True Meaning of 2nd Derivative Test | Symmetric Tensors, Spectral Theorem, Principal Axes Theorem

[@copiryte9535]

As a 2nd year undergrad this clears up a lot of my confusion about vectors and covectors, and I didn't even know we were using two different types of tensors.

[@HaramGuys]

Have you checked out any videos on covectors/tensors by mathemaniac or eigenchris?

[@copiryte9535]

@@HaramGuys Not yet, but I'm planning to when I learn differential geometry.

[@robertwilsoniii2048]

Stanford math 62cm?

[@HaramGuys]

23:15 eigenvectors dont give a flip about your orientation, absolutely golden!!!

[@xminty77]

holy shit this opened my brain so much such beautiful production too

[@aguyontheinternet8436]

thank you for teaching me that a 3d desmos exists. life will never be the same

[@simdimdim]

Thank you for bringing it to my attention.

[@yds6268]

This was an amazing explanation, especially about the difference between different tensors

[@neoMushroom]

15:32 thats a dopamine molecule!! 15:58 this guy sneaking in GTA casually in a math video!!

[@soupy5890]

This was like, perfectly timed for my studies and coming-up research project, really needed some intuition on some of the presented topics, and I do plan on using this as a source of understanding for my more studious overview of it later.

[@crimsnblade8555]

well put together and covers a lot of ground

[@autumrnk]

I quite enjoy this video, and I understand and appreciate why you made the decision to present tensors as matrices. I have never really seen that representation before, and while it worked for the information in the video, I am so glad that index notation exists. I like matrices as much as the next gal but one must admit that working with indices is so common for a person.

[@yds6268]

Funny, I hate indices. If a tensor is not (or can't be) represented by a matrix, I want nothing to do with it.

[@copiryte9535]

excited for functional analysis videos

[@Khashayarissi-ob4yj]

beautiful video. With luck and more power to you. hoping for more videos.

[@ImMataza]

i really love your videos man

[@dvir-ross]

Amazing content! Thanks

[@05degrees]

17:15 Yeah. Thanks for taking time to talk about this quirk of using matrices for that case!

[@cf6755]

the eigenvectors match up with the princeible directions because it has to not tilt in any direction not parallel so that means that the vector cant tilt it's direction

[@jpphoton]

deep as yet unattained understanding .. must watch many times .. but fair to say you are a Jedi.

[@manfredbogner9799]

Sehr gut, Danke

[@RikoZN]

When I had Calculus, This confused me so much, as they would only show piecewise counterexamples, and at the time, I had "functions" and "analytic functions" confused up (I had no idea what "analytic" meant). I treated piecewise functions not as actual functions, but as "piecewise-functions", like if it were a completely different category.

[@carultch]

Is there a higher-order derivative test that works for functions of 2-variables, in cases where the 2nd derivative test is inconclusive? For instance, consider: z = x^4 + y^6 How would you conclude with a derivative test, that the origin is a local minimum?

[@HaramGuys]

diagonalization of higher rank symmetric tensors, hyperdeterminants, these are all concepts that are fairly niche ideas that are still topics of modern research. but this specific one, supposed 4th derivative test would be inconclusive since the 4th order expansion is x^4. try playing around with quartic functions like x^4 - 6x^2y^2 + y^4 or x^4+120x^3 y+200x^2y^2+120xy^3+y^4 and you would notice that there are 4 "principal axes" in the graph

[@user-mf7li2eb1o]

Wow, how did yt not recomend me til now??

[@user-mf7li2eb1o]

CAN YOU SLOW DOWN, i need to watch it in 0.5 from here…

[@copiryte9535]

wake up babe new epsilon delta video dropped

[@axionbuster]

4:07 in the case of 2x^2 + 3 y^2, since tr H = fxx + fyy, and det H = fxx fyy, what can you say about the polynomial g(x) = x^2 + (tr H)x + det H = (x + fxx)(x + fyy)?

