Gradient→Curl→Divergence. Intuition with John Harland
Channel: Math 4 Wisdom
Date: 2023-02-18

What do gradient, curl and divergence mean intuitively? This is a warmup for one future video where John Harland will explain how to calculate the anti-curl, and then another future video where I will relate exact sequences with the cognitive frameworks that I call "divisions of everything".

John Harland and I both have PhDs in Math from the University of California at San Diego. His expertise was in functional analysis and mine in algebraic combinatorics. John teaches math and statistics at Palomar College. He has a passion for big questions in physics. 

I am Andrius Kulikauskas. I host Math 4 Wisdom, http://www.math4wisdom.com, to relate the language of mathematics and the language of wisdom. In this video, for the first 18 minutes I summarize the intuition that I gained in making it.  Subsequently, in the main part of the video, John defines gradient, curl and divergence and provides intuition to understand the antigradient and then the anticurl. Specifically, if the curl of a vector field G is zero, how do we express G as the gradient of some scalar field F? Similarly, if the divergence of a vector field H is zero, how do we express H as the curl of some vector field G?

Andrius Kulikauskas
0:20 Relation between gradient, curl, divergence
4:50 Example how Gradient measures change in all directions.
5:50 Example how Curl measures change in other (perpendicular) directions
11:30 Example how Divergence measures change in the vector's direction
13:40 Introduction to John Harland and his calculation of the anti-curl
14:50 My interest in exact sequences and divisions of everything

John Harland
17:55 Definitions
29:40 Curl: flowing vs spinning
31:05 Curl identifies a single axis. Gradient identifies a single direction. Gradient will not give loops. Curls will give loops.
33:25 Commutativity of partial derivatives implies that curl of gradient is zero.
36:05 Curl and Stokes' theorem
36:55 Curl and loops. Divergence of curl is zero.
38:15 Divergence: Lines accelerate or deccelerate
42:55 Example of a vector field rotating constantly
49:55 Paddle wheel would flow but also spin
50:35 Can scale to get zero curl if you divide by the right power of r. Global rotation vs. local rotation. How does curl bridge injection and surjection in an exact sequence?
54:20 Compute the antigradient
56:15 Connection with algebraic topology. The line integral.
1:01:15 Can add a constant vector field
1:03:10 The anti-curl. Adding an arbitrary gradient.
1:07:25 Can thus distinguish an object and its environment
1:08:40 Calculate anti-curl. Assume the curl is zero.
1:09:55 Maxwell's equations. Gauge theory
1:11:25 Exact sequence. Fivesome for space and time. Separating object and potential, subsystem and system
1:13:15 Anti-curl has boundary issues. Assume that the gradient vanishes at infinity. Local anti-curl.
1:16:10 Assume vector field is integrable and continuously differentiable
1:18:20 Making our learning process transparent
1:22:00 Starting work on the formula for the anti-curl
1:24:15 Partial integrals.
1:25:05 Derivative of partial integral gives identity.
1:17:25 Can reverse the order. Brings to mind raising and lowering operators.

We're now ready for our video with John on calculating the anti-curl!

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