Awesome video! I am not sure how much of it I understood, but it makes me think of how far geometry has progressed since Euclid's times in terms of its abstraction and technical sophistication.
The idea around 1:49 is really smart: instead of compressing the two semi-spheres into 2-D circles, compressing the southern one into a 2-D circle, and then cutting and stretching the northern one onto the same 2-D plane so that the central circle is left as a hole (which is already occupied by the southern). Then since the northern pole is mapped to infinite numbers of points at an infinite distance, only it is not mapped onto the 2-D plane. Thank you for your video.
If you want to hear audio from both sides on your computer: turn on the Mono Audio setting in your desktop settings, which then uses equal output for both sides of your headphones/speakers.
Oh dang, this was such a good higj level overview. Really appreciated your visual and the cadence of your explanations. Susceibed and excited to see what elese you do with this channel
A point on the animations--k-forms should be thought of as paralellopipeds, not simplices. Consider ||v wedge w||---it is the vol of the paralellopiped, which is twice the vol of the simplex.
@@quantumsoul3495 I'd argue you kinda do, because it reminds you that the differential form is not commutative. But yes, if you're not planning on changing order of integration and just stick to canonical orientation, than it's not necessary
@@MessedUpSystem Yes I think it's clearer for instructional video. But when it's integrals, you just pick the canonical orientation en.m.wikipedia.org/wiki/Differential_form#Integration
This is differential topology, not differential geometry. Stokes theorem is definitely cool and used from time to time in diff geom, but defining the exterior derivative does not require the existence of a metric
I would like to ask, how is topology and geometry different ? Edit: A Google search basically said “Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus.
@Natsu tsuu That is not quite true. Geometry says a square and a triangle are different in some respects, while Topology says they are equivalent in other respects. There is no conflict.
@@goldplatealuminum1102 topology is is concerned with things invariant under purely topological notions (continuity, homeomorphisms, homotopy, isotopy, etc) while geometry is generally concerned with metric structures. Many metric structures are topologically equivalent but geometrically distinct. Stokes' theorem does not depend on the choice of a metric tensor but does require smoothness (or at least C^1) and is thus considered part of differential topology.
@@goldplatealuminum1102 Geometry concerns the "rigid properties" of shapes and spaces (examples: angle, length, area, volume, and curvature). Topology concerns the "flexible properties" of shapes and spaces (examples: dimension of the space, the number of 1d holes, the number of 2d holes, etc.).
You voice only comes out of the left channel! Also, consider getting a stereo lapel mic and using beam-forming, this will result in much better audio-quality.
I'm so glad that there are brilliant people out there who make life easier for the rest of us. If progress was dependent on me, we'd still be wearing loin cloths and using spears to hunt our food.
Your due East line shouldn't be curved, because travelling due east or due west are not paths that fall on a Great Circle; they are generally called Rhumb lines or Loxodromes
I'm confused why that means "shouldn't be curved". Any "line" on the surface of a sphere will appear curved from most viewpoints. A great circle looks perfectly straight if you're viewing it from directly above, but not from any other perspective. This is because the viewpoint (and view direction vector) is outside the circle, but in the same plane as that circle. To know that the great circle curves, the viewer would need to measure distances to it in a few directions and see that those distances are inconsistent with a straight line. With the "in the same plane" definition of "above", your Rhumb lines will also look perfectly straight - but again, will look curved from any viewpoint (or with any view direction vector) outside the plane of that Rhumb line. In fact if you make the fairly conventional assumption that the center of the sphere is in the same plane as the viewpoint and view direction vector, great circles are the ONLY "lines" that can ever look perfectly straight - Rhumb lines cannot be completely inside that plane, and thus cannot appear perfectly straight. The arrows shown aren't remotely the correct curves, but they also aren't remotely correct distances either - they're described as 1,000km each, the first apparently takes the person from the south pole to a point a little north of the equator, but the distance from the south pole to the equator is approx. 10,000km. In other words it's not meant to be an accurate diagram, only to give the basic idea.