[@HaramGuys]

Eigenvalue eq

[@10-den-see]

0:45 which movie scene is that?

[@EpsilonDeltaMain]

Gravity by Alfonso Cuarón

[@JennaHasm]

​@@EpsilonDeltaMainYou have beautiful video. Can you sell a course on how to make videos like this? You need to minimise the audio staccatos.

[@tonibat59]

What book does this come from? (14:47)

[@EpsilonDeltaMain]

Mechanics of Materials by Beer, Johnston, DeWolf, Mazurek. mechanical engineering major standard

[@tonibat59]

Great ! Thanks for the beautiful vid

[@giannih2014]

At 2:24 you confused the laplacian operator notation with the hessian matrix. You denote it as H only not ▽^2 f , whereas in 3:07 you could replace Tr(H) instead with ▽^2 f, or keep it as is.

[@HaramGuys]

▽^2 for laplacian is a poor notation that has its roots in quaternion algebra. original maxwell's equations was written in quaternions instead of vectors, where (ai+bj+ck)^2 = (-a^2-b^2-c^2). In modern differential operator perspective, ▽^2 should naturally mean second gradient. no sane mathematician uses ▽^2 to mean laplacian, only in sophomore to junior level undergrad physics curriculum stick to such outdated notation.

[@bra1nwave172]

@@HaramGuys Is that true? Then what notation does a well experienced mathematician use for the laplacian and why?

[@HaramGuys]

@@bra1nwave172 standard is Δ in pretty much all of partial differential equations and differential geometry community. but even this has room for confusion with students from elementary calculus education, where the symbol is typically used for a change of a quantity. and other subjects do use Δ in various different ways. but of course its all about the context of the subject. the point i was trying to make was, op calling a notation wrong because that's not what you are used to seeing is just plain ignorant. its no different than how people get offended by innocuous foreign words.

[@Filup]

I am nearing the end of my combined degree which includes a BA in mathematics. I have never seen or used a tensor and it makes me sad. Can you recommend any resources that would be an excellent place to learn more about them?

[@Filup]

@howdy832 Thanks! I really want to learn more physics, but there is so much more than just the math that makes it too time consuming to self study.

[@SharpyWitted]

The most inTENSE test…

[@CarpediemXI]

Finally found home

[@leolacic9442]

Pa evo da kažem da dalje nisam ni htio razmatrati, ali računam da će svatko tko treba kupiti barem 2 Ulaznice. :D A ako nebudu, onda ide Kids Center, Conference Composition, etc. . :D

[@mariotabali2603]

the emperor will show you the true meaning of the second derivative

[@itumekanik]

Actually the stress rectangle is not a body but only denotes the opposite sides of a single cut plane

[@randomchannel-px6ho]

Cant wait for someone to crack spontaneous supersymmetry breaking

[@frogstud]

The Hessian is famously not a tensor

[@HaramGuys]

Thats like saying that sum of the angles of a triangle is famously not 180 degrees.

[@wargreymon2024]

Griffith is famously not an apstotle either 😭

[@etto487]

​​@@HaramGuys the hessian does not transform like a tensor under change of variable. It beave like a tensor only with linear transformation: Hij = d/dx^i (df/dx^j ) If we change the variables: y^i=g^i(x^j) We get: Hij= d/dx^i ((dy^k/dx^j) (df/dy^k))= (d^2(y^k)/(dx^idx^j)) (df/dy^k) + (dy^r/dx^i)(dy^k/dx^j)(d^2(f)/dy^rdy^k)

[@etto487]

​@@HaramGuys its not a tensor, it doesnt transform like a tensor under change of variables. Take: Hij= d^2(f)/dx^idx^j Change the variables and see what you get

[@HaramGuys]

@@etto487 all about context. Its a "tensor" under linear transformation in each of its respective tangent space around the critical points. What it is not is a "tensor field" in a general curvilinear change of coordinates, which is easily fixed with covariant derivative

[@usermlgbzzcnm]

Fourth comment!