your Calculus teacher lied* to you

[@mizarimomochi4378]

Reminds me of the joke Mathematicians: dy/dx is not a fraction! LaTeX: \frac{dy}{dx} Mathematicians: 😮

[@scottgoodson8295]

\frac{\mathrm{d}y}{\mathrm{d}x}

[@jacobwikowsky3535]

the mere fact that you can legit make people laugh at a joke where the punchline is raw text from LaTeX makes me laugh even harder

[@BrianGriffin83]

@@scottgoodson8295 Ah, I see you're a man of culture as well

[@kilian8250]

@@scottgoodson8295that would be physicists, not mathematicians.

[@wernerviehhauser94]

​@@kilian8250 erm.... no.

[@ingobojak5666]

Physics: If it looks like a fraction, swims like a fraction, and quacks like a fraction, then it probably is a fraction. Maths: Fine. I define it to be a fraction.

[@Jessica-kv2ob]

Now that's a good thumbnail. As a physicist I clicked immediately 😂

[@yruijnaosd6645]

Same here! XD

[@dukenukem9770]

Same…

[@kevchar177]

How to attract physicist fr fr

[@jacobwikowsky3535]

there's dozens of us!

[@kpritc1024]

Yep, it got me too!

[@larzcaetano]

“Prime” notation isn’t Newton’s. Newton’s notation is done by writing dots over the function. Lagrange’s notation is the one you talked about in the video, which is using primes (f’, f’’, f’’’).

[@idontwantahandlethough]

Joseph-Louis Lagrange is my favorite mathematician because whenever someone says his name, the ZZ Top riff from the song _La Grange_ plays in my head. Probably makes him sound way more badass than he actually was. I can't ever look him up because I don't want to find out that he actually didn't have sunglasses and a cool moustache.

[@larzcaetano]

@@idontwantahandlethough I think he was definitely more badass than we can actually think of. That’s a nice way to picture him, as ZZ Top is amazing!

[@johnholmes912]

dots are only used when differentiating w.r.t . time

[@peterbrockway5990]

Leibnitz thought of dy/dx as the ratio of two actual (although infinitesimal) things. Abraham Robinson showed (1961, much later!) how you could add to the real numbers a value whose magnitude was less than the magnitude of any nonzero real and come up with something that was as consistent as real number arithmetic; another _model_ of the real numbers if you will. His approach provided the rigor that had been lacking in Leibnitz' intuitively appealing approach and turned out to be widely applicable to mathematical systems with the right (rather simple) order properties.

[@karelvanderwalt3625]

sometimes i feel we are still struggling with the axiomatization of analysis

[@matsogren7143]

Still, even with Robinson's approach, the derivative is not really a fraction. Rather, it is the common standard part of a set of nonstandard fractions.

[@gabrielgauchez9435]

@@karelvanderwalt3625 well im struggling with natural axiomatization

[@darksecret965]

Hyperreals?

[@johnholmes912]

Real numbers are anything but real

[@MatthijsvanDuin]

It's worth pointing out that partial derivatives are kind of horrible syntax, since it obscures the fact that the meaning of ∂/∂x depends not just on x but your entire coordinate system! For example, say we define a := x² + 3 y then da = 2 x dx + 3 dy and thus ∂a/∂x = 2 x. However, suppose we then define z := x + y, and use (x,z) as our new coordinate system, then a = x² - 3 x + 3 z and therefore da = (2 x - 3) dx + 3 dz. But now the conclusion is that ∂a/∂x = 2 x - 3 instead of 2 x, even though we have not changed the definition of a nor x. Working with differentials themselves does not have this issue: the two expressions for da are indeed equal to each other.

[@archismanrudra9336]

Yeah, comes up in statistical physics, eg, maxwell relations

[@dantemancini911]

You wrote _da = (2 x - 3) dx + 3 dz_ No: da=(2x-3)dx+3(dx+dy)=2xdx+3dy because z is not an indipendent variable, since it depends on x and y. If you claim that (x,z) is a coordinate system, then z cannot depend on x. You can set a new coordinate system with x=u and y=-u+z, then a= x² + 3 y= u² - 3u+3z. Hence da=2 x dx + 3 dy=(2u-3)du+3dz. Note that (2x,3) and (2u-3, 3) are covariant vectors, so their components transform one another as the differentials of a(x,y) with respect to the two systems

[@MatthijsvanDuin]

@@dantemancini911 which one(s) you consider dependent or independent is of no importance here. the equation z=x+y (whether it's a definition of z or simply an equality that holds) implies dz=dx+dy and therefore 2xdx+3dy=(2x-3)dx+3dz. The way to properly formalize this is by treating x, y, z, and a all as functions from a smooth manifold X (which happens to be isomorphic to the 2d plane) to ℝ, these are also known as the 0-forms on X, and dx, dy, dz, and da are all 1-forms on X. A coordinate system for X is just any pair of functions that give an isomorphism from X to ℝ², i.e. uniquely represents each point of X by a pair of real numbers. (x,y), (x,z), and (y,z) are all valid coordinate systems, but (a,y) isn't. If (x,y) is a coordinate system then { dx, dy } will be a basis for the space of 1-forms. Of course most smooth manifolds aren't globally isomorphic to ℝⁿ, but you can also do similar things locally near a point.