I was completely lost at 6:30 with line X = ∂/∂x + x ∂/dy. It looks like differential operator created as combination of multiplication, addition and differentiation named as X. But I don't understand how it related to visualized vector field or any vector field. The operator after application to some function of two variables gives gives just function, not two functions of vector components. Also voice description become ambiguous because "X" and "x" sound the same. And I don't understand everything after, because it based on this. For example, the next slide shows equality ∂/∂x(x ∂/dy) = ∂/dy But ∂/∂x(x ∂/dy) equals to (∂/dy) + ∂/∂x(∂/dy) by differentiation of multiplication... And how it related to the vector field is still non-clear. Next slide, some "forms" things are used without explanation what the forms are... And I lost again. The "dx" for me is "hieroglyph in the integral notation to tell what variable is used for integration" or "hieroglyph in the differentiation operator to tell what variable is used for differentiation" with some vague relation to infinitesimally small piece in the definition of integral and differentiation by limits. Or related to intuitive understanding of integral as "sum of small pieces dx" or differentiation as "division by small number dx", but it is intuitive, not formal, and I am not sure this "small piece" is "form". So, maybe this video is useful to those who already know the subject to recall the whole subject, but I couldn't extract any knowledge after 6:30 because lot of unknown or implicit assumptions. For example, it hard to tell is empty space between letters means application of operator or is it multiplication when you are not in the context, because you want to learn the context. Still, it was very interesting and useful part before 6:30 to see how arbitrary manifolds are tied to functions and researched by local "maps" of these functions. Thanks for great work anyway, I think if you consider that some implicit things are not evident for newcomers, it will make great educational video for newcomers too.
I'm sure you're fully aware of this now. Nice explanations and nice visualizations, but you have a mono microphone plugged into one ear and you're screaming into that ear because the microphone is bad.
please what do you mean on taking "high values and low values" ( of the explanation on differential forms, if it is perpendicular or parallel to dy) how do you define high or low, that was the only thing that was not clear to me, thank you for the video!
A covector takes vectors as inputs and outputs numbers. A 1-form, such as dy, is a covector field, you have a covector assigned to each point in the space. If I input a vector field to dy, then at each point, the covector gives me the number which is the y component of the vector at that point. So by “high values and low values” he just means greater and lesser real numbers.
0:20 or 1+1/(2kpi) (for k being a positive integer) miles under the north pole. He walks up one mile, walks k times along the north pole, then walks down to where he started 2:!3 what about pooints in the enighborhood of the north pole?
You can perform the projection again using the South Pole as reference. Now you have two maps (known as coordinate charts) that cover the entire globe.
@@qilinxue989 And for a journey from one pole to another, I guess you can make the "jump" at the equator, where both maps map it to the same point, so there's no discontinuity. Very nice.
@@mujtabaalam5907 Basically! Note that the “jump” can happen smoothly everywhere that both charts covers. If f and g are maps that take points from the manifold (sphere) and outputs values in flat space (R2), then you can define the transition function to be f(g^-1(x)) which takes points in R2 and map it to points to R2. This is a smooth function, so it allows you to transition from one map to the other map.
My left ear says this was an amazing video! it's so excited to explain it to my right one tomorrow.
Lol
😂😂😂
I thought my right airpod was broken... im glad you posted it
I watched it first time without earbuds and thought it was some kind of differential geometry joke that I didn't get. Now I laughed when I saw this :)
We need global best comment awards.
Flashback to my Tensor Analysis class, taught by a physics professor. This is much better!
Turning on mono audio fixes the audio. Good content!
Big brain
Awesome video! I am not sure how much of it I understood, but it makes me think of how far geometry has progressed since Euclid's times in terms of its abstraction and technical sophistication.
Such a great video with beautiful animations! Thank you Qilin!
Man, you teach a Semester of DG in 15min. You are a genius
Brilliant opening experiment. Got me hooked right away.
The idea around 1:49 is really smart: instead of compressing the two semi-spheres into 2-D circles, compressing the southern one into a 2-D circle, and then cutting and stretching the northern one onto the same 2-D plane so that the central circle is left as a hole (which is already occupied by the southern). Then since the northern pole is mapped to infinite numbers of points at an infinite distance, only it is not mapped onto the 2-D plane. Thank you for your video.
Windows Settings > Accessibility Options > Hearing > Turn on Mono Audio
Cool man, please post more videos...amazing attempt...thanks..
Your video is great! Please consider making more videos.
If you want to hear audio from both sides on your computer: turn on the Mono Audio setting in your desktop settings, which then uses equal output for both sides of your headphones/speakers.
Great video, helped a lot to understand this concept, I’d love to see you cover other subjects!
I sincerely appreciate your work. Thank your for the great insight and inspiration!! 😻😻
Oh dang, this was such a good higj level overview. Really appreciated your visual and the cadence of your explanations. Susceibed and excited to see what elese you do with this channel
My left ear enjoyed this, really cool!!
Great work, great explanations. You gained a subscriber! Hope to see more ^^ ( With stereo ahah )
Great. Keep making such great contents.
This is very helpful! Thank you!!
amazing work, keep it up!
