1/22/2026 6:01 pm
on a sexier note, two things i've been satisfied with lately. 1) a lovely little connection between fixed point iteration and chaos theory! if you tilt your head so that you align with the line y=x, you can use the same logic of stable/ unstable roots you would with a velocity type problem :-} 2) not so much an answer, but i've been delving into function spaces and spaces in general that we play in for calculus, and there's fascinating stuff going on. trying to coax out a nice general working understanding of the structure that we give/ don't give and how inner product spaces and normed (?) spaces enter the discussion. it will be a thoroughly satisfying cultivation of this little seed which i get to plant at the start of the book eeee

1/22/2026
i've slowly been working on stuff. .. ... i'm sort of in a weird place right now. i'm reaching my hands out into the dark and properly touching and feeling things. i've been wanting to do it for a while, but i wasn't even in the right room before, but now i'm here. it's slow going; work between the lines of everything else i'm doing. i have a lot of thoughts.
i guess in general i've been thinking about the unique spot i'm in as someone who wants (? do i really want to? in a way i do) to write curriculum.. i'm an artist, an actor, a dancer, an engineer, a mathematician, a queer trans man, a coder. i'd like to think i've done a lot (save for sports, but i enjoy that as well through lion dance/ wushu and some casual stuff with friends) and that feels really livewire to me at the heart of why i'm struggling with curriculum stuff. the whole reason i write this rn is that i met a guy who finally, FINALLY resonated with even a fraction of how i'm trying to write hypercalc. and it's great, i really love talking with him and he seems to be really excited to talk to me too. but another professor that i work with, who i respect very deeply as the closest thing i've ever had to an aengineer professor, met him and seemed to not think very highly of him/ his research. and... i don't know. there's something to all this that's very raw, like my knowledge is the only credibility i have, and i'm constantly under fire because i don't have a formal education in it (the graduate stuff), but i'm just trying my best, and i think this guy is trying his best too. i'm not sure if people want to realize it but none of us are fully intimately math itself. like, to truly be a disciple of math so much os that you are able to take on his namesake as a halfgod or whatever, i don't even know if that's possible. and if you're not at that level and i'm not at that level, all of us are on the same level to me. an incomplete picture is incomplete, that's the reality of it.
it's sad, i guess. i'm used to people dismissing my personal mathematical beliefs, but to turn up your nose at a professor who's doing published research into areas no one else is doing...? and from our position in our stupid ass hand waving shoddily bullshit applied math classes ? i think we ought not to throw a stone. in fact i think we should take that stone and start to make shit. this guy is trying something, and i'm gonna stand with people who are trying, even if it's not perfect. urgh.

11/28/2025 - 1:26 PM
going to be working on fixing all of "differential analysis" for my fluids class (not a fan of the name... but actually, i misjudged it, there isn't really a better way to say it, haha). currently trying to reconcile this velocity field with a zero-curl but a non zero exterior derivative... yeesh.
ok well i was just doing it wrong, but there's something cute about taking the jacobian of the velocity field where the diagonal (trace) denotes if you have continuity and symmetry on the off-diagonal means you have irrotational flow... something something if you have a jacobian that represents a symmetric bilinear form you get a euclidean surface from it perhaps? cause our velocity fields should always satisfy continuity, but they may not be irrotational, if they were then we would be doing symplectic mechanics on the config space ig since we have an anti-symmetric bilinear form that we could do geometry with.. so interesting (i'm just spitballing, it might not be a symplectic form necessarily but since it represents the velocity change.. ah im not sure, that would be like an acceleration and dont have second derivative information in hamiltonian config spaces hmmm hmmmmmmm)
actually i guess we don't even need to have an anti-symmetric bilinear form there, so maybe its just what the hell ever hahaha

10/11/2025 - 11:35 PM
grabbed an empty big lecture hall with my friend and played professor while we tried to geometrically prove the umlaufsatz ^_^ turns out you don't really need forms? well, at least not to prove that we integrate the curvature to find the rotation index! technically we didn't get to proving the umlaufsatz bit which is saying that if you're a simple closed curve (a regular homotopy of S1.. i think..) then your rotation index is always +- 1.. difficult. got down to saying something about the perimeter needing to be 2pi but that's really non concrete and we were both unhappy with it . lol! something something degree of the curve maybe. waves hands. we'll get it soon LOL

