the Wave Equation. We’ll end our discussion on differential equations by looking at Sturm-Liouville equations and Green’s functions.
Our last topic is a departure from the previous two; we’ll turn our attention to complex analysis. While a vast topic on its own, our focus will lie primarily in looking at elementary complex functions and how they are similar/differ from their real counterparts, looking at complex differentiation and hence also at the Cauchy-Riemann equations (thereby also looking at what it means to be holomorphic), and finally we’ll study complex line integrals, culminating in the proof and applications of Cauchy’s theorem.
Course objectives
At the end of the course, students will be able to analyse a wide-range of mathematical problems on ODEs, PDEs, or complex derivatives/line integrals, conclude which solution method is the most applicable, and after solving it, verify that their solution is correct.
Students will be able to apply their knowledge of linear algebra to solve constant coefficient linear differential equations.
Students will be able to construct the Fourier series of a suitable function and check the convergence of the Fourier series (both pointwise and L^2 convergence).
Students will be able to proof various properties of the Fourier transform (such as properties of its inverse, convolution, etc.) and be able to directly calculate the Fourier transform of suitable functions.
Students will be able construct solutions to various boundary value problems via the Fourier transform, including in important cases such as the Wave Equation.
Students will be able to apply the Cauchy integral theorems to solve complex line integrals, and apply the Cauchy-Riemann equations to be able to verify whether a function is complex differentiable or not.
Schedule
The timetables are available through My Timetable (see the button in the upper right corner).
Teaching method
Lectures and exercises classes.
For more information see Brightspace
Assesment method
The final grade is determined by a weighted combination of:
homework
midterm written examination with short questions
final written examination with short questions*
Note the midterm and the homework are optional and have no retake opportunities, but the examination is mandatory and has a retake a few weeks after the exam. Your grade is the highest of 4 schemes:
100% exam
70% exam, 30% midterm
90% exam, 10% homework
60% exam, 30% midterm, 10% homework
Resit, review & feedback
Examinations are held twice during the academic year for each component offered in that academic year. Midterm tests cannot be retaken. The Board of Examiners determines the manner of resit for practical assignments.
For review and feedback, see Brightspace.
Reading list
We’ll be using free course notes, available electronically via Brightspace. Depending on interest, copies of the notes can also be printed and bought within the first few weeks.
Registration
Enrolment through MyStudyMap (button in upper right corner) is mandatory. General information about course and exam enrolment is available on the website.
Contact
For substantive questions, contact the lecturer(s) (listed in the right information bar).
Remarks
This course is taught in English, but you can submit your assignments / write your exams in Dutch.
Students who are doing a double bachelor Natuurkunde en Wiskunde or Sterrenkunde and Wiskunde, take the course Analyse 3 with Mathematics.
Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.