Early view of next video

I have to say, this video taught me that I really don't know anything about mathematics. I know just a fraction of a drop from an ocean. But man is it an infatuating subject! Thank you so much for this video.

Funny, I was thinking the exact opposite. Letting the animation alone serve to show the (obvious) opposite case struck me as a clever “less is more.”

Don Sanderson

Just from an aesthetic perspective, this video is gorgeous.

Alex Loftus

Numberphile discussed a similar thing today: <a href="https://www.youtube.com/watch?v=6H6EP-AmMFM&amp;feature=youtu.be" rel="nofollow noopener" target="_blank">https://www.youtube.com/watch?v=6H6EP-AmMFM&amp;feature=youtu.be</a>

Daniel Brahneborg

If you have a video about dot and cross product, it might be helpful to link to it around 14:44 or so.

Kevin Iga

Small typographical issue: sometimes you have div F (a space between "div" and "F") like at 6:20-6:51, 7:35, 8:00, 9:14, 9:48, and other times no space "divF" at 6:02, 6:14 and 17:01, 17:22

Kevin Iga

Great video!

Rodrigo Ângelo

Really beautiful video (both visually, and in terms of intuitions conveyed by it)!

Great video! Only note I have - at minute 3:46, where you give examples of vector fields, you start off with fluids, then talk gravity and magnetism. Have you considered swapping the examples around? I think gravity is a much better known and more intuitive example of a vector field, so might be better to anchor people using that. Off-topic question - how much physics have you studied? I know a bit of math (much less than you of course), but when you talk physics, I'm completely lost, and am wondering just how much exposure you've had to it.

Edan Maor

While I like "Visual complex analysis", I think it's use of traditional geometric arguments is sometimes more complicated than necessary, but the visual intuition is really quite helpful.

For me, divergence and curl made more sense in the context of differential forms. Interpreting a vector field as a certain differential form means that both divergence and curl become the same as computing the derivative of differential form.

I think Grant meant that they don't exist in Maxwell's model of the universe. I mean technically light isn't perfectly described by Maxwell's wave equation either, since photons are better modeled using quantum mechanics (wavefunctions, measure algebras, etc.).

Overall it looks really good, and as always it gives a more visual take on math concepts completely out of my league! But unless "Magnetic Fild" is a term I don't know, you might want to look at that (9:32)

Just a nitpicky detail: In 2:10 you mentioned the horizontal lines being Electric Potential lines, the voltage drop was not constant. V went from 11.9, 11.6, 11.3 then 11.1, 10.8, etc... Guess it is a rounding issue but maybe using integral values would be better.

You should think of illustrating Maxwell´s equations one day. The hints included in this video already give great intuitions.

Daniel Armesto

impressive. Great material!

Daniel Armesto

I grasp the concepts better than the actual math. I think I need to go and watch some videos on Khan Academy so I can follow with the math as well.

Awesome!!!!

Ah thanks for this. I’m defending my thesis on Monday, and a significant portion of it uses Hodge Decomposition. This video made for some good cramming. ;D

Jacob Mirra

Amazing. I'm so happy that I could see this earlier

Good to keep in mind. I’ve seen it used more generally, but perhaps in less standard contexts.

3blue1brown

You may want to make sure that your use of the term "phase space" to describe the space of pairs (rabbit pop., fox pop.) is standard. In my experience with dynamical systems, this would be called "configuration space" and phase space would consists of tuples of configurations along with their "velocities" (or "momenta"), so in the population example, I would have called the 4-dimensional space with coordinates (rabbit pop. velocity, fox pop. velocity, rabbit pop., fox pop.) "phase space." More formally, configuration space is a manifold of configurations, and phase space is the corresponding tangent bundle.

I’m lost for words... Beautiful animations and great explanation. I’m super exited for what follows!

Corvinus

When I saw you were talking about divergence and curl, I said to myself, "I wonder if he'll talk about Maxwell's equations." Excellent. Those equations really made the concepts of divergence and curl (and the nature of light itself) click for me.

That was fabulous. Thank you for that! I think I remember an explanation of curl for being on my wish-list during our first phone conversation, so I feel like I just got my Christmas gift! Think about linking to your dot product and cross product lectures from the linear algebra series. Can't wait for the rest!

Burt Humburg

Speling erurs at 10:21 "field" not "fild."

Burt Humburg

Seemed a bit weird when you didn't speak while demonstrating inward flow as it relates to dot product

Jonathan Gjertsen

Well, some transitions feel rough, but apart from that, I didn't find any problem with it (first seeing)

I don't know if you still have a chance to fix typos. At around 9:30 you have "Fild" instead of "Field" twice on the screen.

The pauses between the 15 minute mark and 17 minute mark feel a bit long

Really good. I'm very interested in complex analysis but have never fully grasped the fundamentals. I have Tristan Needham's 'Visual Complex Analysis' which is recommended by Roger Penrose in 'The Road to Reality', but I just can't get through it. Yourimages work much better for me.

Doug Fort

Technically, we can't say that magnetic monopoles don't exist, only that we haven't detected them, since several theories posit their existence.

Hey Grant, love the vid so far but just one thing I noticed (more physics than math) near the beginning and end. When a conductor (e.g. a copper wire) becomes polarized, the negative charges move around but they don't quite aggregate to one section like displayed in the video. Rather, the whole "electron sea" is shifted slightly in the reverse direction of the electric field arrows as to create an excess on one side of the surface and a deficiency on the other. This pic might help some: <a href="https://i.imgur.com/NeAMw4T.png" rel="nofollow noopener" target="_blank">https://i.imgur.com/NeAMw4T.png</a>

Jonathan Fuzaro Alencar

Love it! Great topic!

Awesome vid. I tried to find something odd but didn't really find anything. If anything, 2:29-2.31 transitions felt somewhat unsmooth. Also I didn't spot the white arrows appearing at 5:52 first time around.

Brikir

Around 9:36, is the heading supposed to read "field"?

wye