HTML5 and CSS3 Beginner Tutorial 19 Div and Span

>> YOUR LINK HERE: ___ http://youtube.com/watch?v=B2PJh2K-jdU

Part 6: • Lie algebras visualized: why are they... • Can we visualise this algebraic procedure of adding diagonal entries? What is really happening when we add them together? By visualising it, it is possible to almost immediately see how the different properties of trace comes about. • This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. • Files for download: • Go to https://www.mathemaniac.co.uk/download and enter the following password: traceisdiv • The concept of the whole video starts from one line the Wikipedia page about trace, and I am surprised this isn't on YouTube: A related characterization of the trace applies to linear vector fields. Given a matrix A, define a vector field F on R^n by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its divergence div F is a constant function, whose value is equal to tr(A). • Actually, this is one of the last concepts in linear algebra that I really wanted a visualisation for, the other being transpose, but this is already on the channel: • The deeper meaning of matrix transpose • Chapters: • 00:00 Introduction • 00:48 Matrix as vector field • 02:24 Divergence • 04:50 Connection between trace and divergence • 10:12 Trace = sum of eigenvalues • 13:32 Determinant and matrix exponentials • 15:15 Trace is basis-independent • 18:10 Jacobi's formula • Further reading: • Trace (the origin of the whole video): https://en.wikipedia.org/wiki/Trace_(... • Divergence (more qualitative, and subtly different from the video): https://en.wikipedia.org/wiki/Diverge... • Jacobi's formula (more formal proof): https://en.wikipedia.org/wiki/Jacobi%... • Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: • https://forms.gle/QJ29hocF9uQAyZyH6 • If you want to know more interesting Mathematics, stay tuned for the next video! • SUBSCRIBE and see you in the next video! • If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. • Social media: • Facebook: / mathemaniacyt • Instagram: / _mathemaniac_ • Twitter: / mathemaniacyt • Patreon: / mathemaniac (support if you want to and can afford to!) • Merch: https://mathemaniac.myspreadshop.co.uk • Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] • For my contact email, check my About page on a PC. • See you next time!

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