IMPOSSIBLE INTEGRAL? Here's how to solve it and its properties 

You can't just casually mention that this madness has application in statistical mechanics and not be prepared to show some of them.

the integral sign ∫ is an elongated "S", for sum using it for product integrals is a clash of flavors, like using salt in place of pepper

NOT THE DX IN THE EXPONENT 😭😭😭😭

I lost it at sin^dx(x) lmfao Cool video btw

Therapist: Cursive π/2 isn’t real, he can’t hurt you, it’s all in your head Cursive π/2:

Mathematicians: God damn, physicists! Stop treating dy/dx as a fraction, that's not how it works! Also mathematicians:

This is the absolutely best integral you solved. Thank you for this innovative integral and proofs.

Its inverse function is the product derivative f*(x) = lim (f(x+h)/f(x))^(1/h), which is df^(1/dx) (x), and is equal to exp(f'(x)/f(x)) 😊

It makes no sense to use the integral symbol for this. The integral symbol is a vertically elongated S (and this is also how the lowercase s was written a few centuries ago) which recalls the word "sum", which is the operation that appears, discrete, in its definition. ALL types of integrals can be traced back, in some way, to a direct sum of values ​​of the integrating function. Here, however, we have an object that arises from a product, so it should be indicated with a symbol that recalls the letter P (capital or lowercase).

How about the derivative? I actually derived that myself a few days ago. I have started with a loose formulation of the fundamental theorem of calculus - "the sum of all small changes on a shape is the total change on the shape boundary". I've started applying it do different ideas of "change". Standard derivative defines the change as the arithmetic difference normalized with respect to the change in x. I thought to myself - why don't we use normalized ratio as the change? So I have came up with this: [f(x + dx)/f(x)]^(1/dx) (the brackets here only to show clearly the order of operations) Using this as my starting point I had written down what the fundamental theorem should look like - the product of all small ratios on a segment is ratio of the borders. Through this I have derived the integral you are showing here. So the question is - what about derivative? P.S. the fundamental theorem holds for many more notions of change. You can describe it for discreet functions (from Z or N to R usually). That's called "forward difference operator". You can also use graphs. For the graph (V, E), where V is the set of vertices and E is the set of directed edges connecting them we can describe some function f: N -> V that would describe something traveling along the graph. Than the derivative would the function df: N -> E that defines the edges on the path. Than the "integral" from a to b where a and b are natural would be the path from f(a) to f(b) that the function had took. Idk if this useful or not, but it works.

Is there any geometric interpretation of a product integral??? what can we evaluate with it? great video btw

And I thought I discovered this operator😅😅😅. Ty for letting me know it's already done😊😊😊.

I took graduate level stat. mech. and absolutely loved the math that was involved; there are some really intricate manipulations which result in simple expressions for such complex systems. I ain't never seen no exponentiated dx though, but my first thought was to take log of it.

why does the sum turn into the product for the riemann sum

Since the integral sign represents a long S, the product integral/ geometric integral is representent by a long PP

Why didn't you use the Pi symbol (related to the product) instead of the elongated S symbol (related to the sum?

nice one! rediculously awesome indeed Kamal!

Cool, now do the integral of f(x) tetrated by dx

Awesome 👍 After a long time I will see something else NEW in the integration related vedio ....!!! Thanks for sharing with us ....!!!

Absolutely awesome! I'm so happy now. I will expect next video!

However, the biggest disadvantage of the integral is that only positive functions f(x) > 0 can be considered, as the ln is only defined for these. This is a very, very, very strong restriction.

Dr. Peyam discussed the sqrt(dx) a long time ago. Is there anything you can add to this?

that's sooo cool! especially the distributive property (if you will)

Another really cool weird integral is the normal integral (not product) of f(x)^dx-1 this quantity approaches 0 so it would make sense when you apply an integral it reaches a number, you can actually find out that it’s equivalent to the integral of ln(f(x)) if you multiply and divide by dx and notice the inside is a limit, to be extra sure you can put it into the summation definition for the integral and evaluate it to get the same result.

Can we replace the numbers a and b with the functions f and g in normal integal if so would the formula change ?

So can we generalize this by saying: Given an operator X and a product integral Pi, X Pi = Pi X I.e. the operations commute? Would this be the connection to Statistical mechanics?

I remember I once wondered what the equivalent of Π is in continuous functions just as Σ has ∫ I thank this video for finally answering that question many years later. However, what would be the inverse function of this "product integral"?

