oh my god, your videos are so unbelievably amazing it's unreal. I'm currently learning from Spivak's Calculus on Manifolds and holy crap your explanations are so enlightening. It makes all the tricky proof questions in Spivak so much easier knowing the correct intuition ahead of time.
Your points about k-form and k-form fields was something i always wondered and the part about k-forms and differential k-forms was also something that made it very difficult to understand in other contexts. So glad you made this video! It helped me very much
Welcome Back 00:40 Motivation: Applications of Differential Forms 01:15 Where Are We Going Next? 03:00 Recap: Exterior Algebra 04:26 Recap: k-Forms 06:06 Exterior Calculus: Flat vs. Curved Spaces Differential k-Forms 07:31 Review: Vector vs. Vector Field 08:09 Differential Form 09:40 Differential 0-Form 11:23 Differential 1-Form 12:23 Vector Field vs. Differential 1-Form 12:58 Applying a Differential 1-Form to a Vector Field 14:45 Differential 2-Forms 16:45 Pointwise Operations on Differential k-Forms Differential k-Forms in Coordinates 19:39 Basis Vector Fields 21:33 Basis Expansion of Vector Fields 24:17 Bases for Vector Fields and Differential 1-forms 25:55 Coordinate Bases as Derivatives 27:23 Coordinate Notation-Further Apologies 28:04 Example: Hodge Star of Differential 1-form 33:08 Example: Wedge of Differential 1-Forms 36:26 Volume Form / Differential n-form 39:15 Applying a Differential 1-Form to a Vector Field 42:35 Differential Forms in R^n - Summary 43:20 Exterior Algebra & Differential Forms-Summary 44:48 Where Are We Going Next?
After thinking of differential 2-forms for years and drawing ugly sketches, your pictures for them make me so damn happy. Ugh. How easily extensible is your code for generating them? I read in your FAQ that you build your viz very ad-hoc and with care.
I believe in this case I plotted the 2-forms in R^3 using Mathematica, then exported to Illustrator to make some adjustments. But this feels like something you could write a nice interactive tool for pretty easily (e.g., in three.js). Just provide a function that returns the coefficients of the 2-form in the standard basis. To draw, sample this function and draw little parallelograms according to these coefficients. The challenge (as with vector field visualization) is to give people a good set of tools to "slice" through the 3D volume and see what's going on.
Really interesting! I wonder how much it would help to visualise vector direction with red and green channels for the x and y axes respectively, instead of arrows. Colour multiplication is very intuitive.
Can you re-write every math discovery ever made with sensible and consistent naming conventions? Jk but seriously your clarifications on notation + connections between fields are hugely overlooked by a lot of material and its really time consuming when doing cross-domain research to double back on a concept to make sure an assumption about a small notation variation doesn't actually mean something else and derail later. If git had existed in the 18th century I swear we would all have flying cars and live on the moon by now.
Actually never mind, they probably would've invented AGILE development next and Euler would've spent half his life in stand-ups and sprint planning and estimating whether or not a theorem is worth 3 or 5 story points or if they should just switch to estimating hours since they have to update a burndown chart every day anyway. Scrum master reminding everyone at retrospective that Riemann's Zeta story still isn't meeting the definition of done but aint no one got time for the backlog because everything seems to be working good enough. If Atlassian had existed back then we would have been set back at least 100 years, or equivalently 40 story points at least
So basically, differential k-forms are used to prod and compare physical space with a hypothesis about space at some point of interest, like for example to see if space is curved by some hypothesized amount around some location in space. I would assume from this lecture we might use trial and error in creating different differential forms with different angles to prod space. Or, maybe we use partial derivatives for this in order to measure the orthogonal distance between a tangent plane and a service as you move further away from the point of interest in order to assess the degree of curvature from that point around that point along some distance away from that point for all points. And this is basically a measure of correlation between the surface and the differential form (field).
33:08 Okay, im still obviously missing something about the hodge star. Applying linearity, sure thats fine. But i thought the defining characteristic was that when you wedge it with the original form, you get the "standard basis form" but in this example, wouldn't you get something like 1-2x+2x^2 times the "standard basis form". Not really sure where my breakdown is happening
41:52 "Measuring a vector field with a differential 1-form gives us a scalar function, and at each point the value of that scalar function tells us how well [the vector field] lines up with [the differential 1-form]" I'm not so sure of this. I mean, the resulting scalar function in this example is independent of y, which implies that along any y-axis the "measure of alignment" of the vector field and the 1-form shouldn't change. However, at the top edge the vector field and the 1-form are both pointing "to the right" (well aligned), whereas at the bottom edge the vector field and the 1-form have an angle anywhere from 45° to 90° (not aligned at all). Wouldn't it be more accurate to say that the 1-form does not measure alignment with the vector field, but instead measures change of the vector field along the direction of the 1-form? More concretely, the 1-form a=x dx measures how a vector field changes when you take a small step into the dx direction (and then scales that measurement by x).
I disagree that basis of vector fields shouldn't be thought of as derivatives: that's what they algebraically are. When we do determinants to find area at each point, the calculation is specified by the partial derivatives as unit coordinates in the tangent space. This then becomes the jacobian matrix, and its determinant representa the oriented area (relative to the point of focus and a standard euclidean coordinate reference frame at each point (including slope).
i dont get it, why at ~19:00 alpha(X,Y) makes sense? shouldnt we have alpha^beta(X,Y)? alpha is 1-dimensional "ruler", while X,Y is two dimensions. I thought its necessary to have the same dimensions, no? what am i missing? can you point me to mm:ss in some lecture so i can rewatch?
