The intuition and implications of the complex derivative
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- čas přidán 31. 05. 2024
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I like informative channel like this 3blue 3brown and many more but youtube algorithims keep recomending dumb stuff and i am praying it will understand and recomed good channels , i found this channel by quora
Awesome content subcribed
complex intgerals explained visually?
@Ashton Smith not that i struggle with it,just curious
Check out one of the easy tips in mathematics.czcams.com/video/D1K-x9Vw4OA/video.html
the virgin regular derivative vs *THE CHAD COMPLEX DERIVATIVE*
finger:
This man👆
yup
Virgin lol
I wonder if there's a reason he chose blue and brown as the colors for the curves. 🤔
3blue1brown?
This is what people will wonder in a hundred years from now about curves drawn in instructional media contents everywhere. The movement that started it all will be forgotten but the tradition will live on.
There are mostly three colors being used here. Orange (yes, brown is orange), teal, and chartreuse.
Tertiary colors are just popular right now. I wouldn’t read too deep into it.
@@MrAlRats nice . We should make a religion out of this .
@@luke_fabis the only two colours I understood from your comment was brown and orange.
If videos like this were available when I was taking Calculus I think I would have appreciated it more and understood it more deeply... it's sad that 30+ years later I have forgotten most of the topics and this serves as a refresher course as well as provide insights I don't think I'd ever connect. I don't know, if this would have changed the course of my life and career, but understanding mathematics and fundamentals like this sure give you an advantage! Thanks for posting and I look forward to your next video of applications.
MATH problems
Best Integral question
czcams.com/video/Wc-U1OPdKJc/video.html
See one time.
Yeah I often wonder how my life might be different if I had the resources available to people today. For example the reason I'm not currently working in digital media as a concept artist (or more realistically just another art grunt for some giant company) is because my portfolio was never really professional quality as I've suffered from shaky hands since I was a young teenager and my drawings were never that crash hot with all the mistakes of unsteady hands. But I fiddled around with a digital drawing app and pad the other day and WOW the level of work im capable of making with them is exponentially superior to anything I can do with my hands alone. My portfolio could have been much bigger, much more polished and I wouldn't have wasted so much of my income buying expensive alcohol markers, paper, erasers, felt pens, paint, canvases etc.... ive spent a significant amount of money on those and if id been born 15yrs later I never would have had a need for them. I could have used that money to do something more useful AND had a great shot at being someone worth employing long term doing something I enjoy.
But im too old to "break in" to that industry now. Itd be competitive enough if id had those 10ish years of industry experience, so without them I can't afford the risk of shifting careers because if it doesn't work out, ive got no safety nets.
But thats how it is. I still remember my mother contemplating how her life might have been different if she'd grown up in the world I did - I suppose its just a matter of hindsight. I used to tell her she should just go for it and pursue what she wanted, but being single with 2 kids I totally get why she didn't, now.
Still... the future is only what we make of it from this point onwards, so the fact that these resources exist is still an excellent thing.
Now if only I was any good at mathematics so I could comprehend this video...
I highly recommend 3blue1brown's intro to calc playlist
Same here ... after 30 years I have full scale passion to conqour all fields of mathematics that once I heard of during the wasted years of college.
Same!!
You read my mind. I've been wondering about non-integral derivatives.
MATH problems
Best Integral question
czcams.com/video/Wc-U1OPdKJc/video.html
See one time.
Amazing timing, Zach. We literally covered this in fluid mechanics today. Thank you and may the Rankine half-body bless your singularities.
now do quaternion derivatives, octonion derivatives, sedenion derivaties and so on, also is there like generalized numbers? like n-nions?
@Adrian Martinez Dorsett which one ?
@Adrian Martinez Dorsett nice dude haha I was wondering if it was a Hamilton reference indeed...
This makes me so *drah*
DRAHHH
Drah
*B* rah
*hard* os em sekam sihT
U make no sense
This is the most intuitive explaination of complex derivative and derivative in general i've ever seen. This needs to be seen in the first lesson of calculus 101
I'd definitely recommend the book "Visual Complex Analysis" by Tristan Needham, if you want to learn more about this.
