The Leibniz rule for integrals: The Derivation
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- Опубликовано: 29 сен 2024
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Today's video is going to be exciting! W are going to derive the Leibniz Rule for integrals in its whole form! It's one of the most powerful tools of integration, so be prepared! :)^
Quick note: The Integral I(x,t) is just in terms of t, so it's I(t) thanks for noticing! Even though, it doesn't quite matter, since we are just differentiating by t^^
Visit my website! =)
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Flammable Math: Can we interchange the limit and the integral? We are going to assume that we can.
Me: *Cringes in Monotone Convergence Theorem, Fatou’s Lemma, and Dominated Convergence Theorem.*
Isn't the Dominated Convergence Theorem enough for interchanging the limit and integral?
It is, but you first have to demonstrate that a dominating function exists.
@@LuisFlores-mu7jc Okay thanks! Because over here at my engineering college, we do not cover the theorems and lemma of what math majors usually go through so I do not know much about other theorems ): I want to major in math but the closest thing to a math major in my country is an engineering major.
For anyone who is watching now and wondering about whether the limit and the integral can be exchanged: wikipedia article on Leibniz Integral Rule, under the section "proof of basic form", has the details. You'll need analysis to fully understand, but basically assuming the partial derivative exists and is continuous, it holds.
@@thegigachad1254 why are you so mad😂
Thank you!! This is one thing about the video that was annoying me, but now I can go check out the wiki page!
I thought because the integral was a definite integral in terms of x, that I() was only dependent on t, not both x and t.
Erin Cobb Yes, I should be a function of t only. A minor mistake.
@@bggbbdg5625 Not a mistake. I is both dependent on x and t, but the integral was given in terms of x alone. The fact that the integral of I(x,t) is done over x does not mean that I is only I(x). That is why he started with dI/dt, which clearly means it is not zero. There was no mistake sir.
@@murtithinker7660 if the function is integrated in terms of x and the limits of integration are in terms of t then when they are substituted in for the x values the entire expression would be a function of t and so I(x,t) would actually be I(t) so the way it is written at the beginning does appear to be erroneous.
Papa flammy’s voice was so deep damn,
Also, who else is here watching prerequisite videos for log gamma video
Great video, very clear explanation! It's just given in Strauss in the appendix as a theorem but never really proven. This is a shame since it's such an important result and people often forget to differentiate the bounds of the integral which is problematic in many situations in engineering physics and differential geometry. It's also not clear how to handle it when the derivative may not exist but the integral does (ie integrating around a pole/residue etc.) Just some ideas for future videos, maybe take this on in the context of an analytic function or a Feynman path integral, or a problem that needs renormalization... I'd prefer you didn't wave your hands while doing it but Andrew may be watching so just do what feels right!
Great video. I actually understand most of it this time.
ruclips.net/video/vFDMaHQ4kW8/видео.html 💐..
Thank you very much for this explanation! From Russia
Watching this late at night while wearing headset in loud volume then suddenly the sound. Haha. My sleepy feeling gone. 🤣
I'm glad I discover this channel. greetings from Chili :D
Write the integral as:
I(a(t),b(t),t)
and use the chain rule to obtain:
I'(a(t),b(t),t)=\partial_{a}Ida/dt+\partial_{b}Idb/dt+\partial_{t}I(a(t),b(t),t)
and simply write down the terms.
Thank you!!! This proof was annoying me because I couldn't find a nice explanation for it. Finally get it now. I literally have an undergrad in maths and have never seen this, strange. But ya it's really cool. thanks again
Flammable math: Can we interchange the limit and the integral?
Me: Should I consider Riemann integration or Lebesgue integration??? .....and after a while I cringe in monotone convergence theorem, change of variables in lebesgue integration ,fatou's lemma and dominated convergence theorem....and the list continues..
ABSOLUTELY BRILLIANT !!
but I'm really curious about how quickly you swap the board and you don't hurt your fingers. that's incredible!
be careful man, we need you.
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How does he know that I’m watching this in the morning when he says good morning😰
Around the 5:00 minute mark, how does the linearity thing work for integrals? Why can you split the integral up like you did? Can someone pls explain?