[@Heulerado]

There is a notation that specifies the entire coordinate system: (∂/∂y)ₓ,ₜ meaning "Vary y while keeping x and t constant". I've mostly seen it used in thermodynamics.

[@Tzizenorec]

@@Heulerado It sounds like (∂/∂y)ₓ,ₜ(z) would be equivalent to (dz/dy)|(dx=0,dt=0) [where "|" means "given that"].

[@KJParadise]

I think of dy/dx as "the ratio of an infinitesimal piece of y to an infinitesimal piece of x", which then makes them formal things that you can add, subtract, multiply and divide.

[@major__kong]

Man, I have an advanced degree in engineering heavy in math. But I didn't understand that at all. All I know is bprp is going to be up late tonight updating his " dy/dx isn't a fraction video" :-)

[@jbragg33]

What do you by an 'advanced degree' ? In France we don't have that concept so I'm curious

[@pseudolullus]

Tensors are used in some engineering branches, but this video is more about tensor calculus/differential geometry stuff.

[@MusicEngineeer]

Differential forms (aka exterior calculus) it what comes after vector calculus. The latter works only in 3D whereas the former works in nD. You'll get a proper n-dimensional generalization of curl there. ...and the amazing generalized Stokes theorem - which is, in my humble opinion, the most beautiful theorem in all of math. ...even though I also don't really understand exterior calculus very well - but just enough to appreciate the magnificence of the theorem. All the classical vector calc integral theorems, some complex calc integral theorems, the fundamental theorem of calc - they all fall out as special cases of that one single (and simple!) theorem. ...well "simple" in the sense of being short to write down. The content is quite dense, though

[@FrederickTabares-kj1pl]

Bachelor's is 4 years and isn't considered an advanced degree, Master's is 2 years plus a Bachelor's and is considered an advanced degree, Doctorate is 2 years plus a Master's and is an even more advanced degree. The number of years is a cultural memory idea and doesn't actually correspond to reality for a lot of people. Speaking as an American.

[@major__kong]

@@jbragg33 Master's degree. Between bachelor's and PhD. Heavy in numerical methods and vector calculus for my thesis.

[@foogod4237]

I still distinctly remember being in high school (a long time ago) in a precalculus class, working with derivatives, when I actually solved a math problem by splitting the dy/dx and moving the "dx" to the other side, which made the whole thing much easier to sort out (I forget exactly why). I came up with the correct answer to the problem, but my math teacher marked it as wrong, and just told me "dy/dx isn't a fraction, so you can't do that" (which really annoyed me, because it's a complete hand-wavy answer in the face of an _actual demonstration that it did actually work_ ). Then a couple of years later in college, I took a differential equations class (which is a whole branch of mathematics based around the idea that "dy" and "dx" are separate actual things, and "dy/dx" is, in fact, just a fraction made from those things), and finally had confirmation that my old teacher was just completely full of it.

[@nosuchthing8]

They go into shock over take it to approach zero, thus dividing by zero

[@TonyFisher-lo8hh]

When splitting hairs, a differential form is NOT a derivative. It is shorthand form of writing "delta forms", and it gives the same results.

[@krabbediem]

Isn't it Lagrange notation that uses the prime-symbol. Newton used dots... are those also called 'prime symbols'?

[@jbragg33]

I think you meant Leibniz notation

[@bjornfeuerbacher5514]

@@jbragg33 No, he is right. Leibniz used the notation dy/dx. Newton used dots. Lagrange introduced the prime.

[@aMartianSpy]

^^^^^

[@ScienceTalkwithJimMassa]

@@bjornfeuerbacher5514 Newton called it the flux or fluxion.

[@bjornfeuerbacher5514]

@@ScienceTalkwithJimMassa Yes...? That wasn't the question here. The question was about notation.

[@Stabacs]

I never even learned to work with dx/dy. it always was f‘(x) And then dx randomly appeared in integrals and was never explained. But then we also never learned about partial differential equations…

[@sidviscous5959]

I'm afraid I lied to my calculus teacher when I said I understood what he was doing most of the time . . . .

[@jacemandt]

It's not a fraction, in the sense that "normal" division is an operation performed on two actual real numbers, and dy and dx are not actual numbers. But the reason that the notation dy/dx is so widely used is that when we apply the laws of fractions to it, the result suggests mathematical processes that are still valid. "Separation of variables" actually works to solve some differential equations, for example. Pretending it's a real fraction and applying those laws, in other words, reflects intuitions that are mathematically true, so it's useful to continue pretending it's actually a fraction.

[@ziervhannyozzy]

it's a ratio dude, still fraction

[@MatthijsvanDuin]

Multiplication is defined in way more contexts than just for numbers, and x / y is simply defined as the unique z such that z * y = x, if such a z exists and is unique. When multiplication isn't commutative (i.e. x * y isn't necessarily equal to y * x) such as with matrices, sometimes you also see the syntax y \ x for the unique z such that y * z = x.