Video last month 'Why imaginary numbers are needed to understand the radius of convergence' is exactly the example in Needham's book. So I bet Zach already referenced from it. Good choice. :)
Thank you so much for naming the book. It was very useful.
I remember randomly getting it out of our school library when I was ~15 because I thought the pictures were cool. Didn't understand a word but it was fun to look at!
I hated that book as an undergraduate and did not find it helpful
It’s funny how pirates spend their entire lives looking for treasure when the true treasure was in the friendships they made along the way...
Is this a reference to Lemmino's Cicada documentary ?
Math teachers are pirates. They're always trying to find the X
@@GameJam230 yes lol
My brother and people in 4chan came up with that phrase friends we made along the way although idk if u r using that as a reference. It was referring to the wall trump was saying he was going to built.
@@xxnotmuchxx that has been a saying for like a super long time???
So interesting to see how we have the understanding of the ins and outs of complex concepts. Great work!
This has helped me put together many of the concepts I've loosely learned on calc 3 these past few weeks; some of those even from today's class. You've released this video at a great moment for me!
I haven't seen this topic covered a lot. I'm transferring to EE and will have to take another math class involving complex derivatives and integrals so this is great!
I have been studying complex derivatives and integrals over the last couple of weeks. Thank you for making this!
I love learning about stuff no body else knows, and I understand these topics well because i have watched tons of such videos and have a good base
Zach, you are a most talented teacher. I enjoy listening to you and thinking along. Your diction is easy on the ears. The whole package is a rare gift.
I recommend Herbert Gross' MIT lectures on complex analysis. Great stuff.
The legend!
Awesome video! This is such an interesting topic.
Damn this is perfect timing. Just started my complex analysis course at uni 2 days ago!
I really love channels like this.❤️ Keep it up plz
I can follow along with this because it's making something abstract real, described in terms of movement with actual visuals. I never understood a single concept of calc because it was just a flat nonexistent thing in a textbook. :0 This is awesome.
I am so grateful for this! Thanks!
You made me appreciate math class way more man! Great work!
Very clearly described -- congratulations!
Ordered the martin Gardner the collasal book of short puzzles 🧩 book you recommended 👌🏿cant wait to start it lol
I wish you had posted it when I was in engineering. I barely passed my maths.
But now I love it.
I'm taking calc 1 rn and thought this would be a good thought exercise video for me. I'm crying now
Wow this is a great, amazing introduction to complex differentiation.
Wow this was amazing
Thank you sir. I wish if you would consider making a similar video on Complex Integration.
Thank you for making this video.
I think it is kind of gap when you jumped to that two curve parts while explaining the conformal mapping, you should have explained the constant associated with two curves' intersections are being mapped to the output spaces and then that rest of movement in input and output is all right. And I have a question, if we move along one curve say 1 of two members from intersection point and then separately from junction point along 2, the angle between them is equal to the angle between the output movement along transformed curve? And this property is not holding if instead of moving along 1 or 2 , we move along other paths as those two paths are not going to be level curve of each other? And yes, I was surprised to know about that equality of derivative in all direction instead of different phase stuff. I like this video, but pls clarify my doubt.
Complex derivative? More like "Completely delightful!" This video was so fun to watch, and I love how you explain things. Thanks so much for making and sharing it!
I'm watching all this as a high schooler and it's going over my head, watch me actually need to come back to these videos in college
Two Zach videos on the same day!
I registered in your sporser. The reason was primary to support you! Great job
im pretty new to calculus, but complex numbers and how they relate to the subject have been bothering me lately, so thankfully I found this vid
Awesome plan to use this complex system in analyzing and optimizing geometries to make fea better I think there its applications can increase the optimization for forces
thank you for your information
Very useful, thanks!
I was waiting for this topic in Internet
Thanks to Zach star.