I think I(t) not I(x,t) since since x is integrated out
I don’t understand 11:01 , if we r integrating w respect to x, why is it that the boundaries are inputted to the t variable of primitive F? Shouldn’t it be F(b + db, t+ dt) - F(b, t+dt)?
Around 15:35, regarding the first integral, when the limit of an integral is the integral of the limit, what theorem is used there?
Siu Kwan Yuen Actually ， should use lebesgue dominated convergence theorem. To be honest， math people never interchaning limit with integral without justify. This video is for engineering probably
Thank you for your presentation style..sehr gut ..immer...
Thank you. Best video, I've seen today .
I like your video so much thank u so much!!!
Best explanation ever!!!..... but can anyone explain why interchanging the integral and limit really worked
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Damm this is back when papa was a lil dry 😪 but it's okay now bc he's saucy teacher now very moist ⚡⚡⚡
14:00 How do we take the derivative inside the integral? After all, the limits are still a function of t. It may be illogical but I just don't get it.
As far as I know, one can bring the differential operator inside the integral (integration wrt 'x') when the limits of integration are independent of 't'.
aren't you assuming that the limit approaches 0 by the positive side? in 6.33
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It seems that I has only one independent variable which is t since x has been integrated out. So what do you mean by I(x, t)?
I think because when I is first defined, the integral hasn't yet been evaluated so it is still in terms of both x and t?
It's to be integrated wrt x what is t
This helped. Thank you!
What would this look like if you were to extend to two and three dimensions?
I was wondering where you'd get to talk about uniform convergence, and I see you basically didn't since that's a topic on a different level than most math youtube videos. Given that you're only talking about definite integrals (using a closed interval), and are assuming an antiderviative exists (and thus f is continuous), then that implies that the function must be uniformly convergent. This doesn't work though if you're talking about improper integrals in general, which is where I first saw this idea used on youtube. In that case though, the function was uniformly convergent since it had an exponential whose power was going to negative infinity as the input went to infinity, and was bounded from the left by 0, so it satisfies the definition of uniform convergence despite not being on a compact space. Still, I'm sad to see that I never really get to see anyone wade into the weeds of analysis to *really* solve these issues instead of just handwaving it and saying "trust me, you can change the order of the limits".
At 9:56 why doesn't b + delta b replace x instead of t?
ruclips.net/video/vFDMaHQ4kW8/видео.html ...💐
Ur head is covering the board
11:05 sir shouldn't it be f(c, t+∆t)(∆b)
Excellent video, just a little question, shouldn't the limits of the integrals be from a to a + da, then from a + da to b and the other one just like it is? Thanks
At around 10:00 mark, shouldn't you plug those bounds for x instead? I am confused :(
Oh, you bad boii 0 this all equals to
f(c,t)db.
It's all correct except for the last limit. As b(t) is supposed to be derivable (and so continuous), when dt -> 0 also db -> 0. So, the entire limit simply goes to 0. In fact, you have to do the limit not in this moment, but later, when you divide by dt: in that case, when dt -> 0, f(c,t+dt) db/dt -> f(b,t) b'(t). In the hypotheses, omitted from the beginning, we have to suppose a(t) and b(t) as derivable functions of t, and f(x,t) as an integrable function with respect of x ( for example f continuous w.r.t. x ), and derivable function with respect to t.
👍🏼
Flammable Maths I thought you were the mistake!
The whole proof thus has an error
Is he a student or a professor?
he is so much more dude...so much more :')
Student studying to be a professor IIRC.
Yes.
@@aryan040103 you re s
Yes
10:35 there is confusion about the integral of f to F is subject to x (first variable), then the central theorem derivative is about second variable t
Thanks for pointing that out. I didn't know how to fix that, but thanks to IITrojo's comment. I think the correct one should be: the integral equals F(b+delta b, t+delta t)-F(b, t+delta t)=dF(c, t+delta t)/dx *delta b= f(c, t+delta t) delta b. Just for everyone's convenience.
I was going to point out too, then I read supergeek supergeek's and ERik's comments below: at 9:40 you plug the boundary values into the t-part of f(x,t) instead than into the x-part. Then sending things to zero collapses the mistake.
Great channel anyway, keep going!
After this video I'm convinced that what I like is Algebra, Number Theory and Discrete Maths
Be sure whatever alarmed you here would follow you in Analytic Number Theory, at least.
