Man

Bhai 😕🤑🤑

Wut 🤣

Considering how expensive college is, and how good 3brown1blue is at explaining math, this is quite literally a good financial decision

YESSS... I would get on my knees for that

lmao

UGHHHHH!! so good

Please never comment again. 😂

badum tss!

pun intended

I say same with you!

blackpenredpen you learned calculus from watching KZfaq tutorials, you don't know how to add fractions, but you know laplace transformation

San Samman you can add fractions by multiplying both sides of both fractions by the base of the other fraction. After that both fractions will have the same base and you can add the top part( don’t know the English word) together. An example would be: 3/4 + 2/3 We expand the first one by 3 and the second one by 4. 9/12 + 8/12 Now that the bases are equal we can add the top parts (9+8)/12 =17/12

Da Real Doenertier i was kidding xd

Da Real Doenertier it's "numerator / denominator"

I share your opinion. I dare to state that I had good teachers... but NOW I know that they didn't have the right tools with the right ways to explain this material. Hats off for the approach, for the quality and the efforts behind them.

I'm on Patreon immediately

He can't just take all your money. He can only take a small amount of your money over a small amount of time.

ha ha i get that reference!

iabervon you need that small amount of time to approach 0

The_StarByte, True! For the circle, you could express it as two separate functions of x, but not all implicit curves are so easily separated. You could have something crazy like x^2 + y^2 = tan(y/x).

And I thought that we will use some integral criterion to derive ln(x) but instead of that you jumped right into multivariable calculus ... that's unconventional and kinky at the same time! :D

I still remember the first time I encountered epsilon delta. Frustrating times.

Are you going to make some videos about differential forms? V I Arnold in his book Ordinary Differential Equations says, that differential equation is just relation between differential forms.

3Blue1Brown : please start making pi creature t shirts. All you followers would buy them!

praanav mahadev menon i assume you are in class 11 like me

Aryaman Manish Joshi yes I am. kinematics chapter rn :)

praanav mahadev menon you can criticize the system afterwards but right now focus on your studies I have just passed 12th and regret not studying much. avoid this at all cost

praanav mahadev menon bro, dont generalise it to india, there are many good teachers all over india really teach the basic stuff (atleast in my case)

Vishwas Dubey ik but still. cbse2is very dry and bookish in its teaching. there are several teachers out there but only a handful ate good as you say.

The fact is that: they are right. dx/dy should not be threated as a ratio, but is a totaly different concept. Likewise, you can't proof anything with "tiny nudges" and small increments, you would need a whole lot of theory and rigourous theorems. However, wether or not these exaplainations (that are not proofs) are to be considered acceptable, depends on your course of study: a math student will not like this series, but as an engineering student I'm more than happy to think of small increments (without forgetting that there is a whole lot of theory behind that I don't need to remember), and I find these videos as an awesome way to understand complex concepts that can (should) be properly formalized.

@@maurofoti526 i heard it was possible to use infinitesimals and prove calculus rigorously that way. Hyper real numbers and something

Im trying to self learn it before I take the class, If I end up with the integral OF dx with respect to dx, what does that even mean?

@Boundary Theory The actual important thing to remember, though, is that the "d" is not a variable, despite the fact that they write higher order derivatives in a way that totally makes it look as if it is, even down to the algebra of how they are defined. "dy" is at least variable-like, and so are "dx", " d(2x+5)" and "d([insert anything else])", but the "d" is not. I don't think it's really a function, either, though I'm not sure. It's almost more like subscript.

@Boundary Theory I wasn't implying anyone else thought it was. It's just that I have questioned whether it was or not before, and thinking it is would be a continuation of the idea things which are considered just symbols actually being exactly what they look like. That "d" is not what it looks like.

frogstud ...Reaallyyy..

what does leibniz have to do with this video? xD

He was

@@SuperKnowledgeSponge discoverer and inventor of calculus

He wouldn’t

IRisingFuryI There's always Wikipedia, aka the Bottomless Abyss.

IRisingFuryI even professional mathematicians get lost in Wikipedia sometimes, lol.