@@zayncharania9182 Do what?
Shoutout to my boy Jay Z
very good content!
We definately want more videos on complex analysis it's so cool and has direct applications in math like in transforms ...
Great video
Thanks for video but I have a question.
So when you make the transitions on conformal map, what does graphs during transitions mean?
Like, on 10:24, what does this graph means and how did you get it?
Can u do a video on how hard is to discover something in math? I have seen some simple problems with simple answers and all the way to Riemann hypotheses.
Speaking of xy = n and x^2 - y^2 = n graphs, I noticed that they are both hyperbolas.
Even algebra can prove this! If we map x' to (x - y), and y' to (x + y), which results in a 45° conformal map, we actually get two conjugates.
x'y' = n
(x - y)(x + y) = n
x^2 - y^2 = n
Learned a new way of visualising differentiation
How did you do grid lines transformation in steps? What were functions in between?
I like these visualisations a lot :)
Wow Zach here is a master of mathematics on CZcams commenting on your channel.
Very nice thankyou.
amazing!
I feel like this was a 15 minute long commercial for the globe behind him hypnotizing us with flashy numbers and lines that zoomed in.
Hey, i would like to know what u use for animation
This derivate analogy is clear for stock derivatives.
Your t-shirt quote pi^2 = g really got me thinking... 🤔
I absolutely don't yet understand this kind of maths but im enjoying it💕
Did anyone else see the optical illusion when scrolling past the thumbnail of the video
In fact we can think of ordinary derivatives as a particular case of complex derivatives
F'(x)= lím h--0 Im{F(x+ih) / h } try this with any function like ln(x) you will be surprised. Congrats to Zach amazing videos
This really helps visualize the curvature of space.
I wouldn't have thought to find another channel with a quality like 3Blue1Brown, but this one is on par :)
Please make more videos on complex numbers
11:46 was kind of difficult to wrap my head around. But its a very neat way to analyse the functions!
Good. New ideas added
I'd like to see a video with more examples of useful conformal mappings.
Why don't YOU play yourself with conformal maps? That the best way to understand the topic. I once implemented a conformal map grapher using desmos (inspired by the 3B1B video about Pythagorean Triplets), and I'm sure there are easier alternatives online to graph them. Then strat playing with them, see which properties the have. It's a pretty fun and different way to think about functions
What program do you use to animate this stuff?
Great man! I think you should talk about cellular automata 😬
fantastic
love it
Please make more videos about visual complex analysis.
Thanks
What software do you use to make these visualizations
Someone please tell me which software is used for this visualisation
Amazinggggg
Is the complex derivative the same as the tangential derivative in Geometric Calculus? How does the complex derivative compare to the Vector derivative in Geometric Calculus and fractional derivatives?
I want to define x^2 in terms of triangles instead of squares with three dimensional axes on a 2d plane at 60 degree angles. How will this affect things since the dot product between the axes is no longer zero because they are not orthogonal?
Could you comment on the complex derivative in terms of a quotient of 2 directed integrals?
May I please know what the derivative of the fn being zero has to do with conformity?
If only this video came out 4 years ago, when I was in high school
Please, I didn't get understandable solution to this integral: integral of e^[x²] dx.
I get erf function from Wolfram Alpha. What is it?
the definite integral cant be expressed with normal functions. Its a very famous integral in mathematics. Surprisingly, forms of this integral can be solved from -infinity to infinity, but not any general bound
Most functions where it makes sense to talk about the area under the curve don't have an integral with a closed form. This means that you can't write it as some combination of trig functions, exponentials, logs or powers of x with a finite number of terms. The error function (written as erf) is a shorthand for that integral, and there are many other functions that are solutions to integrals or differential equations that don't have a closed form that we give names to. It makes it easier to work with them
@@henryginn7490 Thanks, I get the same answers on Quora. It is nice to see active people contributing to something far more far fetched.