Still calc all the way
Flammable maths, it'll be great if you can create playlist for all ur integral techniques especially for high school students like me. I just completed A levels. I also completed calc 1 and 2. I've been interested in integration lately. I need such playlists
Heh... it's all the same thing at the end of the day... only difference is notation. Look at operator notation in QM, QED and particle physics, it will make you feel better and worse about all things discrete. Look at Noether's theorem and the Lorentz group.
@@jcd-k2s yes from a technical perspective no from an intuitive. Also where that analytic stuff leads is not something to sneeze at look at renormalization group flow look at universality etc. as well as the work on what is and is not possible in different dimensions/spaces. Most people get stuck on the quantum "paradoxes" that lead to all the wave particle Copenhagen handwringing. This is unnecessary if you ground your intuition in the algebra and statistics instead of the geometry. The geometry follows from the algebra and the paradoxes ARE part of that emergent geometry not the other way around. Gain an intuitive understanding of this and quantum mechanics becomes about as confusing and difficult "understand" as any other physical theory. No one wastes any time worrying about why the order of rotation of a physical object about 3 axes determines its orientation and which face it presents. Some things commute some don't and this has consequences in how they project on to the chosen basis of measurement. As for the born rule that's a different can of worms but one that's no more mysterious than the Pythagorean theorem or the boundary conditions canceling the contribution of the retarded potential, or any similar conservation law. The point is we learn techniques and many different ways to approach looking at the same problem without deference to how the mind that evolved to solve calculus and geometry problems in a 3d world can be recruited and changed into a mind capable of thinking about and "understanding" problems outside the realm of common perception othen than trough cheap projection or the parlor tricks of metaphorical representation. That's a problem straight out of differential geometry. How to make the brain think in manifolds about manifolds that exist in topological or algebraic spaces it does not currently possess. The math and pedagogy is a means to physically grow a physical representation of itself in an object embedded in a discrete 3d space... yeah that's something that makes my head explode to consider and the hard part is definitely not the differential geometry part... we get that for free from things like vision, facial recognition, hunting, throwing, and knitting skills.
thorough ground-up explanation that tied in the relevant theorems and definitions. thank you so much :)
Glad you enjoyed it! =D
at 10:00 when it said F(x,db)-F(x,b) i spent a solid half hour trying to figure out how that was possible. turns out it was a mistake on his part lol
you could have looked at the video though
Stopped the video exactly like you and was also baffled until I read your helpful comment.
I am confused. What's the error
@@flutterwind7686 Error is that the function was in respect to x so the bounds that he substituted in should be in the x part
10:39 When you are integrating f(x, t+delta t) with respect to x why are you substituting the boundaries on the t variable? If you are integrating with respect to x the boundaries should be added to x, right?(Even if the upper bound and lower bound are in terms of t)
I was just thinking about this too... Is there any particular reason why its like this?
Note that delta-b is actually (b(t+delta(t)) - b(t)) and same for a.
Also there's probably some conditions on which you can replace the integral and the limit on the 1st term, most likely that it converges. Since you require the definite integral between a and b, this is most likely to happen.
Who's here because feeling guilty of not knowing the Leibniz rule after having whatched today's video (26^{th} June 2018)?
me
MadSideburns Me, but I mean its understandable. Us young mathematicians and young physicist barely got into advanced Maths. The leibniz rule is probably from now on my favorite rule due to how powerful it is. Any integral that cant be solved with any sub or series, easily solved by the leibniz rule, its just so elegant and powerful. Papa flammy ' s proof is also my favorite.
Well I myself dont know much about the rule, i know the process and proof and its use, but its great and one importance is that, any integral that was "impossible" to solve could now be solved with this technique which could POSSIBLY come useful when studying physics if you do stumble across something that is "impossible" to compute. I REALLY doubt it though since physics uses the lightest of mathematics and doesn´t go too deep into a certain technique. Like Feynman said, "The physicist is always interested in the special case", so a professor giving a lecture will never cover a general case on when you can use this since you will most likely never use it. This technique works really well if you are doing a hard problem just for the fun of it, it leans more into the pure maths side of importance.
Ardian Np, I myself dont know physics(yet) at all since im doing self study. Im currently finishing up differential equations and calculus 3 which are the prerequisites to even learn Classical mechanics or really any undergrad physics course.