The exponential form of the product integral kinda reminds me of the formula for the factor of integration for first order, linear and nonhomogeneous differential equations, wonder if there's something there.

Wonder if there would be a good definition for ∫f(dx), something like ∫sin(dx)

I enjoyed this video indeed

How exactly did you get a product from taking the integral of your function raised by dx? Shouldn’t it just be the limit of the sum from i=1 to n of f(x_i)^(delta-x_i) as n goes to infinity?

У меня сразу появилась подспудная мысль, что без логарифма здесь не обойтись.

This doesn’t appear to be the same as the kinds of product integrals mentioned in Wikipedia, and I seem to recall that your previous video agreed with the Wikipedia version. Maybe I’m misremembering? To be sure, Wikipedia indicates that there are multiple versions of the notion of a product integral, but I don’t see the definition you’re using there. Here is a link to the Wikipedia article for your convenience. en.wikipedia.org/wiki/Product_integral I wish you had included a reference so I can trace its provenance Anna so we can all learn more about this. The Wikipedia article cites Volterra for their version, and that’s quite a traditional origin. Who introduced the world to this product integral you’ve discussed?

Bad notation. The integral sign is a typographic variant of the letter S, as the sum sign Sigma also. Product should use different symbols.

Yeah bad notation, you could integrate f(x)^dx -1, and that's just the integral of ln(f(x))dx, but for a product integral draw a p instead of the s...

If you had to do this integral in the reverse direction, would you have xd in the exponent or the base?

Hi, "ok, cool" : 1.42 , 6:26 , 6:45 , 8:19 , "terribly sorry about that" : 8:49 ,.

Yeah I do know the statistical mechanics meme! Now it's everyone else's turn to know it.

Every calculus student's dream, Integral of f.g = intgeral of f times integral of g!

infinite products are awesome

Hello, how can i start watching your videos. I'm a class 12th student, us there anything i can understand. If you know plz tell me😢

on who's regard did you take the initiative to replace the summation with a product?

Is it possible to request an integral?

F = int(f^dx) F' = f^dx

Fun stuff 🎉

0:15 LIES! Well okay, maybe it is a thing but it makes my head hurt.

Could you please tell me what application you use for this? I've been trying to find a nice whiteboard like app for a while.

Now do product derivatives

I wonder if Mathematic can compute this forms

but why is it an integral sign if it is not defined as a sum?

So can you take the dx root of dy?

Absolutely crazy 🤣

This video is clickbait until you give a valid reference for your use of dx in the exponents. No where in the wikipedia article for Product Integrals is the integral symbol used while simultaneously the dx is an exponent. When the integral symbol is used, dx is a factor. When dx is an exponent, a product symbol is used instead of the integral symbol. The video is interesting in that it shows how arithmetic mean correlates with the integral, and how we can use that correlation to generalize the discrete geometric mean into a continuous analogue. But an integral with dx as an exponent is clickbait until proven otherwise.

Why product and not sum? Isn't integral is always a sum?

Where did you meet such in stat. mech.?

Wow interesante

What if divide by dx? int((f(x))/dx)

to be even more ridiculous in the last step you should've turned e^-sin(x) into 1/e^sin(x), so you would have A*(sin(x)/e)^sin(x)

Product derivatives?

At 2:55, it is said: “we know that this product now turns into a sum”.. Could someone please explain why ?

How does product term becomes summation term ??

can we say (f(x))^(d/dx)=lim_h->0((f(x+h)/f(x))^(1/h) then ln(f(x)^(d/dx))=ln(lim_h->0((f(x+h)/f(x))^(1/h))=lim_h->0((ln(f(x+h))-ln(f(x)))/h)=f'(x)/f(x) f(x)^(d/dx)=e^(f'(x)/f(x)) ... (Ac^x)^(d/dx)=e^(ln(c))=c No idea if this is legit, and the notation is questionable.

❤

purple is hard to read

I never learner this in calc? When is this even taught?

wWhat does that even mean?

just take a log

Poor Ehrenfest. He had some demons to fight. He lost.

Third

8th 😎 wait I mean 7th

firsties

"and it's properties" AAAH, it should be "and its properties".

BRO BE WATCHING THAT ONE MATH GUY BRI! 🗣️🗣️🐐💀☠️🔊🗿💥💥🗣️🗣️☠️🐐💀🐐🐐💀💀🔊🗿🗿💥💀📢💀☠️💀🗣️💥🔊🔈💀

what is the equivalent of this property in the physical world