Adityadhar Dwivedi tbh I think everyone has the same thoughts when they first see something like erf, li, gamma, J, etc so it isn’t far fetched at all, just a natural part of the discovery process
@@henryginn7490 yeah, discovery process will be integral to our independent idea.
1:00 fuck no I was staring at that g approximation. Nice one!
Complex analysis
How did you plot complex numbers on desmos?
Please make a similar video on complex integral.
The only thing I would add is that for the derivative to exist, it is not sufficient that the limit exists and is the same in all linear directions. It has to exist and be the same along all non-linear paths of approach as well. This suggests how the complex derivative existing is similar to the output of the function being "locally flat" aka Euclidean at that point (think of splitting it into two surfaces for the real and imaginary parts), which is why the existence of the complex derivative is such a nice condition and much stronger than the existence of say, the gradient of real-valued multivariate function.
Total derivative and Wirtinger calculus next please!
I'm just starting complex analysis with an awful professor. This is amazing.
It really be like that sometimes
The part about the conjugate function blew my mind. I might be wrong, but can the same function also be described using vector-valued input and output while still being non-differentiable?
Can you make one video on circle having complex intercepts and parabola having complex roots
Ah, yes. Let's start the day just before going to work by watching a video on stuff that made me drop university.
Since when does math have intuition? What are we? Intuitionists???
Some are. Heathens. I say! Heathens!
Chad Platonic Idealism vs. Virgin Intuitionism
¿ Cuánto tiempo demoras preparando un video de estos?
What software did you use for showing these transformations
desmos
Can you suggest some good books for complex analysis?
It was easier for me to understand the complex derivative after learning that:
1. The transformation of multiplying by a complex number let's say a+bi looks like multiplying a vector by a matrix of this form
a -b
b a ...which only rotates and scales the plane
2. Jacobian matrix of an R2 to R2 function is the linear transformation that most nearly resembles the function at a given point
3. If a jacobian has the same form as the matrix above, then at any given point it only looks like a scaled rotation, which means, that angles are preserved around that point, and it can be described as multiplication by a complex number!
Now take the conjugate function for example, it flips the whole plane, so all angles are inverted, it can't be described by a rotation, so it can't be described by a complex number, so it's not differentiable
The glove seems to be spinning in the wrong direction: Why is that? Is the north at the bottom or is it simply wrong?
we need to cover some topics like .
stochastic process , stochastic calculus , Ito calculus
these animations remind me of the nyquist diagram in control systems
Did you use Desmos to make this? Because that looks a lot like Desmos.
Also, my favorite thing is how you can fairly intuitively derive Euler’s formula from this idea of complex derivatives and the fact that e^z is the derivative of itself. Just pick any number of points around the unit circle, draw where the real and imaginary differentials are for each point, and Euler’s formula jumps out at you. The key observation here is that the imaginary differential is always tangent to the unit circle.
Yep! All desmos.
Zach Star Ack, I was editing while you were replying XD
Which software do u use to explain all this???
desmos
_”Cooler than the regular Derivative”_ ........ I’m confused and lost already. I want to get good at maths just for its own sake and see what higher mathematicians level are at.....but damn! I do t think I’ll reach this level. 😬
It's probably best to have a basic understanding of differential calculus before attempting to understand this, because this video does assume that you do know how derivatives work and such
If you take your time you'll get to whatever level you want. Embrace the confusion and frustration that you'll experience along the way. Study from multiple textbooks. Do as many exercises as you can. Have fun. And keep at it. That's all there is to learning math.
"Complex number fundamentals | Lockdown math ep. 3" ~ 3blue1brown
czcams.com/video/5PcpBw5Hbwo/video.html
Check out Grant Sanderson's channel, too.
I am teaching myself math and later I want to solve unsolve problems in math. I am reading a textbook on complex analysis call complex variables and application and its free online. It deals with imaginary numbers/ complex numbers. Best to learn calculus first. I gotta revisit calculus, series, etc too.
@@barca928 “to whatever level you want”. Lol, best bullshit ever.