[@petervinella5545]

@@jacemandt Separation of variables is a calculation convenience which is justified by the existence of anti derivatives, the chain rule, and integration

[@ExistenceUniversity]

What is a "real number" in your context?

[@quantumsoul3495]

@@ExistenceUniversity an element of the set R

[@FatherGapon-gw6yo]

This whole explanation makes me feel very tense

[@Thesaddestmomentinourlives]

As I recall, dy/dx is not, in fact, a fraction in 'standart calculus', but it is in the theory of differential forms, it really depends on the field

[@Rockyzach88]

I remember mumbling something in my diff eq class about how it behaves algebraically and the professor goes "what was that?" because he didn't hear me. I said "nothing, nvm" because I was too scared to bring it up in front of the whole class. I've heard people comment on that before a few times since.

[@itzakehrenberg3449]

The tangent line to a curve y=f(x) in the plane is the intersection of all (topological) closures of secant pencils through the point. The symbol dx can be any non-zero change in x; the symbol dy the corresponding change in the y values of the tangent line to the curve. The fraction dy/dx is the ratio of two real numbers representing the slope of the tangent line to the curve. Notice that dy differs from ∆y=f(x+dx)-f(x).

[@Chris_5318]

That's the way I learnt it.

[@jbragg33]

As a physicist, there is never a time when dy/dx is not a fraction, and there's nothing you math nerds can do about it ! :D JK, nice video, very clear, thanks. I would actually really like to dig deeper into this topic, but everytime I try the terseness of math textbooks stops me...

[@MatthijsvanDuin]

Except dx and dy also show up in integrals, and not as part of a fraction :D (And yes those are the same objects).

[@MatthijsvanDuin]

Differentials also help to generalize some stuff that may otherwise only be familiar for special cases in 3D, e.g. the gradient, curl, and divergence operators are exactly the exterior derivative operators from 0- to 1-forms, from 1- to 2-forms, and from 2- to 3-forms respectively, but curl and divergence are dependent on your coordinate system and curl is 3D-specific while the exterior derivative does not depend on coordinate system and applies to any manifold of any dimension. Similarly the cross product is a 3D-specific and basis-dependent version of the exterior product, which itself is general and basis-independent. (The reason for the cross product being such a special case is that it relies on identifying the spaces of 1-forms (its arguments) and 2-forms (its result), which is only possible in 3D because both happen to be 3-dimensional in that case). Some more stuff that's simplified and generalized are that ∇·(∇×A)=0 and ∇×(∇f)=0 are both just instances of d^2 = 0, and Stokes' theorem generalizes/simplifies to: the integral of some differential ω on the boundary of X equals the integral of dω on X (see "Generalized Stokes theorem" for the precise statement).

[@FatherGapon-gw6yo]

Thats how we end up with whacked out nonsense like relativity, the big bang, string theory and other drivel

[@johnholmes912]

NOT even wrong; you obviously weren't paying attention in kindergarten

[@toby9999]

​@@FatherGapon-gw6yoNot really.

[@TaladrisKpop]

What allows the division at 15:30? Without a course in abstract algebra, this is pretty much like saying "dy/dx is a fraction because I say so"

[@wargreymon2024]

"Without a course in abstract algebra..." You have no idea, look no further than his channel playlist👀

[@ExistenceUniversity]

So because you refuse to look at the subject that you know exists, he is using an argument from authority, because you are lazy?

[@quantumsoul3495]

Both numerator and denominator are 1-forms ie functions from M to T*M evaluated at a point p (so actually they are function from TM to R). Since division is defined in R, the defintion of the fraction f/g where f,g:A->R is (f/g)(x)=f(x)/g(x) and is well defined.

[@TaladrisKpop]

@@quantumsoul3495 I know that. My problem is that it is should be in the video

[@TaladrisKpop]

@@ExistenceUniversity I know de Rham cohomology very well. The issue is that nothing is explained in the video. The video only says "du and dv are functions, not exactly but ok. Then we divide dy by dx". OK, but if it is not said what dy and dx exactly are, how can one know we can divide them? And that dy/dx is a fraction as claimed?

[@jlmassir]

Any sane presentation of differential forms should explain why dx and dy (or du and dv, like in the video) are defined that way. It is a violence against the student to "just define" them. There's a way to connect this definition as linear functionals in the tangent space to the intuitive notion of infinitesimals which is the working notion for physicists, engineers and even mathematicians. Just ignoring this working notion makes differential forms something esoteric. Many mathematicians reason with infinitesimals but translate their reasoning to differential forms, and it would be "dishonest" to hide this connection from the student.

[@djsmeguk]

He did. There's a whole video series from a few years ago where he goes thru it in detail.

[@jlmassir]

@djsmeguk Good. I will keep my comment so that people trying to understand differential forms don't feel bad for not having a clue while watching this video. You could also help them by posting links to the video series.

[@preoalex8298]

the prime notation is actually Lagrange’s notation.

[@krzysz5023]

When I took business calculus I could not understand Leibniz Notation and I noticed a lot of fraction rules carried over but my professor didn't know what I was doing or how I got the right answer after trying to nail down this method. I knew it wasn't novel but since then I keep looking for validation that it works (other than my A in the class) so thank you! 😅

[@thetinkerist]

I understood only a fraction of it.

