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:

Introduction

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.


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Let’s get started

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


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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.


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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.


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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.


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


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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.


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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.


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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.


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