The Leibniz rule for integrals: The Derivation 

@@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!

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.

Glad you enjoyed it! =D

he is so much more dude...so much more :')

Student studying to be a professor IIRC.

Yes.

@@aryan040103 you re s

Yes

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.

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.

@@benoitavril4806 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.

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

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.

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Flammable Maths damn you really do care about the comments, this made me happy :)

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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

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=)

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I was just thinking about this too... Is there any particular reason why its like this?

Comments that aged poorly

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@@PapaFlammy69 i am studying this right now do you have vectors as well on your channel?

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

I was just going to ask the same question. That could be a small error which does not contribute very much to the conclusion. in 9:50, you should first rewrite the integral f(x,t+delta t) as integral of f(x,t)+f'*delta t (definition of derivate) then use the mean value theorem which gives you f(c,t)*delta b +f'(c,t)*delta t* delta b, then dividing this term by delta t and sending it to zero you get f(b,t)*b'(t) +f'(b,t)* delta b, where delta b is infinitesimally small and therefore zero. So you will get the same result f(b,t)*b'(t)

@@Gossamer2288 yep same question on my mind as well

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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

If a or b go to infinity you can do this exact process. Just have on the outside a limit saying that a or b goes to infinity or minus infinity. Then when the process is finished you can distribute the limit back into the answer.

@@antonioromerio5555 okay thanks. Do you have an idea of my other question?

@@kwameawereohemeng3931 I believe any such restrictions should be the same as those already placed upon a definitely integrable function. f(x,t) has to be continuous on [a,b] and definitely needs to be differentiable with respect to t. a(t) and b(t) must also be differentiable, but these things all just follow from the result

👌😂

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

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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

Zonnymaka eh! I don't even notice that myself lol.

Zonnymaka .......*Isn’t it ?* ; D

Bei Gott, what a coincidence.