A point on the animations--k-forms should be thought of as paralellopipeds, not simplices. Consider ||v wedge w||---it is the vol of the paralellopiped, which is twice the vol of the simplex.
You’re right, that’s my bad!
Great job.
Top Tier stuf. Hope you make more videos
YES!!! SOMEONE THAT DOESN'T OMMIT THE WEDGE PRODUCT INSIDE THE INTEGRAL!
Given canonical orientation, you don't need the wedge right ?
@@quantumsoul3495 I'd argue you kinda do, because it reminds you that the differential form is not commutative. But yes, if you're not planning on changing order of integration and just stick to canonical orientation, than it's not necessary
@@MessedUpSystem Yes I think it's clearer for instructional video. But when it's integrals, you just pick the canonical orientation en.m.wikipedia.org/wiki/Differential_form#Integration
哇塞，這個視頻製作的真的超精良誒！
Awesome vidéo !!
This is differential topology, not differential geometry. Stokes theorem is definitely cool and used from time to time in diff geom, but defining the exterior derivative does not require the existence of a metric
You’re right, but the course name was differential geometry so I had it there for consistency.
I would like to ask, how is topology and geometry different ?
Edit: A Google search basically said “Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus.
@Natsu tsuu That is not quite true. Geometry says a square and a triangle are different in some respects, while Topology says they are equivalent in other respects. There is no conflict.
@@goldplatealuminum1102 topology is is concerned with things invariant under purely topological notions (continuity, homeomorphisms, homotopy, isotopy, etc) while geometry is generally concerned with metric structures. Many metric structures are topologically equivalent but geometrically distinct. Stokes' theorem does not depend on the choice of a metric tensor but does require smoothness (or at least C^1) and is thus considered part of differential topology.
@@goldplatealuminum1102 Geometry concerns the "rigid properties" of shapes and spaces (examples: angle, length, area, volume, and curvature). Topology concerns the "flexible properties" of shapes and spaces (examples: dimension of the space, the number of 1d holes, the number of 2d holes, etc.).
You voice only comes out of the left channel!
Also, consider getting a stereo lapel mic and using beam-forming, this will result in much better audio-quality.
Hi, I love the graphs. What tool did you use to create them? Thanks!
Bro this was a good content. But can you fix the audio please?
I'm so glad that there are brilliant people out there who make life easier for the rest of us. If progress was dependent on me, we'd still be wearing loin cloths and using spears to hunt our food.
Thank you!
Thank you for tNice tutorials, tNice tutorials was a huge help.
Your due East line shouldn't be curved, because travelling due east or due west are not paths that fall on a Great Circle; they are generally called Rhumb lines or Loxodromes
I'm confused why that means "shouldn't be curved". Any "line" on the surface of a sphere will appear curved from most viewpoints. A great circle looks perfectly straight if you're viewing it from directly above, but not from any other perspective. This is because the viewpoint (and view direction vector) is outside the circle, but in the same plane as that circle. To know that the great circle curves, the viewer would need to measure distances to it in a few directions and see that those distances are inconsistent with a straight line. With the "in the same plane" definition of "above", your Rhumb lines will also look perfectly straight - but again, will look curved from any viewpoint (or with any view direction vector) outside the plane of that Rhumb line. In fact if you make the fairly conventional assumption that the center of the sphere is in the same plane as the viewpoint and view direction vector, great circles are the ONLY "lines" that can ever look perfectly straight - Rhumb lines cannot be completely inside that plane, and thus cannot appear perfectly straight.
The arrows shown aren't remotely the correct curves, but they also aren't remotely correct distances either - they're described as 1,000km each, the first apparently takes the person from the south pole to a point a little north of the equator, but the distance from the south pole to the equator is approx. 10,000km. In other words it's not meant to be an accurate diagram, only to give the basic idea.
Cool video; I subbed
Good Job it helped a lot thanks
I was completely lost at 6:30 with line X = ∂/∂x + x ∂/dy. It looks like differential operator created as combination of multiplication, addition and differentiation named as X. But I don't understand how it related to visualized vector field or any vector field. The operator after application to some function of two variables gives gives just function, not two functions of vector components. Also voice description become ambiguous because "X" and "x" sound the same. And I don't understand everything after, because it based on this. For example, the next slide shows equality
∂/∂x(x ∂/dy) = ∂/dy
But ∂/∂x(x ∂/dy) equals to (∂/dy) + ∂/∂x(∂/dy) by differentiation of multiplication... And how it related to the vector field is still non-clear.