Oh, yes it´s bottomless. I like the kind of ever-branching definition trees you get on wikipedia. Wikipedia, what´s etale cohomology? Well, for any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X... Schemes? Wikipedia, what is a scheme? An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring... Wikipedia, what´s a locally ringed space? A locally ringed space is a ringed space such that all stalks are local rings. Ringed space? That´s a topological space together with a sheaf of rings. Topological space? A topology T on X is a subset of the power set of X, so that the empty set and X are in T, and T is closed under finite intersections and infinite unions. Ok. Then let´s return to the ringed space. Wikipedia, what´s a sheaf? Let X be a topological space and let C be a category. A presheaf F on X is a functor with values in C, that sends open sets to objects and inclusions to morphisms. What´s a category? A category consists out of a class of objects and a class of morphisms between those objects. Each object has an identity morphism and there´s an associative composition of morphisms. Ok, then let´s get back to the sheaf. Wikipedia, what´s a functor? Functors are just the structure-preserving functions for categories. You can call them category-homomorphisms if you like. Ok. But wikipedia, you still haven´t explained to me what a sheaf is. You only explained presheaves. A sheaf is a presheaf in the category of sets satisfying the locality and the gluing axiom. Then let´s get back to ringed space. Wikipedia, what´s a ring? Abelian group with an additional multiplication operation that satisfies associativity and distributivity. Ok. Let´s get back to the locally ringed space. Wikipedia what´s a stalk? The stalk of F at x is the limit over all open sets U containing x of F(U). It´s basically a construction capturing the behavior of a sheaf around a given point. I´ll pretend I understood that. Back to ringed space. Wikipedia what´s a local ring? A local ring is a ring with a unique maximal ideal. Maximal ideal? That´s an ideal whose quotient ring is a field. Ideal? Kernel of a ring homomorphism. Kernel? The preimage of {0}. Ring Homomorphism? Structure preserving map for rings. Then back to maximal ideal. What´s a field? A ring, but multiplication is commutative and invertible. Then let´s get back to the scheme. Wikipedia, what does isomorphic mean? It means there´s a bijective homomorphism Bijective? Injective and surjective. Injective? f: A → B injective iff ∀x∈A∀y∈A:f(x)=f(y) →x=y Surjective? f: A → B surjective iff ∀y∈B ∃x∈A:f(x)=y Homomorphism? Structure-preserving map. For example if you have an operation + on your algebraic structure, then a homomorphism f would satisfy f(x+y)=f(x)+f(y). Then let´s go back to the scheme. Wikipedia, what´s a spectrum? The spectrum of a ring is the set of all prime ideals. It is commonly augmented with the Zariski Topology. Prime Ideal? Ideal whose quotient ring is an integral domain Integral domain? Ring without zero divisors. Back to the spectrum. Zariski Topology? And it goes on like that for quite a bit longer.

Woah

Why

Yeah I learned it the same way! I was taken aback when I saw the video implicitly differentiate without y being dy/dx but it makes so much more sense now

differentiating y as dy/dx and x 'normally' is the same as shown in the video, but then dividing every term by dx.

Personally I find 3b1b’s way a lot more straight forward, if you’re working with many variables. I frequently forgot to put the dy/dx after the y, and I didn’t really get why I did it, until much later. This video would have helped with that so much, but I guess we all have our preferences.

@@kikkukun Thank you! This is the piece of algebra I was missing. Now my textbook makes sense.

@@kikkukun multiplying* ?

same here bro

It's like when the T-Rex shows up to save the day at the end of Jurassic Park.

I just got to that. I was like “OHHH! So that’s how you get 1/x”.

it took me a minute not gonna lie

awesome :)

I would guess that you are already aware, but when you say "learning their origins rigorously," these videos are far from rigorous.

Yeah true, I should really say "intuitively". I wish they'd taught all this stuff back in school. I suppose I could already prove the product rule rigorously from logarithmic differentiation, but like it's really cool to see the way he did it in one of the videos. Though the video on the power rule did prompt me to think "Oh shit!" and come up with a proper proof for it with binomial expansion!

Th3Muzza Ignore all the pseudo intellectuals telling you that learning mathematics riigorously is separate from or superior to seeking an intuitive grasp of the concepts. The symbolic formalism they mistake as "rigour" is little more than an efficient method of organizing and easily manipulating concepts you are able to mentally visualize. By focusing on simple examples you can easily visualize alongside their symbolic representations, you learn to automatically associate one with the other in the future. Without that effortless fluidity of thought between symbolism and the mind's eye, "learning" math would be as empty and unfullfilling as memorizing the Chicago phonebook.

+Philip I will accept that insult, but please don't view me as someone who holds the beauty of viewing things the way 3Blue1Brown presents as inferior to rigorous proofs. I would rather be recognised as the smartass faking intelligence that I am than viewed as someone who cannot appreciate the value of intuition.

Caldrago Haha, sorry if I came across a bit harsh. Your comment was actually totally reasonable, and I have strong feeling about the topic probably made me a bit overzealous. The pseudo-intellectual jab wasn't at you or anyone in specific really, just at my general frustration at the lack of substance in modern math and physics education. Watching 3B1B videos just reminds me of what education could and should be centered on, and the fact that there are so few creative visual and interactive resources that could catalyze math and physics education, despite modern technological resources being virtually limitless.

Skillmau5 not to mention that it was addressed as the former question in that specific scene

it wasn't a play on words

@@defectus1769 But it was beautiful nonetheless

Right exactly what I thought, especially with my calculus teacher XD

so did he.

you would probably be having a very painful time understanding him as a student, just like me right now. i know he could be the best one to explain this but i just can't see things as smooth as he is trying to show us (its driving me crazy)...... ( its seems like we have got to go through lots and lots of (blindly working with derivatives) and then figure out after a punch of years that this was the beautiful meaning of everything we used to blindly study.

how's it going? good luck brother!

