What are /r/3Blue1Brown's favorite Products & Services?

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Ohhh, gotcha, you can write this as

a^(b) ≡ a (mod b).

Other than that, great! This is actually a version of Fermat's little theorem. I would recommend becoming familiar with mod notation before getting too far into it, but the proof for this is quite neat. More generally, it states that:

a^(p-1) ≡ 1 (mod p), where a is any number relatively prime to p, i.e. a can be greater than p if you wish. Props to you for finding this yourself, modular arithmetic is hella cool.

You can try to compile the Python code to webassembly. But honestly, I don't think manim is a good choice here.

You want:

1. nice graphics
2. interactive
3. fast to update (I assume, otherwise interaction is not fun)
4. in the browser

Manim can do the following:

1. nice graphics

For all the other points you would have to fight really hard or make compromises.

I think you are better of using d3js or something that suits you. If you really want to copy the 3b1b style (which is great, but not the only visual style out there that looks good) you can most certainly achieve something similar with most libraries out there.

im assuming you mean that the position (p) of a particle changes in time, and that p(t) = e^t(1+i) so by multiplying that out we reach p(t) = e^it+t. that can be rewritten as p(t)=e^it X e^t . so as you have seen from the videos, we know that the e^it part will control rotation. as t increases the position of the particle will rotate around the origin due to that term. Along side this however the magnitude of the distance to the origin is controlled by the e^t part.

we know that the position is rotating around the origin in a circular arc. but also getting exponentially further away. so the trajectory of the position of the particle is a giant arc who's distance from the origin is exponentially increasing (see this graph for what it would look like)

now onto its velocity, even though its position has a complex number in the exponent, the how you differentiate it does not change. so since p(t) = e^t(1+i) v(t) = (1+i)e^t(1+i), here we can see that the velocity is just a constant times by the position. and as before we can use this to work out a graphical viewpoint as to whats going on.

multiplying by a complex number has the effect of rotating the number by its angle, and stretching it by its radius. so we can right v(t) = (1+i)p(t) this means that the velocity "vector" will be at a 45 degree angle to and 2^0.5 times larger than the position vector at that point.

this does also make some intuitive sense when looking at the graph of the position, the velocity is at an angle less than 90 degrees (since it can't "pull" the position round into a perfect circle) but is defiantly curving it round by a constant angle.

hope this helps

I think it might be interesting to dispel some myths about infinity, such as the common misconception that infinity is just a really large number (like when kids say "Infinity plus one!")

Brilliant has a good write-up on infinity that isn't too advanced, i.e. is perfect for a casual informative speech. https://brilliant.org/wiki/infinity/

doing an animation to show how this works would be pretty cool. It uses hypocycloids,or the trace of a point of a circle rolling inside another circle.

https://www.desmos.com/calculator/xbjepscoyr

I once made a Desmos graph to show myself the cartesian multiplication of complex numbers, and I was surprised to see how polar it looked. I remembered that multiplication by a constant yields that constant as a derivative, so my graph also demonstrated what complex derivatives look like. From there I recalled that e^(x) has a derivative of itself, and realized that the graph of that must form circles. Then I remembered that e^(0) = 1, which gave me a way to orient where the function sat on the plane, and then I quickly realized where Euler's formula came from, because the circle intersecting e^(0) is the unit circle. It was a most insightful afternoon!

I lost that graph when my old high school shut down my email account there, but I have recreated it:

https://www.desmos.com/calculator/3zdukamxxw

Would you be open to using textbooks? I used this book during my undergrad and found it really helpful! It has tons of practice problems for you to try out. I'm not entirely sure if they follow the videos in the playlist 1:1 but you could always just skip to the relevant sections in the book. I haven't personally used it myself but have also heard great things about this. Hope it helps and good luck!

If you are impatient to learning and/or reluctant to Python, I suggest Geogebra. It has a graphical interface and is available as an online app, so you can just jump into it. It's a great tool for learning geometry overall.

VM means virtual machine. Essentially installing a program that will simulate another computer in software. You can use this to set up a virtual Linux environment

https://www.virtualbox.org/

You should find guides for windows installation there.

But this may just complicate the process further, you'll have to commit to learning some rudimentary Linux and bash stuff to do this as well.

GA was what turned me from a game tech into a lifelong mathematician. Chris Doran and Anthony Lasenby are to blame… it was their book I discovered on a shelf in the office by accident and spent a year banging my head against until things started clicking.

There was a text I used when I was tutoring Trigonometry years ago. The truth is, there are only like four or five things in trigonometry to explain, and the rest just comes down to twisting and turning those ideas to get the rest of the course.

From what I remember, most of these ideas come down to two or three figures, which you can find in this text: https://www.amazon.com/Proofs-Without-Words-Exercises-Classroom/dp/1470451867/

I vaguely recall Concrete Mathematics covering Josephus and its generalizations in one of the early chapters. You might want to take a look at that.

The factorial function satisfies:

a!/(a−b)! < (a+b)!/a!

(This is called log-convexity.)