Next slide, some "forms" things are used without explanation what the forms are... And I lost again. The "dx" for me is "hieroglyph in the integral notation to tell what variable is used for integration" or "hieroglyph in the differentiation operator to tell what variable is used for differentiation" with some vague relation to infinitesimally small piece in the definition of integral and differentiation by limits. Or related to intuitive understanding of integral as "sum of small pieces dx" or differentiation as "division by small number dx", but it is intuitive, not formal, and I am not sure this "small piece" is "form".
So, maybe this video is useful to those who already know the subject to recall the whole subject, but I couldn't extract any knowledge after 6:30 because lot of unknown or implicit assumptions. For example, it hard to tell is empty space between letters means application of operator or is it multiplication when you are not in the context, because you want to learn the context.
Still, it was very interesting and useful part before 6:30 to see how arbitrary manifolds are tied to functions and researched by local "maps" of these functions. Thanks for great work anyway, I think if you consider that some implicit things are not evident for newcomers, it will make great educational video for newcomers too.
Pl suggest me how to learn differential geometry and tensor I I v poor in maths
More videos needed
Nice video, thank you for explaining this. One caveat: it is really distracting to only hear your voice on the left channel!
There are actually an infinite number of places where you can travel north 1000, east, 1000, and south 1000 miles to end up in the same place.
TY TY SO MUCH!!!
Thank you sir
Please make more
Impressive
How can you make a video with such good animations but the audio is abjectly horrible?
Are the sample softs there when you open the software or do you have to download them from sowhere
😂😂😂
I'm sure you're fully aware of this now. Nice explanations and nice visualizations, but you have a mono microphone plugged into one ear and you're screaming into that ear because the microphone is bad.
nice video, king
wow fancy seeing u here !
please what do you mean on taking "high values and low values" ( of the explanation on differential forms, if it is perpendicular or parallel to dy) how do you define high or low, that was the only thing that was not clear to me,
thank you for the video!
A covector takes vectors as inputs and outputs numbers. A 1-form, such as dy, is a covector field, you have a covector assigned to each point in the space. If I input a vector field to dy, then at each point, the covector gives me the number which is the y component of the vector at that point. So by “high values and low values” he just means greater and lesser real numbers.
Nice video but you need a better mic
thanks
are maps manifolds ?
The audio quality is so good! Where did you get the microphone?
Couldn’t agree more. I need that microphone asap
Agreed, have you found out where he got the micorphone yet? I'm still waiting
great
another one video banger and left 😂😂
thanks, Thanos, glad you're getting into soft instead of...well...
Stereo is the future.
Unfortunately, the video has audio in only one channel
0:20 or 1+1/(2kpi) (for k being a positive integer) miles under the north pole. He walks up one mile, walks k times along the north pole, then walks down to where he started
2:!3 what about pooints in the enighborhood of the north pole?
You can perform the projection again using the South Pole as reference. Now you have two maps (known as coordinate charts) that cover the entire globe.
@@qilinxue989 And for a journey from one pole to another, I guess you can make the "jump" at the equator, where both maps map it to the same point, so there's no discontinuity. Very nice.
@@mujtabaalam5907 Basically! Note that the “jump” can happen smoothly everywhere that both charts covers. If f and g are maps that take points from the manifold (sphere) and outputs values in flat space (R2), then you can define the transition function to be f(g^-1(x)) which takes points in R2 and map it to points to R2. This is a smooth function, so it allows you to transition from one map to the other map.
Plz reupload with good audio
Thanks so much for tNice tutorials video, it helped so much!
不错
code of video?
All youtube videos are now 3Blue1Brown animations.
cultured me clicked because of the thumbnail 😕
Accidentally, My left earlobe is not working , so to me, this video has no audio
Heyy buddy
man can you fix the audio problems? it is extremely frustrating to have only one ear heard.
👀
#SoME2
Wasn't intended for that actually, just had to pump out this video quick for my final project lmao
@@qilinxue989 oh
He told manifold is nothing but surface and start using manifold everywhere. Its confusing
Tybg
for the love of god convert your audio to mono
I love you
the audio in the left earphone is triggering
Please fix the audio
Wait.
Fine exposition...but terrible sound quality, i.e., far too much 'noise resonance'.
My na is Michael to
Please fix the audio and reupload
just say no homo then its fine
Why it was saved in just left ear? It becomes very tiresome!
Now get a degree in theoretical physics and you might become bigger than Einstein
türkçe altyazılı olmalı😭
Voice was not loud.
You need a better mic or to back up or something. You keep peaking.
This was just way too fast paced for me
Audio issues, good content but hard to listen to. You should fix and repost. It hurts your brand.
Bad and sad