0:27 “This isn’t because the values don’t exist, but because they’re not real.” Hmmm okay mister

This always seemed obvious logarithms are just roots for when x is the exponent instead of the base just like square root of -1 except a bit fancier

Letting the log have multiple values doesn't break the math.

0:38 this identity isnt actually eulers but still but the TRUE eulers identity is (sigma n=1 to ∞) (1/n²)=π²/6

Thanks this was helpful

The only question i think is the prime of j(e=0)=0 implies that the "principal of least action",which means the path will choose the stationary point of j (tangent line), so the prime of j(e=0) is equal to 0

thanks.. that was amazing

e^(I*pi) = -1 Ln both sides Pi*i = ln(-1) Pi = ln(-1)/i I proved that pi is rational in the complex plain, is this true or did I do a wrong step?

This is the best explanation of variable calculus so far on the internet

it would also be good to have more explanation of why there is a total derivative wrt x instead of a partial derivative

a few mistakes in the integration by parts spoil this video unfortunately.

What is the intuition behind hyperbolic trigonometric functions?

you actually need to do a series of calculus of variation and optimal control theory. Thank you very much

😮

😊very clear! Brilliant!

I waited and waited for the narrator to say that f(z)=ln(z) is literally the inverse function of f(z)= e^z . But he never got there.

very nice question

Loved it

Beautifully explained...

If the angle was measured from the top would it be R(cos(t)-1)?

6:43

Thanks💖

now optimize multivariable functions on manifolds

Awesome video! Note: The domain shown at the end isn't really standard, because you can't define a continuous version of Log on all of C. If you make both inequalities at 3:26 sharp, it will become holomorphic, which is much nicer in proofs and calculations.

thats is not a good enough proof. doing functional analysis you have to take into account for the domains and range, you have not put up condition. if you said f is continuous over all R then your statement would be considered true, but for example if the function blows up to infinite at the specified point and is obviously not continuous and differentiable then your proof would be wrong. All in all nice video but you took a pretty general case and did not specify the generality

Excellent job, Xander! Thank you

u run so fast that I can't keep it up but awesome!

Actually your result is wrong. Logarithm of a negative value is a multivalued function. That means it can have multiple values. For example: 3i*pi is also a solution.

So this is how Pure Mathematicians developed the construct of a Complex Logarithm Mapping. Maybe this is the process that Euler used.

Thank you for this video, it provides a clear derivation of the Euler-Lagrange Equation. However, to be rigorous, at 4:18 du should be equal to d/dx (∂F/∂ȳ') dx = (∂F/∂ȳ')' dx and not equal to d/dx (∂F/∂ȳ'). What I mean is that du is not the derivative of u, but is instead the differential of u, which, by definition, is given by du = u'(x) dx = (du/dx)dx. Similarly, dv should be η' dx and not η'. Despite these innacuracies in the screen at 4:18, we can see that u, v and du are substituded correctly in the next screen at 4:31, so the proof of the Euler-Lagrange Equation is not compromised. en.wikipedia.org/wiki/Differential_of_a_function#Definition I also have a suggestion regarding the presentation. I think you should add some visual queues in the screens where only expressions appear when reading out loud the contents of the expressions, so that the narrator's voice is followed in the image as well while possibily inserting additional information. For example, in the screen at 4:31, it would help a lot to add some colored brackets below each part of the expression stating each element of the integration by parts formula (u, v and du) and make them appear in the screen as you are reading the formula. Apart from these details, great work with this video! I hope you continue doing more videos :)

Amazing👍👍👍👍👍

(1 + 2n)Pi*i, where n is an integer?

that's what i was looking for

Going nowhere? If y is a curve, does J then just depend on x? Confusion

When you say "nowhere", you mean that f does not go up or down? I can work some of the problems in Mary Boas' book, but I found this lecture to be heavy-going.

This channel's gonna blow in 3,2,1...

U R amazing!

شكراً جزيلاً 🤍

Kindly speak allowed and in a slow paced way for non- natives.

Cool Video!

Thank you for a fantastic derivation of Hyperbolic Functions.

What is the intended audience for this? Is it people trying to watch a mathematical equivalence between a starting equation and final equation? Certainly, the pace combined with the formulas suggests it does not matter WHY this is done. Just as Einstein earned a Nobel Prize with a three-page paper but my 80-page thesis was not worth printing except to get me an advanced degree, I perceive this video as testament that someone likes to talk a lot. WHAT is the goal of the video? What IDEAS guide you to the goal? I don't see the latter question being answered. Rather, I see someone showing how mathematical equivalence works during manipulation of equations. Call me disappointed - by almost every Lagrange video that I am finding.

find the optimum of J=int[x'^2(t)-2tx(t)]dt please

great video !!

The flow of the exposition is good but showing only one formula at a time on the screen makes this really hard to follow the sequence of steps in the derivation. One needs to keep going back and forth all the time to make sure no tiny nuance in the semantics was missed.

The complex definition of sine and cosibe is no the same McClaurin Series as always?

The most simplest introduction ever!!

Well made video, but as others have already commented you moved too quickly. Would have liked to see this video at a slower pace with pauses to digest and breakdown what has happened in previous steps.

Thx Xander Gouws for the Proof of Langrangian eqution . Very nice done with min. amount of time explaining the procedure how to get there. I may add it seems to me : Although Calculus of Variations is very usefull and efficient Method showing the Proof, represents however : A littel of Eulers application regarding to Subject- Matter. Hope you find it interesting enough to investigate. cheers🍻

Amazing 😄