Manifolds 25 | Differential (Example)

📝 Find more here: tbsom.de/s/mf
👍 Support the channel on Steady: steadyhq.com/e...
Other possibilities here: tbsom.de/sp
You can also support me via PayPal: paypal.me/brig...
Or via Ko-fi: ko-fi.com/theb...
Or via Patreon: / bsom
Or via other methods: thebrightsideo...
Join this channel on KZbin: / @brightsideofmaths
💬 Access to the community forum: thebrightsideo...
🕚 Early access for videos: thebrightsideo...
❓ FAQ: thebrightsideo...
🛠️ What tools do you use: thebrightsideo...
Please consider to support me if this video was helpful such that I can continue to produce them :)
Each supporter gets access to the additional material. If you need more information, just send me an email: tbsom.de/s/mail
Watch the whole video series about Manifolds and download PDF versions and quizzes: tbsom.de/s/mf
Supporting me via Steady is the best option for me and you. Please consider choosing a supporter package here: tbsom.de/s/sub...
🌙 There is also a dark mode version of this video: • Manifolds 25 | Differe...
🔆 There is also a bright mode version of this video: • Manifolds 25 | Differe...
🔆 To find the KZbin-Playlist, click here for the bright version: • Manifolds
🌙 And click here for the dark version of the playlist: • Manifolds [dark]
🙏 Thanks to all supporters! They are mentioned in the credits of the video :)
This is my video series about Manifolds where we start with topology, talk about differential forms and integration on manifolds, and end with the famous Stoke's theorem. I hope that it will help everyone who wants to learn about it.
#Manifolds
#Mathematics
#Differential
#LearnMath
#Stokes
#calculus
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[@wes9627]

Here is an interesting application of differential manifolds in its simplest form. An ideal kinematic mechanism has a mobility number, m, equal to its independent Degrees of Freedom (DoF), which is also the number of orthonormal constraint manifold surface tangent vectors. Let n by m matrix N represent coordinates of these m orthonormal vectors. The n entries in the jth column of N represent direction cosines of the n mechanism joint variables relative to the jth constraint manifold tangent direction, and the sum of squares of these n direction cosines is 1, the full unit mobility in the corresponding orthogonal tangent direction on the manifold surface. In a similar manner, the m entries in the ith row of N represent direction cosines of the ith joint variable relative to all orthonormal constraint manifold tangent vectors. The sum of squares of these m direction cosines is m_i, the ith joint variable's effective mobility number tangent to the mechanism's constraint manifold surface. The sum of all n m_i joint mobility numbers is exactly m. This indicates that a kinematic mechanism's mobility, m, is exactly distributed among its n joint variables according to their mobility numbers, m_i. The matrix version of the Implicit Function Theorem states that (n-m) dependent joint variables may be expressed as functions of the remaining m independent joint variables, provided an (n-m) by (n-m) Jacobian matrix is invertible. The theorem does not say how to obtain a suitable partitioning of the variables into dependent and independent sets, only that it can be done. This is where the mobility numbers, m_i, come into play. In general, 0 ≤ m_i < 1; the closer m_i is to 1 the more independent the variable is, and the closer m_i is to 0, the more dependent the variable is. So it is safe to set (n-m) variables as dependent, whose mobility numbers are the smallest, and set the remaining m variables as independent, whose m_i are the largest. In general, the higher a joint variable's mobility number, m_i, the easier it will be to move and control the kinematic mechanism through that DoF. Mobility numbers are related to mechanical advantage, where the higher an input's mechanical advantage, the easier it will be to move and control the kinematic mechanism through that DoF.

[@Agus-of6rh]

I don't get the equallity of the right at 2:01

[@Agus-of6rh]

oh nevermind, you clearly explain it ten seconds before 😅.

[@Adsgjdkcis]

I think the example at 9:00 is wrong, because the chart described for the target S1 mapping S1 to a real interval is not a homeomorphism!

[@brightsideofmaths]

Why not? It's not mapping the whole S1 :)

[@Adsgjdkcis]

​@@brightsideofmathsah, you're right, of course! I didn't pay enough attention and missed that you omitted a point from S1!

[@Adsgjdkcis]

@@brightsideofmaths thanks for making the best videos on differential geometry I've found btw. You've covered some topics (eg tangent spaces) far better in 10-20 minutes than I've seen others do in 1-2 hours!

[@brightsideofmaths]

@@Adsgjdkcis Thanks a lot :) And thanks for your support!