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.
Timetable
Schedule
For detailed information go to Timetable in Brightspace
In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.
Additionally, you can easily link MyTimetable to a calendar app on your phone, and schedule changes will be automatically updated in your calendar. You can also choose to receive email notifications about schedule changes. You can enable notifications in Settings after logging in.
Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.
Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.
Mode of Instruction
Lectures and exercises classes.
For more information see Brightspace
Assessment 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
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
As a student, you are responsible for enrolling on time through MyStudyMap.
In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.
There are two enrolment periods per year:
Enrolment for the fall opens in July
Enrolment for the spring opens in December
See this page for more information about deadlines and enrolling for courses and exams.
Note:
It is mandatory to enrol for all activities of a course that you are going to follow.
Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.
Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.
Contact
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.