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
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