Yep, good catch.

True! Good catch.

In my opinion, it's not true. If you take into account the limit of x/sinx for x that tends to 0, the limit is equal to 1. So the equation y^2 = x/sinx is true only for the values 1 and -1 when you go around 0.

Davide Bellucci None of what you said is relevant. (0,y) satisfies the equation for any y, so the graph of the equation contains all points on the y-axis.

Slarti Bartfast You are wrong. If you isolate the y and take the square root, the result is two functions which cannot have more than one value for each value of x. So you have to accout for the limit.

BumDeeDum lim x-->0 (1/x)

Vishwas Dubey does not exist ;)

never...

That's the point

SKO oh! i realized it now. XD

Yes, please!

Great, now that's all I hear ,too!

wtf dude! now i cant not hear it. thanks

Yammy Hammy subtitles say ladder.

I was also confused 😂

The algebra approach isn't "better" so much as it is a way to reinforce the concept of implicit differentiation on a simple equation. Since the equation in question is a polynomial, it is trivial to find a derivative without relying on calculus techniques. Much how we can take derivatives of polynomial functions using the difference quotient. It becomes less trivial when applied to transcendental functions. For instance, sin(xy) + e^(3x) = 7y Using implicit differentiation, we can easily reach cos(xy)(x dy + y dx) + 3 e^(3x) dx = 7 dy Expanding: x cos(xy) dy + y cos(xy) dx + 3 e^(3x) dx = 7 dy Rearranging: y cos(xy) dx + 3 e^(3x) dx = 7 dy - x cos(xy) dy Factoring: [y cos (xy) + 3 e^(3x)] dx = [7 - x cos(xy)] dy Dividing: dy/dx = [y cos (xy) + 3 e^(3x)] / [7 - x cos(xy)] This would not be a 6-line problem using algebra.

+Uejji Note that we know that sin(x+dx)=sin(x)+cos(x)dx due to derivatives. So it's doable (but annoying).

Is that so? Let's check it. Let x = pi/2 Let dx = pi/4 x + dx = 3pi/4 sin(x + dx) = sin(3pi/4) = sqrt(2)/2 sin(x) = sin(pi/2) = 1 cos(x) = cos(pi/2) = 0 sqrt(2)/2 = 1 + 0(pi/4) = 1 So, this is not true.

Let's elaborate: sin[(x + dx)(y + dy)] = sin(xy + xdy + ydx + dxdy) Using our angle sum rule for sine from trigonometry ( sin(a+b) = sin(a)cos(b) + cos(a)sin(b) ) we can (through some rigor) come up with the following sin(xy)cos(xdy)cos(ydx)cos(dxdy) - sin(xy)sin(xdy)sin(ydx)cos(dxdy) - sin(xy)sin(xdy)cos(ydx)sin(dxdy) - sin(xy)cos(xdy)sin(ydx)sin(dxdy) + cos(xy)sin(xdy)cos(ydx)cos(dxdy) + cos(xy)cos(xdy)sin(ydx)cos(dxdy) + cos(xy)cos(xdy)cos(ydx)sin(dxdy) - cos(xy)sin(xdy)sin(ydx)sin(dxdy) And that's just the expansion of *one* term. Good luck removing all of the dx and dy from those trig functions in order to group, factor and divide out. Now, maybe this can be done with algebra through some ingenious steps (although I strongly doubt it), the point wasn't that it's impossible, it's that saying that the algebra way is the "better" way is fairly naive and really only applies when examining nicely structured polynomials, which in the world of higher mathematics we cannot limit ourselves to.

as dx tends to small values, the equation becomes true.. afterall thats what "d" means in "dx"

You aren't looking at the average speed over a finite interval (e.g. 4 seconds), you are looking at the speed at a given instant in time. At a different instant in time, the speed will be different so you can't extrapolate one speed over a full interval of time.

You are absolutely correct, now that the ladder slides down the wall in a straight line and thus the movement of the ladder is no longer described bij a circle, hence X^2 + y^2=5 is not a valid description of this movement/problem. However if they would pulled the wall away in a split second and the ladder would have fallen without sliding back on the floor (along the x-axis) then the movement would have been a circle and the formule and dx/dt would have been correct.

@@TensegrityEnergyThere is still a theoretical circle here, but its center is moving along with the base of the ladder. If we fix the camera to focus on the bottom of the ladder, we will see the top trace a circular arc.

YUP

You're in our tute Andre lol

Most people are probably not coming from Kyle's channel, not yet.

@Keith Barndo His videos go through more examples though. The 3Blue1Brown videos just explain the intuition.

@@thomassogg2920 true

1Darco1 He didn't explain why each side is differentiated separately in general. He merely explained it for the simple case of a circle, because one side is constant. The general case follows pretty naturally, since you could always subtract everything onto one side, making the other side 0 (which is constant).

QED appropriate channel name lol

u know his name???

you can check out his voice and look at the first video on multivariate calculus on khan academy, and you can see his name at the end of one of the videos from this series