10/10/2025 - 5:03PM
been a hot second since i worked on proofs LOL im 22 now hi (grins 22ly). anyway my sights are on the umlaufsatz with differential forms next (the forms should not be surprising at this point). i did a really elegant thing for index notation (at least i like to think so) and i'm hoping that translates over to the umlaufsatz too... there was this stupid "wronskian" jumpscare which was absolutely definitely not a wronskian we pulled in the rotation index proof, so i'm gonna untangle that first for everything WRT alpha and i have a feeling it'll illuminate.. something? there's something enticing about this idea of 1D -> 2D where you have to pump up a pushforward using second derivatives in order to lift the space. i'm cooking maybe. i'll have to see though- i really do think pushing forward into new maps is just as easy as pulling back though, so umlaufsatz should be relatively beautiful? the weird thing is that i can only find proofs which discuss degree anyway. grrrrrr. at least with my prof's i have something to sink my teeth into... i don't care about winding numbers, we're trying to prove rotation index regular homotopy bullshit man!! get the winding number OUTTA here !!

9/30/2025 - 7:22PM
made some real progress on the continuity eqn ^_^ i feel like i'm happy with it! rn my preferred intrepretation is density as a pullback of mass into the volume space- there are certain physical maps we can't really interact with (mass, temperature, energy etc) because they shift too much for us to properly work with them, so we need to quantize them in the form of pullbacks into volume space. i want to look into how densities arise more (like especially with composite / anisotropic materials.. would we ever want to design a material or even object with a certain density form? could be interesting) to make sure this interpretation feels Right, but rn it does. there's also this divide between the tope and the quantity which is very interesting and subtle. when we say density, do we mean the full form rhodV, or do we mean the imposed function rho on the volume tope? i think i might start referring to these as the density tope and the density function, but I have to see if that's consistent with burke (who, by the way, is an absolute visionary in my mind, holy shit. that dude was goin places i want to go). there's also this thing about the inner product of topes and how twisted topes inner products with themselves shake out... twisting in general is something i need to study more as well, like can we see symmetries arise from simply pullback of quantities or do we have to prescribe the behavior? i think this would change our ability to use symplectic manifolds or reimannian ones (which. ok common theme but i need to get into those more as well teehee). anyway yeah! fairly happy!

9/28/2025 - 12:20am
working on my continuum equation. extremely interested in the result that rho = dM/dV... you can divide forms? what does that mean? are we embedding them somehow? you can't divide vectors either, unless somehow we mean component wise? ... my solution is .. hehe ^_^ ( definitely not thinking about fucking gongcheng & shu at the same time to see if i unlock any dlc to help solve this )
edit: 12:47am
ok so i came to a place of peace with rho = dm/dv being slightly valid as an existing Thing and not a mathematical exercise only, but i'm still really grappling with what object rho is. the dv needs to stay a dv if we want to hide away our dm in . every time it exists ever HAH (which is so sexy... just hide it under the rug babe dw about it, i love that). so we can't emdy away dV but a) what does it mean to divide a form and b) what does it mean that we're then applying that to a form... urgh. wait. wait wait wait wait waaaait omg. i think this has something to do with the extremely known output of dV holy shit. it's a 1d map. it only ever emdys to a number . wait . waitttttt. ok that doesn't exactly help us in the division department but there might be something special about 1d output maps. that could be so real. i do really like this as an excerise because it's forced me to abandon my preconcieved tope preference (dV is not superior to dM and is actually worse because keeping it around in rho STOPS us from abstracting dM) and also the 1d thing. that seems really exciting.

9/28/2025 - 12:10am
i decided to make a page here so you all can witness my descent into madness. let me do a quick recap for you. in the spring of 2025, i took a course on.. "vector analysis". at the time, i bemoaned the professor. i should probably get down on my hands and knees and thank him deeply, really. even through his extreme tangle, the alluring gleam of differential forms caught my eye. a generalization of calculus. how elegant... i took to them instantly, and became obsessed with their logic. fast forward through a summer of writing the first (?) (perhaps, just "a") textbook on teaching calculus using only differential forms. i completely shed the sticky, heavy armor of .. multivariable calculus, and embraced the wavering instabilities of hypercalculus, its sprawling maps and twirling emdyed fields. thrown into the deep end of translating differential geometry, partial differential equations, and .. extremely elementary fluid mechanics (tch) into the notation, it crumbled beneath my fingertips and it has sent me into a magikdrunk fever. little interests me in my problem sets except a deeper understanding of the physical objects of the universe- symmetrical tensors on sympletic manifolds, 3forms and gelled tangent spaces, star-spangled maps of my own greedy creation. it's the ultimate hedonism and i'm literally addicted. it's really hard to exist in a space where everyone is denying me of the exaltation of these objects at every step of the way (shoutout my fluids prof who i really like and respect saying that people would only like to see surfaces "in class" and that "2d projections are what we use for analysis" as if that's not the most backward logical process i've ever heard), so i made this page. bask in it with me please ^_^ iloveyou