[@peterbogardus1560]

😂

[@DrAndyShick]

I remember when I first saw "Now multiply both sides by dx" and was like "wait a minute!"

[@curtiswfranks]

I do not even know what a fraction is.

[@dominofan238]

What I really dislike about the dy/dx notation is that it implies that the variable name x has a special meaning. That's not true. For example, the functions f, g: M -> N, defined by f(x)=x^2 and g(t)=t^2 are precisely the same. We need to choose a variable name to write down what the function does, but that choice is completely arbitrary and the variable name has, purely mathematically speaking, no meaning on its own. I will admit that the Leibniz notation is convenient and intuitive and that it can be very useful especially in natural sciences, but in my opinion it should be introduced to learners more carefully as it could easily lead to conceptual misunderstandings.

[@MatthijsvanDuin]

The syntax makes more sense if you think of y and x as functions from your space M (as abstract smooth manifold without any innate coordinate system) to ℝ. So x is just a particular function on your space that you've chosen to use as coordinate function, and then (if your space is 1-dimensional) { dx } should be a basis for the 1-forms on M hence dy will be divisible by dx for every function y. (This does require suitable choice of x, not every function is usable as a coordinate, e.g. x² will not do!). But that viewpoint has its own limitations, so it's common to just be sloppy with notation when needed as long as it's clear enough what you mean. Every syntax for derivatives has pros and cons.

[@viliml2763]

"it implies that the variable name x has a special meaning" No, it doesn't. That just a you problem.

[@dominofan238]

@@viliml2763 To make it clear, I mean as soon as you write dy/dx you introduce the arbitrarily chosen variable name into the derivative notation. Of course you can do stuff like dy/dt, so there's nothing special about x of course, but the point that an arbitrary choice of variable name makes its way into the notation for the derivative function persists. As a side note, I find it annoying how passive aggressive your very first reply sounds when apparently you haven't even understood what I was saying.

[@cameronhayward7858]

@@dominofan238Well yeah, we have to name the independent variable. Later on if we have multiple variables, df/dx and df/dt may be completely different things. I might be misunderstanding you, but whether we call the independent variable x, t, or anything else makes no difference so long as there’s just one. In that way it IS special.

[@quantumsoul3495]

Actually, M and N are manifolds, if p is a point of M, phi:M->R^m is a local chart at p, and phi(p)=x ,we have (d/dx^i)_p = phi_{*p}^{-1} (e^i). When f:M->N, and psi:N->R^n is a local chart we have (df/dx^i)_p = psi^{-1}_{*psi(f(x))} (d(psi f phi^{-1})/d x^i) phi (p) iirc

[@benarcher372]

What a great video. Thx!

[@saltandvinegar-h78]

Getting my bachelor’s in mechanical engineering, we treated it like a fraction all of the time.

[@rahulvats95]

There are practically zero instances where it behaves otherwise, I almost forgot that it wasn't a fraction after all the time I have seen it acting like one.

[@richardchapman1592]

Loving this way of defining derivatives as fractions but too simple to work out if dx/dy for a spiked function f(y) is meaningful and whether there may be a grey area, that converges to your definition of a zero function, to treat spiked functions.

[@richardchapman1592]

For R² tangent planes, wouldn't it be more accurate to use nonlinear functions as curved tangential generating derivatives at a point. Makes little difference for smooth functions but could get approximately mathematically treatable for spikes and step functions.

[@byronwatkins2565]

From the definition of the derivative, certainly dy/dx is an improper fraction before taking the limit dx-->0... i.e. before 'canceling' the zeroes.

[@cmdrblahdee]

I think why it's emphasized to not be a fraction is how some people will "cancel the d's" algebraically; at least when they first start learning calculus. Also, idk that "dy/dx" is what they say isnt a fraction. It's "d/dx". Them emphasis is that it's an operator on y. However... the physicist way of handling calculus is way way easier and more inuitive, imo.

[@jprogsie]

i clicked on the video and though it must be some whiteboard software type shii but turns out its the ultimate classical chalk board the king of boards

[@walterbrown8694]

Took my first calculus course in 1955 at Purdue University. In the derivation of the derivative it was obvious that dy/dx was a fraction, and this was never an issue. Never had any problem with the notation as an engineering undergrad, or in my career in the defense industry afterward.

[@ernststravoblofeld]

Calculus gets easier when you discard all the unnecessary philosophical baggage. Differential is definitely a fraction while you're using it. And you can divide by zero if you don't look directly at the zero.

[@stevehorne5536]

The reason for the philosophical baggage is because there's cases where you get absurd results otherwise. Don't ask me about them - AFAIK they're rare and obscure enough that you can go a long way without seeing one, but that can just mean you have the confidence that you can't possibly be wrong when you're finally wrong. Probably a topic for introductory analysis courses, along with the epsilon delta definition of a limit.

[@topilinkala1594]

@@stevehorne5536 The theory of differentials is as rigorous as any mathematical theories and it has restrictions. If you have partial differentials you just can't assume they work as a "fraction". What you need is to calculate the total differential and work with that.

