Distributions

Here, you find my whole video series about Distributions in the correct order and I also help you with some text around the videos. If you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

Part 1 - Motivation and delta functions

Distributions is a video series I started for everyone who is interested in calculating with generalised functions in the realm of partial differential equations or in physics. We will start with an overview and quickly define the new concepts in a rigorous way. It will be helpful if you have some knowledge in the basics of Linear Algebra and Real Analysis. My video courses can help you there if you want to refresh your memory. Use them while watching the videos about distributions whenever you don’t understand a term we use. On the other hand, if you are interested in the topics about distributions, you might also be interested in my Functional Analysis and Measure Theory series. However, they are not necessarily needed to understand this course but still some videos could help you for getting more deep insights into the field of distributions.


YouTube PDF Quiz


Part 2 - Test functions

After this short historic overview, we can go into the mathematical definitions. We start by defining the so-called test functions:


YouTube PDF Quiz


Part 3 - Convergence of test functions

Now you know the space of test functions as a set. However, they have also some important properties, we should discuss. One that is immediately used is a notion of convergence.


YouTube PDF Quiz


Part 4 - Space of distributions

At this point, we are ready to define the important space of distributions. It turns out that this is not so complicated when you are already familiar with linear maps.


YouTube PDF Quiz


Part 5 - Regular distributions

Next, we look at more examples for distributions. Most importantly, we will define so-called regular distributions. This will be the starting point for a lot of definitions that will follow.


YouTube PDF Quiz


Part 6 - The delta distribution is not regular

We have already introduced the famous delta distribution. Now, we can show that this is not a regular distribution.


YouTube PDF Quiz


Part 7 - Distributions form a vector space

Let’s go one step back and look at the space we have defined. Distributions are linear functionals with an additional property: the particular continuity defined above. However, the space still, very naturally, forms a vector space. This means that we can add and scale distributions.


YouTube PDF Quiz


Part 8 - Multiplication with smooth functions

In the same way, we could define the multiplication of distributions. However, it turns out that, in general, the product of two distributions cannot be defined in the useful manner. Therefore, we restrict ourselves to the multiplication of a distribution with a smooth function.


YouTube PDF Quiz


Part 9 - Coordinate Transformation

Since we are working in the n-dimensional space $ \mathbb{R}^n $, coordinate transformations are very natural. The simplest ones, can be described by invertible linear maps. Let’s see how distributions should change under such transformations.


YouTube PDF Quiz