There is only one way to extend (or interpolate) the factorial function to the positive reals such that that inequality holds. This unique extension, surprisingly enough, has ½!=(√π)/2. Using the rule n!=n·(n−1)!, we can extend this backwards to (−½)!=√π.

You can graph x! at https://www.desmos.com/calculator to see what this interpolation looks like.

Made what u/Holobrine said into an interactive on Desmos: https://www.desmos.com/calculator/hrowua3uwg

(drag the f slider around to change the frequency you're looking at, or change the function h(x))

I believe from googling, once you setup a python environment in Visual Studio, you can install manim in the terminal in Visual Studio. You can use pip3 install manimlib (if on Linux/MacOS) or pip install manimlib (if on Windows) in the terminal to install manim.

https://code.visualstudio.com/docs/languages/python

Coming from a Mac user, I’d recommend using Chocolatey to install python, pipenv (makes it easy to set up virtual environments and load them up with packages), gtk-runtime (for cairo), miktex (for latex), ffmpeg, and sox.

Hello! You can use ApowerShow in creating video clips. This program can also edit, record and convert videos. I always use it when it want to make stunning videos in just few seconds.

brilliant.org

The courses aren't free, but there are lots of community-made problems that are genuinely creative and challenging. The best part is that you can add your own problems, too.

What you are describing – including the name "computational notebook" – is the concept that was introduced by Stephen Wolfram with Mathematica in 1988. Over the last 30 years, notebooks have evolved into versions more suited for data science such as (python-based) Jupyter, versions more oriented towards math like the one you describe (take a look at https://www.sagemath.org/ for instance), while Wolfram Mathematica is designed for both. As always, Wikipedia is a good place to start https://en.wikipedia.org/wiki/Notebook_interface .

Yes, if I got you correctly. These partial derivatives span what is known as the "tangent space" - you can think of this as taking the tangent lines to the coordinate curves and "smearing" one of them along the other one.

> But since its a plane we have a single tangent line

Wait can you iterate on this and what you mean?

Maybe this helps: https://www.geogebra.org/m/avbdq8my So we have two parameters u and v, those define a surface via this mapping (u,v) -> (u, v, f(u,v)). The partial derivatives give you the two red lines as tangents to the coordinate curves b and c. And the jacobian is exactly the matrix that takes the unit vectors in the domain to the tangent vectors that span this tangent plane.

It can easily get confusing since there's a bit more happening and some spaces get implicitly identified with one another (basically because we can interpret points in R^n as both points and "vectors"). If you have the option to take a look at the book "A Visual Introduction To Differential Forms And Calculus On Manifolds" (maybe the author has published a pdf online) it may help to clear things up for you

He is defining a multivariable function in two variables: S(x,y). This function, in 3d, would look like this:

https://www.geogebra.org/3d/s8xqqbax

The blue plane is the constraint, and the projection onto the (x,y) plane of intersection between S and the plane is a circle. The constraint, k, is the square of the hypotenuse of that circle.

Could you create a video that visually shows why the abc formula works the way it works?

I'm not talking about some visualizations about completing the square and then deriving the rest of it, I'd like to see a full geometric intuition on it.

For example, when I play around with the first and third form of a quadratic equation on https://www.geogebra.org/m/EFbtkvVP, I can visually understand what all the symbols are doing.

For example, with the first form: a is width, b is a side step left or right with some parabolic biased step up or down and c is adjusting for the parabolic bias by stepping up or down.

With the more intuitive third form: a is width, h is a side step left or right and k is a step up or down.

Is there a nice visual intuition about why the abc-formula is the way it is? I get the algebraic interpretation, I visually understand why completing the square is the way it is but I wonder if there's a complete visual understanding of the abc-formula.

Just to add to this, section 3.F of Linear Algebra Done Right gives a great discussion of dual maps and how they work. It has an accompanying video on youtube.

I'm a fan of Ian Stewart and David Tall's book on the subject if you can get your hands on it.

One of the neat things is how you prove that the integral of a differentiable function around a closed loop is zero, if the function is defined everywhere inside that loop. You break up the area inside the loop into triangles, so that your integral is the sum of the integrals around each of those triangles. A differentiable function is one that's roughly linear at small scales, and linear functions have antiderivatives, and the integral of something with an antiderivative around a closed loop is 0 by the Fundamental Theorem of Calculus, so the integral around each small triangle is gonna be roughly 0. And so, adding all the triangles together, the total integral is 0.

(You need to keep careful track of the epsilons and such to make that rigorous, but the point is that the integral over each triangle is 1) small because the triangle is small and 2) small because it's roughly linear, so it's like doubly small. So it stays small when you add them all up)

I'm sorry, that wasn't totally coherent… but read the book, it'll make sense

hi, I'm a developer and recently I found myself facing a very curious mathematical problem: on the play store I found this game and I was wondering if there was a mathematical rule to determine if a maze is solvable or not

Game link: https://play.google.com/store/apps/details?id=com.crazylabs.amaze.game

It's a very popular game so I think it can be a good idea for a video ��