Uploaded July 2020 | Updated September 2026, 1 week ago
Video explaining the curl formula: youtu.be/b5VJVa5q3Oc
Video on surface integrals: youtu.be/hVBoEEJlNuI
It's possible for the boundary of a surface to have multiple separate parts. It turns out that, in general, if a surface has a boundary, then that boundary is made up of closed curves! This is why I use the closed line integral notation during the derivation.
For the example of a cylinder, the boundary is the two circles at the top and bottom of the cylinder. Each circle is a closed curve, so we can use the curl theorem with those circles as the boundary!
Explanation of Stokes' theorem, also known as the curl theorem or the Kelvin-Stokes theorem. We go through an intuitive explanation to understand the curl theorem using the definition of curl as a limit of closed line integrals.
Full Valuable Vector Calculus playlist: youtube.com/playlist?list=PLug5ZIRrShJHgsWPng59fFFoqn183aO-1
0:00 Theorem Explanation
7:34 Orienting the Curve
10:14 Example Problem
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Music: C418 - Pr Department
Video explaining the curl formula: youtu.be/b5VJVa5q3Oc
Video on surface integrals: youtu.be/hVBoEEJlNuI
It's possible for the boundary of a surface to have multiple separate parts. It turns out that, in general, if a surface has a boundary, then that boundary is made up of closed curves! This is why I use the closed line integral notation during the derivation.
For the example of a cylinder, the boundary is the two circles at the top and bottom of the cylinder. Each circle is a closed curve, so we can use the curl theorem with those circles as the boundary!
Explanation of Stokes' theorem, also known as the curl theorem or the Kelvin-Stokes theorem. We go through an intuitive explanation to understand the curl theorem using the definition of curl as a limit of closed line integrals.
Full Valuable Vector Calculus playlist: youtube.com/playlist?list=PLug5ZIRrShJHgsWPng59fFFoqn183aO-1
0:00 Theorem Explanation
7:34 Orienting the Curve
10:14 Example Problem
New math videos every Monday and Friday. Subscribe to make sure you see them!
Music: C418 - Pr Department
![F[x]-Module Derivation of Rational and Jordan Canonical Forms](https://i.ytimg.com/vi/q5uj4o0O5R0/mqdefault.jpg "F[x]-Module Derivation of Rational and Jordan Canonical Forms
Similar matrices isomorphism proof: https://youtu.be/-ligAAxFM8Y
Every module is a direct sum of cyclic modules: https://youtu.be/gWIRI43h0ic
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The rational canonical form and Jordan normal form of a matrix are very important tools in linear algebra, but ring theory and module theory give us a very effective way to prove their existence! Here we show that every matrix is similar to a matrix in rational and Jordan canonical form.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Rational canonical form
14:17 Every matrix is similar to RCF
18:33 Algebraically closed fields
20:10 Jordan canonical form
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Music: OcularNebula - The Lopez")