[@stevehorne5536]

@@topilinkala1594 It sounds like you're describing a different issue where partial differential notation doesn't always specify everything it needs to - what is held constant is often left to context (though there is extra notation when necessary). Algebraic manipulation of the partial differentials can move them into a different context where they get wrongly re-interpreted as keeping different variables constant, or just completely lose the context leaving no clues which variables are kept constant. I was thinking of the history of calculus and analysis, where past a certain point, some absurd results did turn up and making the theory of limits rigorous was the fix to at least be able to systematically identify the bad ones. However, I guess I'm making assumptions - I assumed the "philosophical baggage" was there because of something to do with this, based largely on the idea that a differential that's not in a fraction with another one isn't directly associated with any particular well-defined limit. It can be associated with a well-defined infinitesimal if you don't mind non-standard analysis - don't ask me for details, I'm aware of the subject but only just starting really - but non-standard analysis isn't really considered, well, standard. And I'm not going to say it's a magic fix for everything just because it exists - I don't know what this justifies and what it doesn't yet.

[@maymkn]

You sound like a physicist.

[@itzakehrenberg3449]

@@maymkn Or engineer...

[@homerthompson416]

Any chance to get that differential forms playlist completed? It's such a cool subject.

[@jedediahjehoshaphat]

All those years in undergrad, postgrad mathematical analysis class, seminars and monographs only to realise it's a fraction all along 😭

[@TheTerribleSwede]

Unless you learned from an old school teacher.

[@Alan-zf2tt]

A more considered comment runs like this: but first! (dy/dx) (dx/dt) tends to dy/dt by cancelling the dx fractional parts. Sorta explains dx tending to zero nicely. Anyway... 6:08 - nicely done! I hope all listeners/viewers experience thoughts: tied variables? orhogonality? A nice thing about orthogonals is that a lot of messy stuff gets sent into null space no? 12:58 general observation: a robust or stout result in math should really stubbornly present itself however the result was approached upto isomorphisms? Therefore (dy/dx)(dx/dt) is isomorphic to dy/dt and if nicely so it is a 2 to 2 too

[@robfielding8566]

use Johnathan Bartlett's calc notation fix. d[] is the implicit diff operator over *, +, ^ and log. d[ dy/dx ]/dx = d[d[y]]/(dx^2) - (dy/dx)(d[d[x]]/(dx^2)) which is pretty easy to derive yourself, noting that d[d[x]]=0 means "x is a line".

[@igoranisimov6549]

What is he talking about? Derivative by definition is a limit of a fraction. Notation has nothing to do with it. My teachers did not lie to me.

[@Syndiate__]

Well they sure did to me.

[@igoranisimov6549]

What did they tell, let's hear it

[@Syndiate__]

@@igoranisimov6549 They told me that dy/dx was not a fraction and to only think of it as notation, not something that can be arithmetically operated on. I then saw integrals and questioned their position due to the dx

[@igoranisimov6549]

This does not give a definition of derivative. If we have a function y=f(x) the ratio of change of the function vs change of the argument x in the limit when change of x approaches zero gives as derivative. If your teacher did not tell you it is still in any basic calculus textbook

[@Syndiate__]

Bro, I'm not arguing with you on the definition of a derivative or what dy/dx truly means. When you said "what is he talking about?" followed with "My teachers did not lie to me", I thought you meant that teachers do indeed teach it, and that the titles was incorrect. However, this is not what the majority of teachers teach, at least not mine. Which was my point.

[@ajbXYZcool]

There's an image in the math lab at my university that asks if dy/dx is a fraction, asking the 2 Mario brothers. First, Mario goes the long way to say "no", ie what I think is the usual accepted answer. Then Luigi goes "if there's a line, you must divide!"

[@pierreabbat6157]

From the title, I thought it was going to involve Lie groups.

[@selenatang314]

It makes sense that it’s a fraction right? Dy and dx are like delta y and delta x. Dy/dx is a ratio so it can act like a fraction right?

[@topilinkala1594]

Separation of variables in differential equations already assumes that dy/dx is a fraction. On the first course where you meet such differential equations you normally get some hand waving why it works. In multivariable calculus you finally find that partial differentials have problems and you've introduced the concept of total differential which can be shown to work as a fraction in every way fractions work. So one assumes it's a fraction. Then if you ever get analysis you get the formal definition of diffrentials and a rigorous proof that total differential is a fraction.

[@archismanrudra9336]

Goursat, I read the English translation via Dover, actually does an interesting interpretation in the 1-variable case, which is exactly the differential form concept, but understandable to an undergrad. So say y =f(x), then he interprets dy to be a linear function at any given point, x as dy(h) = f’(x) h For any function, y, of x. Since x is also a function of x, he wtites dx(h) = 1 h, since the deriv of x wrt x is identically 1. Then dy(h) / dx(h) = f’(x), and the chain rule is easy to verify

[@TekCroach]

Of course it's a fraction as per its definition itself on one condition - the change in the run approaches zero.

[@MrMrkBo]

From my amateur perspective, I always felt that the only reason we were admonished that dy/dx should never be considered a fraction is because in the beginning it is easy to lose sight of what a derivative is as we work with them, and our instructors didn't want that to happen, because so often, dy/dx is often treated operationally like a fraction, although it isn't.

