The Essence of Multivariable Calculus | #SoME3

In this video, I describe how all of the different theorems of multivariable calculus (the Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theorem, and the Divergence Theorem, plus also the original Fundamental Theorem of Calculus in one dimension) are actually the same thing in higher math. I present this by going through each theorem conceptually step by step (no formal proofs) and then summarize a recurring idea that we observe in each theorem. Turns out, the idea I present is exactly the basis of the Generalized Stokes' Theorem, sometimes called the "Fundamental Theorem of Multivariable Calculus". Hope you enjoy!

Please note a minor error in the video:
At 9:22, inside the dot product of F with r'(t)dt, r'(t) should've been with a vector arrow on top of it to signify that it is a vector (we can't take the dot product of a scalar function and a vector field, we can only take the dot product of a vector field and another vector).

This video was made for the 2023 Summer of Math Exposition (SoME3) run by 3Blue1Brown.

Chapters:
0:00 Intro
1:28 Video Outline
2:07 Fundamental Theorem of Single-Variable Calculus
7:38 Fundamental Theorem of Line Integrals
13:05 Green's Theorem
15:29 Stokes' Theorem
18:08 Divergence Theorem
21:31 Formula Dictionary Deciphering
23:07 Generalized Stokes' Theorem
25:55 Conclusion

Feel free to reach out to me at:

#SoME3
#math
#calculus
#multivariable
#multivariablecalculus
#perspective

[michaelsheard]

Professional mathematician here, who has taught various flavors of calculus a zillion times: This was magnificent.

[Shams-]

My whole electromagnetism understanding has changed now . Like all these rules in school has gotten so much more colorful and fun and meaningful...

[Ultiminati]

As a physics and cs double major student i liked this video :D

[trimuloinsano]

I swear these kind of videos should be watched at lessons in university. They explan so much better what we are studing instead of mechanically doing things and just memorizing theorems without actually understanding what they say. As an engineering student, I thank you and will show to my university mates

[pyfzbuf]

I taught myself Calculus III entirely using Professor Leonard's lectures on YouTube and doing practice problems in my college's calculus textbook. Prof Leonard does a phenomenal job explaining the intuition behind many of these key ideas, so I came into this video understanding the ideas of each theorem pretty well and understood their relationships. However, it was the end of your video that blew my mind. I honestly did not expect to see a very generalized version of all these theorems, and to see that it could be summed up so simply was super cool. Thanks for informing me about the Generalized Stokes' Theorem. I will be taking Calculus III in a month at my college, and I feel even more prepared.

[lipeters]

Please continue making videos sir, you have immense potential for explaining complex things and more importantly for building connections and intuition, I really hope your project is recognised in the SoME3

[ilangated]

In university I took a course about physics simulations. There reached a point where we needed to calculate the mass of an arbitrary polyhedron so we could model its forces properly. I was shown how you could use the divergence theorem to calculate the volume of a closed polyhedron by turning an integral over its volume into an integral over its surface. You could then assume constant density and use the volume to get your mass. I think that was coolest application of multi-variable calculus I've seen. Thank you for this video, it's such a good refresher!

[allanjmcpherson]

On the subject of visualizing higher dimensional integrals, the textbook we used had a great line. Referring to triple integrals, it said something like, "You can think of this as the four dimensional volume under a three dimensional surface. This is not particularly helpful."

[shadowkryans]

Simple but effective. This is a dope some3 submission for sure

[monadic_monastic]

Very nice vid, one of my favorite submissions so far for #SoME3 I feel!

To add to the discussion, you might be interested in what you might view as a possible "sequel" to this: at around the 9 min mark, you mention that we "can't multiply vectors".

Tho, what if I told you that this is totally possible, and in a way where you don't have to resort to the math in general relativity, but can also take a lot of what you learned in vector calculus and extend it in higher dimensions?

(you will need to drop the cross product and replace it with something else that reproduces its properties in 3D while letting it generalize in higher dimensions, called the wedge product. You'll also need to include more "directed objects" instead of restricting yourself to just directed lines, i.e. regular vectors. For instance: directed plane segments, directed volume segments, etc, modeled by multivectors, in the same way vectors modeled directed line segments geometrically).

That subject is called Geometric/Clifford Algebra, and an associated calculus to it called 'Geometric Calculus' in a similar way vector calculus was to regular vector algebra. (There is a related area called "Clifford Analysis" that goes quite in depth with pure math formalism and rigor, but you won't need it just to extend vector calculus).

[TiagoSilva-ycbe]

i love watching calculus and physics videos, and this was by far the best multivariable calculus video i've ever seen!! simple and intuitive explanations of hard topics of high-level math. never have i seen such a clear explanation of the fundamental theorem of single variable calculus it was really astonishing my jaw dropped. the fact that you're from chemistry and still make this video with that passion and beauty makes me wanna learn more about new stuff. I absolute recomend this to who's starting calculus, this is some neat material

[olofvaldemarsamuelsson]

Really great video dude! Just about to take my first multivariable calc course and this has got me all excited to unpack the levels of abstraction in more detail.

[blackestbill]

This was beautiful, reminds me of Poincaré’s quote “Mathematics is the art of giving the same name to different things” or in this case different names to the same thing. Thanks for sharing! <3

[AA-gldr]

Thank you. As a recovering math-phobe, I really enjoyed this. Tremendously helpful and very instructive.

[parkerc.]

Great video. As a meteorologist I enjoyed your perspective and it brought a smile to my face as you simply explained the maths i enjoy analyzing when i look at the diverging wind fields, the upward movement caused by the curvature of winds, over the different surfaces at varying pressure levels of our troposphere.

[peterasamoah]

This video was heat 🔥 we gotta get you more subs. I legit thought calc 3 was beyond me until I watched this and for the first time I actually get it. Keep the uploads coming king 👑

[mustafizurrahman]

Awesome mesmerising superb. I completed my BSC in electrical and electronics engineering from most famous university in my country 23 years ago . Unfortunately I didn’t fathom anything regarding greens theorem during my fields and waves course in BSC. I wish I would have watched this video during my study. Thank awfully for this video

[vishalmishra]

Thank you for sharing this extremely insightful simplification of an otherwise a highly complex topic (perception of complexity of multi-variable calculus). This simplified (geometric) image will likely stick in my mind for years to come. Human mind thinks differently and complex math can be translated into a human-mind-friendly format using these insightful changes in perspective.

[popaandrei]

man i can t believe you explained it so nicely, it s the first watching one of your video, i hope you have more. congrats on you explanations, i can t believe i understood so much while i am still struggling with my PDEs and A level pure maths, etc. very big appreciation for founding your video. lots of thanks

[nobodythisisstupid]

My calculus textbook explicitly summed up how all the concepts of multivariable and vector calculus that were taught were extensions of the fundamental, basic concepts and theorems from the very start of calculus. It made me appreciate the courses more once I saw how seemingly unrelated concepts were simply logical extensions of earlier concepts that carried with them incredibly broad and far reaching conclusions and applications.