@@restitutororbis964 I'll come back here and tell you some cases in which the Leibniz rule is useful when I will study Physics II (in Italy this is the name of the module about electromagnetism). I bet there will be plenty of them.
See you in a couple months ma bois.
At 14:12, the terms f(a,t)Δa and f(b,t)Δb are the results of f(c1,t)Δa and f(c2,t)Δb (you called them both c) when Δt tends to zero, so they shouldn't be placed inside of the limit again as you musn't replace an expression inside a limit whith the value to which it tends. I mean the end product is the same, because you then could've just written f(c,t)Δt/Δt and then calculate the limit and get f(b,t)db/dt, but the way you did it is a bit less rigorous. Or am I talking pure nonsense? What I'm trying to say would be really easy if I were there with you pointing at stuff the on the board lol
Papa: Leibnitz trick
Me: what?
(after the video)
HE'S A CHEATER, THAT IS THE FEYNMAN TRICK!!
Feynman's book (surely you're joking Mr Feynman) : so people like to call it after me.
Me: I'm an ass.
I(t) does not depend on x, you dum-dum. No, wait a minute, at 11:53 you take the limit over delta t and get delta b in the result, yet delta b depends on delta t and approaches zero as delta t approaches 0! I don't even know whether the theorem holds with how you've formulated it, usually one requires a(t) and b(t) to be monotone and simply does a substitution reducing this case to the one with constant limits.
I love how you set it up so perfectly that one could easily derive the formula for when a and b are functions of t from here.
Ich hab das so lange gesucht! Dankee! Es war seeehr nützlich :)
Holy shit i Watch this Video so often i Love it
Excellent Understanding of mathematics by this Young boy. Thank youn for explaining Leibniz rule of Integration.
Just think of the integral as an infinite sum. The derivative of the sum is the same as the sum of the derivatives
I feel like I'm too dumb for a physics degree now
@@PapaFlammy69 U sure Pappy?
Fuck if that ain't a spicy proof that I don't know what is.
Mr. Flammy, A plus delta a is greater than a so you have to switch the linits of integration at 4:19. Also the other inegrand shiuld have linits of b and a pkus delta a..because delta a is between a and b..
👌😂
Correct me if I'm wrong but.. I think by the fundamental theorem of calculus:
[ F(a) - F(a+del_a) ] + [ F(b) - F(a) ] + [ F(b+del_b) - F(b) ], which simplifies to
-F(a+del_a)+F(b+del_b) which is the integral from a+del_a to b+del_b
wat
exactly
It's commonly called "Feynman's Technique" because it was him who popularised it in his lectures on teaching science and maths.
How does one become such a cheeky math monke 😍
Young Papa Flammy is so cute
Papa Flamey, I love you diversification of the channel. You taught my Norwegian wife how to make American pancakes. She only knew how to make the inferior European version. Now you're helping me with my homework for this week. Thank you for the value you have provided me.
MISTAKE!!! AT EXACTLY 9:46. Intigration has the variable x so limits will be placed for x not for t.
Double dragon equals double dragon and I just leave it on, that's really old. Each one is different and inverted unto each other in subdivisions to find answers of their lives. It's also a knowledge theory .
Have you tried differentiating an integral using the multi-dimensional chain rule? It makes the the Leibniz Integral rule obvious.
At 10:31, why does Int{from b to b + delta-b} f(x,t + delta-t) dx = F(x,b + delta-b) - F(x,b) and not F(b + delta-b,t + delta-t) - F(b,t + delta-t) ?
thanks for your help.
I'm student studying Economics in Korea.
Leibinz rule for differentiation of definite integrals was big problem to me.
I can now overcome that thanks to you. ^^
I feel you pain, dude...it's not so easy to record a flawless session. Keep up the good work tho!
(Please, leave the "Umm, what can we do now" catchphrase to RedPen! I'm sure you'll come up with something original which will fit your carachter!)
Zonnymaka eh! I don't even notice that myself lol.
Zonnymaka
.......*Isn’t it ?*
; D
Bei Gott, what a coincidence.
integral_a^(b+delta b) f( x, t+delta t) dx = F( b+ delta b, t+delta t ) - F(b, t+delta t)
Ur proof is not rigorous, nor is your statement of fundemantal theorem of calculus, and mean value theorem
I was with this until 10:10. You plugged the limits of integration with respect to x in for t in when evaluating the integral.
