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.
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.
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 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.
First, tangents are no more just simple line attached to the "surface", since this is not possible in all cases. Tangents are now Derivations / velocity vectors of curves. A simple example: The cartesian coordinates have the coordinate chart φ(x, y) = (x, y). The y-axis is now the curve γ(t) = (0,t) and its velocity at t=0 is γ'(0) = (0,1), which is what we normally also interpret as the y direction. But in another chart ψ(r, θ) = (r cos θ, r sin θ), this curve γ has a totally different meaning: it's the radial component and its direction depends on the position on the manifold. So tangents have two important components: the position and the direction. And the basis of this vector space are ... the partial derivatives ;-) For our regular cartesian coordinates, the basis would be {∂/∂x , ∂/∂y} and for the polar coordinates {∂/∂r , ∂/∂θ}. The change of basis is now the jacobian of the transition map (θ ο φ^{-1})(x,y). Maps and charts are specifically chosen names for this, since real charts on paper behave the same way in the overlapping area. The creativity didn't end here, the collection of charts, that fit the whole manifold, is called an Atlas. A vector field now has coordinates and basis vectors. The Vector Field V(x, y) = (x, y) in R2 points outwards from the origin, but we assumed a basis, when writing it as the components of the vector (x,y). Explicitly written, this would be V(x,y) = x · ∂/∂x + y · ∂/∂y. The same Vector field also would be V(r, θ) = r · ∂/∂r using the polar coordinates. Derivatives are like directions at a specific point (which I omitted here), the long version would be " ∂/∂x|_p " the p.deriv of x at p. I agree, this is a bit of a mess, when starting diff. geo. So I rewrite it a little bit: V(a, b) = a ∂/∂x|_(a,b) + b ∂/∂y|_(a,b) . In Words: The vector field at point (a,b) points in the direction a times the direction x at point (a,b) ... plus ... b times the direction y at point (a,b). The example X = ∂/∂x + x · ∂/∂y is just the Vector field X(x,y) = (1, x) in cartesian coordinates. But the Vector field X acts as u said, as a differential operator on a scalar field on the manifold. The next thing to note here is the Dual Space of V. This are linear function from the Vector Space V to its field (real manifolds, this are the real numbers). In finite dimensions, there exists a natural dual basis. For our example: if { ∂x, ∂y } is the basis for our Tangent space at point p, its dual basis is defined to be { dx, dy }, with dx(∂x) = 1 , dx(∂y) = 0 , dy(∂x) = 0, dy(∂x) = 1, also known as the kronecker delta. Differentials (or 1-forms) measure the corresponding components amount in a vector. So it's just natural that the answer to "How much does ∂x point towards ∂x?" is 1 and "how much does ∂y point to ∂x?" is 0 (in the cartesian basis!). The Tangent Space at p is typically denoted as T_p M and its dual space T*_p M
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.
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.
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.
@@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
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.
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.
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.
The animation of the stereographic projection... is... for me... at least... harder to visualize than having the sphere tangent to tangent to Cartesian plane, at the south pole of the sphere. That might be harder to animate?
I am a structural engineer. I only needed to take vector calculus at uni. The general form of stokes theorem wasn’t a part of our course. It is very satisfying to see this introduction with animations. Only one issue. Why the audio is panned to the left?
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