[@Schraiber]

I'd love some more videos about differential forms. They're so weird to me. There's just something about them that hasn't clicked yet despite my attempts

[@joetursi9573]

It can denote a fraction as well as the derivative.

[@broccoloodle]

probably can define the reciprocal of dx by the ring of fractions over the graded commutative E(R)-algebra of p-forms where E(R) denotes the ring of smooth function on R, multiplication on p,q-forms is the wedge product. that's probably too much for a comment

[@geektoys370]

It is not “a fraction” it’s whatever this fraction approaches when dx gets small.. it’s a limit

[@kchorman]

Slope is Δy/Δx. The slope of the line is the same as the derivative of the line. It's nearly an elementary understanding that tells you it's a fraction. Why this overly complicated way to say the same thing?

[@toby9999]

Is it still a fraction at the limit?

[@perman07]

​​@@toby9999If you formulated the limit in terms of some sequence with n->infinity, then every term in that sequence converging towards the derivative would be a fraction. The derivative would be the value of that limit, and all elements in the sequence would be fractions. Intuitively, I don't see a problem viewing it as a fraction when it behaves like one in every fashion I know.

[@skhi7658]

It is quite easy to understand. The zero in mathematics means not nothingness in an philosophical /ontological sense. The zero in mathematics is a representative of the absence of something present or the symbol for a renormalization. In this respect, it is a virtual /abstract threshold which only knows approach or crossing but does not know reaching. Zero is not a state

[@xinpingdonohoe3978]

It's good to see more differential forms content. Differential topology and geometry is the zenith of calculus and analysis. I always struggled to actually figure out what they meant by that. I guess they mean things like dy/dx+3/7=(7dy+3dx)/(7dx) maybe don't work, but if you treat them as their own separate "class", then some fractional rules are followed. dy/du du/dx=dy/dx (chain rule) d(ay)/d(bx)=a/b dy/dx (by chain rule and linearity of the exterior derivative) d(y+z)/dx=dy/dx+dz/dx (by linearity of the exterior derivative) 1/(dy/dx)=dx/dy (by chain rule) It's actually quite impressive how well it does that. Actually, I think they do work in that fractional example above, now that I'm thinking about it. (7dy+3dx)/(7dx) =(7dy)/(7dx)+(3dx)/(7dx) =d(7y)/d(7x)+d(3x)/d(7x) =7/7 dy/dx+3/7 dx/dx =dy/dx+3/7 All this just by the rules above, I'm fairly certain.

[@QuicksilverSG]

Why this quickly dived into the deep end: It's not just calculus, it's linear algebra.

[@AswathRao]

Probably I am playing a semantics game here. I still claim dy/dx is not a fraction bc it is just a lazy form of the notation d/dx(y). Now the chain rule says d/dx(y) = d/dz(y).d/dx(z). Now the sloppy form of the notation, I could do algebraic manipulations as if they are a fraction. So it is a fraction only in the context of algebraic manipulations and where chain rule applies.

[@SimsHacks]

Finally understood what used to be a complete nonsense in my undergraduate differential geometry class.

[@ferretcatcher2377]

I wouldn’t worry about semantics too much. For me dy/dx is a derivative; it is an expression showing the rate of change of a function wrt to an independent variable. I’ve never considered it to be a fraction. It’s just like in physics an electron is not a particle nor a wave; it’s an electron!

[@Chris_5318]

Nevertheless, if you do, I guess it'd be called Calculus 3 or Calculus 4, then you will learn about differentials. They are very useful when dealing with functions of several variables.

[@larsnystrom6698]

Long after doing any math I accepted that dx were an infinitesimal, and that dx/dy were a fraction between infinitesimals. In physics, we always treated them like that. Math utilized Wirestrass way of making differentials stringent, but it made this very unintuitive. Now when we know that infinitesimals are an actual thing (from Robinson) I wish math books would utilize them. Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler attempting that, but not doing a great job, in my opinion. But, not being a mathematician, my judgement might be flawed. Robinsons way of showing that infinitesimals are an actual thing also lacks the intuitive flavor. I wish someone would write a book making these concepts coming alive.

[@larsnystrom6698]

The key to making infinitesimals comprehensible is, I think, to explain an integral as a summation of an infinite numbers of infinitesimals. Although, not being mathematician enough, I don't really know what kind of infinity is needed here. Wirestrass formalism with limes and such somewhat failes, maybe because the integral becomes an infinite summation of things of zero length, instead of infinitesimals, which aren't zero, Not intuitive, in my opinion, because that sum seems to be zero too. Not that my jugement of these things are reliable.

[@MathCuriousity]

Does Michael penn realize that 1 forms can only be treated as a fraction if we are dealing with everywhere non-vanishing curve?!!!!!!

[@zhelyo_physics]

As a physicist, I do like this video! : ))

[@arielgalles2107]

I did u substitution in calc 1 which requires treating dy/dx as a fraction so I knew it was a fraction since my first semester of calc. I still like prime notion though since it's fewer pen strokes

[@Blackmuhahah]

I'm so Stokes d.