I am this videos 69420th viewer and I think that's beautiful
Does the integral in 2:00 depend on x? Wouldn't it be a function of t exclusively?
Idk if you are going to see this comment, but Im just saying, thank you because I thought your whole channel composed of mostly high level stuff, so I was so happy when you linked this video to a newer one so that I could actually know the “why does this work?” Behind he math you where doing
Flammable Maths damn you really do care about the comments, this made me happy :)
12:45 you sounded like Sheldon from big bang theory😂❤
Your videos are great, it would be awesome to see a collaboration video with Dr. Peyam and BlackPenRedPen over something really awesome. Keep it up!
Comments that aged poorly
the upper and lowe bound will be substituted for x in the integral, not t!!
Still dangerous for me to scratching the board and get catched in the uni😅😅😅
yes there's an error at 10:00 where he substitutes the variables, read the comments to find out how it is solved :o
Could you make some videos about Abstract Algebra?
And what's your name gorgeous boy?
What is it used for? Would you please do an example or two maybe three. And PLEASE SLOWDOWN! I realize that you are excited but you sound like you are about to have a heart attack or stroke. I enjoy your enthusiasm for the subject but it is no fun if nobody understands.
It's used to solve many integrals (he has a lot of examples in other vids) and prove many things. One such thing is the transform derivative principle of the Laplace Transform. Let f(t) be a function and F(s) be it's Laplace transform. Then L[(t^n)*f(t)](s){this is the laplace transform of (t^n)*f(t)}=(-1^n)F^(n)(s) {ie the nth derivative of F to s}. This is extremely important for ODE.
Thanks for all your hard work :D. I'm really looking forward to fresnel integrals > f(b,t)delta(b), why does that delta(b) remain unchanged.
Wouldn't that go to 0 as well if we take the limit as delta(t) goes to 0, since that's what allows us to change c --> b in the first place?
Does rewriting b(t+delta t) as b + delta b only work with linear transformations, meaning delta b = b(delta t), or is it not like that in this case?
No, Δb is only a compact way to write b(t+Δt) - b(t). It doesn't matter what b(t) is except it's differentiable
I don't understand why, at 15:12, we can interchange the limit and the integral ?
is it the same meme guy never thought will come here to actually learn maths for test lol
What is t
for the bounds of integration why do you separate [(a + delta a) to (b + delta b)] as [(delta a) to a] + [a to b] + ..., instead of [a to (delta a)] + ...
I am confused as well
Fantastic
u have made some mistakes while you are computing integrals
15:20 I dont quite get how the integrant becomes dt(f(x,t)), i assume you are using L'hopital yes? So dont you have to differentiate the whole sum in the numerator? Where did dt(f(x,t+dt)) go?
Wauwwww
Hey can anyone please help why we cant interchange limit and integral....please keep it simple. I just entered into the world of calculus
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11:50
I can't not understand...could some one help me?
At 12:00 b is continous function of t so delta of b is actually b(t+delta t) - b(t) so it goes to zero as delta t goes to zero...
He is neither a student nor a professor. He is a studessor
Are you from Germany @Flammable Maths ????
I would not have thought to say a(t+deltaT)=a+deltaA: interesting.
I was a little confused when the integral was split, but I see now the idea is the integral over [a+deltaA, b+deltaB] = integral over [a,b+deltaB] - integral over [a, a+deltaA]
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@@beoptimistic5853 He starts off by saying volume is a triple integral - off to a bad start...
@@vinuthomas2814 for phycisian a mathematics is only tools but we should open our eyes to see it, i think it s verry good for scientist people how want convert a virtual and theory to a reality ; it s amzaing tool (he really hard and experienced scientist ) to resolve equation is the most easy but the important we use for what? .
If you have chance you one day know what you do whith if no like a majority's good look
Wow, sheldon started a RUclips channel!
What if the integrals bounds are defined in terms of t?
I can't stand seing an integral of a sum/difference without parentheses.
me neither
I think you forgot to write the integral at the 6:00 mark, but I may be being a brainlet. Good work though, I'm enjoying binge watching your videos.
edit: Oh my bad it was just combined with an integral with the same bounds, I see.