[@pulsar22]

A fraction is a special form of a ratio. When the units of a ratio where the numerator and the denominator are the same it boils down to a simple fraction. So you are partially correct and partially wrong. When you are dealing with raw unitless numbers, it is a fraction. When dealing with, say velocity of a car, etc they are ratio.

[@radfue]

I am not mathematician so I still don't fully get what they mean by it's not a fraction but it's always a bit weird for me considering that the definition of a derivative is the limit of a fraction and that the chain rule follows the same kind of rules fractions do And it also makes intuitive sense since a derivate is the proportion of the variation of two variables I am guessing that maybe for some very specific things the fraction rules don't quite apply?

[@johnholmes912]

dy/ dx is a form expressing f→f₁

[@metaekpyrosinpalingenesis7040]

How can you divide by dx if dx is a zero-divisor in the exterior algebra?

[@sillymesilly]

Well obviously its a fraction why else you could separate them and integrate

[@aravindhvijayanandan3010]

Is it not a ratio??? For ratios can be represented as a fraction as well. Still this video confuses me more than it clears my doubts of Calc 1 now that the video is titled in such a way

[@johnholmes912]

surely not dy/ dx is a form of notation expressing f→f₁

[@aravindhvijayanandan3010]

@johnholmes912 It sure is a notation in the same way {delta}y/{delta}x is, i.e., slope of the secant line. It is indeed a ratio then.

[@APaleDot]

what I never understood is what exactly is dv/du in differential forms terms? Is it just a function that takes in a single vector? How exactly do you evaluate it?

[@amjadmalik7285]

How about....dy/dx is the change in y wrt x. ....also equivalent to the gradient. Since it is 'a ratio' of the change, it can be treated/ manupliated as a fraction...

[@jamesfortune243]

So, dy/dx walks like a duck?

[@supratimsantra5413]

Wandering explanation...... congratulations 🎉 sir ...

[@elliottwade1901]

So... when is dy/dx not a fraction, now?

[@jtdenton1483]

It's a fraction of infinitesimals.

[@stevehorne5536]

AFAICT in at least two contexts - Leibniz' original non-rigorous calculus, which was heavily based on infinitesimals but didn't question or formalise what infinitesimals are, or modern non-standard analysis which includes hyperreal numbers which (1) embed the reals, which work the same as they do for real numbers, and (2) includes both infinities and infinitesimals, and (3) is a rigorous formalisation of essentially what Leibniz did. Some doubts - hyperreals and non-standard analysis definitely exist as I've described them (and NOT how infinities are often described, e.g. these are NOT like cardinal number infinities - infinity+1 is a different infinity, and even infinity+infinitesimal is a different infinity), but I haven't figured out enough to be sure that we end up which dy/dx as a simple fraction (in particular, I don't think you get a value for 1/0 as a hyperreal, so you still end up looking at infinitesimally different values and then throwing away the infinitessimal terms at the end - in which case dy/dx is a fraction, but not an algebraic solution of the apparent algebraic equation you start with because of those extra introducing-and-discarding-infinitessimals steps). There's also the surreal numbers, and other than that they embed multiple number systems that include multiple styles of infinities (including the cardinal numbers, ordinal numbers, hyperreals and I think more) I know even less about them so far.

[@sergeyromanov5560]

no infinitesimals in standard math, so no, it's not

[@stevehorne5536]

@@sergeyromanov5560 If you mean as opposed to non-standard analysis, I read a quote somewhere that non-standard analysis using hyperreals is just as valid and correct as complex analysis with complex numbers. "Non-standard analysis" is a bad name, because more obvious names were already taken. "Not standard" doesn't mean "wrong" or "not maths". Once upon a time, zero wasn't standard math. EDIT - found that quote. It's in "Infinitesimal Methods for Mathematical Analysis" by J. Sousa Pinto, as translated by R.F. Hoskins, published in 2004. It's in the translate preface from the original Portugese version - "Nonstandard Analysis was the title chosen by Abraham Robinson (1918-1974) for the book published in 1966 in the North-Holland series Studies in Logic and Foundations of Mathematics. For the first time in modern times the rehabilitation of the concept of infinitesimal was established rigorously in this book, using modern mathematical logic, in particular model theory. Nonstandard Analysis might better be called Infinitesimal Analysis: however there is already a classical connotation for the latter which would make it difficult to use in the required context. For some mathematicians the choice of name is not a happy one since the term "nonstandard" suggests opposition to "standard" classical analysis. This is not the case. Nonstandard Analysis simply offers to the mathematical analyst a greater variety of mathematical objects with which to work." Apparently the analogy of complex numbers didn't come from here, or at least isn't as easy to find, but this still makes the basic point. Infinitessimals have been included in a rigorously defined number system that includes the reals (behaving as reals should) since 1966.

[@MrAwombat]

Who cares​@@sergeyromanov5560

[@joshuagabel1717]

This explains nothing

[@SM-ok3sz]

Prime is not Newton, it’s ZZ Top’s La Grange notation.

[@ricomajestic]

Wow! After listening to that song for years I finally know what it is all about.😅

[@MrMooagi977]

As a maths teacher, I'd like to know how you got your writing so neat🤔

[@nosuchthing8]

Rise over run, sure

[@TheDayglowcamo]

Given no intuition as to what forms are other than the character manipulation he showed all this video does is say look i created some rules and they show that the notation shows a division of the symbols. Maybe this video is for people who already have geometric insight into forms thus no reason to review. But for those of us that do not the review has no meaning other than a set of rules. I guess i will go watch his earlier videos on forms and hope they give meaning to the character manipulation.

[@curtiswalker8457]

Change in Y over change in X dy/dx. Rate of change with respect to x. Ez peezy

[@vassillenchizhov290]

Could you provide a reference for the careful definition of the nonlinear differential forms? Are these cogerm differential forms, e.g. one could define the cogerm differential form for the length as ds = sqrt(x^2 + y^2)? And one gets the change of variables factor from these for free.

[@curtiswfranks]

At the end, who says that v depends on u?

[@periodictable118]

While purely mathematically it may not be a fraction I pointed out that so long as you stick with single variable calculus there is zero issues with treating it as a fraction, and my math prof reluctantly agreed.

[@johnholmes912]

But there aren't many ( if any) single variable problems in nature

[@lumina_]

that statement at the beginning is false, the prime notation, f'(x), is Lagrange's, not Newton's 🤦‍♀️

[@oni8337]

Still waiting for that differential forms series finale (generalised stokes theorem)

[@oldcowbb]

another case of physicists making a logical mess and mathematicians fixing it

[@tyandthetymebenders6317]

What makes dy/dx not division. Or is Dy/Dy defined as slope?

[@togoso]

Them’s fighting words

[@curtiswfranks]

For d²: Who says that fᵤ ̦ᵥ = fᵥ ̦ᵤ? If d² = 0, then what about d²f/(dx)²?

[@vassillenchizhov290]

You have to assume continuity of the second partial derivatives and by Clairaut/Schwarz you get the symmetry of the mixed derivatives. Let f : R^2 -> R be a twice continuously differentiable function (i.e. in C^2), then df = f_u du + f_v dv and d^2 f = d(df) = f_{uu} du^du + f_{uv} du^dv + f_{vu} dv^du + f_{vv} dv^dv = (f_{uv} - f_{vu}) du^dv = 0. As far as d^2f/dx^2 goes, this is notation for the second derivative. d^2 f = d(f_x dx) = f_{xx} dx^dx. dx^dx would be zero, but in the denominator you also have dx^2 = dx^dx which cancels it out and you are left with the second derivative f_{xx}.

[@ExistenceUniversity]

Saying it's not a fraction is like saying that 1/2 is not a fraction because it's actually an operator that takes half of the operand.

[@johnholmes912]

no it is not;

[@ExistenceUniversity]

@johnholmes912 yes it is

[@kchorman]

1/2 is not a fraction, it's 2^(-1).

[@ExistenceUniversity]

@kchorman Yeah, exactly, because that's not the same thing at all lol

[@chrisray9653]

The rate of a variable changing relative to another variable. :/

[@dannygjk]

I'm getting flashbacks to 3rd or 4th semester calculus or am I off?

[@ukyoize]

If it is a fraction, then what d/dx f(x) is?

[@ricomajestic]

A fraction. You are just rewritting things.

[@Ymir._]

dy/dx where y=f(x) equals (f(x+h)-f(x))/h. i have yet to watch the video, but this is the definition.

[@regfordca]

You must have mastered the annoying Calculus on Manifolds by M. Spivak.

[@ai3t86]

As a physicist, I like this

[@goodplacetostop2973]

18:24

[@deaccaed]

I do not understand, that somebody learns, it wouldn't be a fraction. In germany/austria, you learn it, that it is a fraction. It is like Δy/Δx, while Δy/Δx = (y_2-y_1) / (x_2-x_1). If you choose x_2 ~ x_1, than you have dy/dx. In school (gymnasium), you have to explore the behaviour of a curve by "using a line" and there are two solutions, which students find: you can use a line, which goes through two lines of the curve (secant line) or you try to use a line with same the same slope in a point (tangent line). Than you calculate the slope by fraction. After that, you learn to use the notation of dy/dx instead of Δy/Δx and a little bit later the f'(x) notation.

[@nbooth]

Δy and Δx are numbers but dx and dy are not numbers. With differential forms you have have a ratio of differential forms like dy/dx that is kind of like a fraction but it still isn't one.

[@jonassattler4489]

In Germany you also go to university and actually learn what a derivative is, instead of having to contend with whatever half baked nonsense you get taught in schools. Something being taught in German schools should be anything but a ringing endorsement of that thing. ∆x/∆y also is a legitimate fraction as long as ∆y is not zero. It is an approximation of the derivative. But dy/dx is just notation. "dx" isn not a mathematical object. It is not a real number and it is not a function. Treating it as such is just very confusing.

[@Roq-stone]

If "dy/dx" means differentiating the function of 'y' with respect to 'x', I don't see why "dy/dx" should be called a fraction. Or else, d2y/dx2 (the second derivative) would be total nonsense then. Anyway, nice click bait.

[@blueslime5855]

11:50 Cauchy's equations or is there more to it?

[@joetursi9573]

My error below. It's a fraction.