4 ms·

This is all really confusing. Why not just represent divergence and curl for what they are? Sources/sinks, and circulations. As someone who had lots of cours

This is all really confusing. Why not just represent divergence and curl for what they are? Sources/sinks, and circulations. As someone who had lots of courses in electromagnetics, I see no basis for physical reality in these diagrams.

bollu 7y ago
While this is true, note that this special relationship between gradient, divergence, and curl only exists in 3D, due to the fact that 2 + 1 = 3, and the presence of the [hodge star](https://en.wikipedia.org/wiki/Hodge_star_operator https://en.wikipedia.org/wiki/Hodge_star_operator) operator that allows us to convert "areas" into "normal vectors to areas". A curl is really a "infinitesimal area", which we then choose to interpret as "a normal vector". This is captured by the [De Rham cohomology](https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/De_Rham_cohomology), whose calculations are sometimes simplified by using the [Penrose notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation https://en.wikipedia.org/wiki/Penrose_graphical_notation) as outlined above. Having used the notation, I don't think it helps for vector calculus. And I 100% agree with you that we should teach the intuition for div, grad, and curl first, but I wanted to bring to the attention of anyone who's never seen the "fully general" form of these objects as to what it looks like.
kmill 7y ago
The paper doesn't really explain everything you need to be able to read the notation, but, being a tensor diagram aficionado, I have to say I found the way they encoded divergence and curl very exciting: it is all consistent and helpful for doing calculations. The loop-around-a-tensor notation is calculating the covariant derivative. For example, if you have a vector field, the covariant derivative is a "matrix field," where at every point in space you have a matrix that can take in a vector there and spit out the component-wise directional derivative. The divergence graphical notation (connecting the two wires together) is equivalent to taking a trace of this, meaning you create a scalar field where at every point you take the trace of the matrix from the covariant derivative. I guess all I'm saying is that the notation for divergence and curl are consequences to a couple of building blocks that have more generality. I'll also add that, like any language, it takes some getting used to. > Why not just represent divergence and curl for what they are? I'm curious how you would represent them for what they are. Maybe you have something in mind that would make calculations easier?
AnthonBerg 7y ago
I have come to believe the theory that people have profoundly different styles of cognition :) Alternate representations are incredibly useful to get as many styles of cognition on board as possible. I found I could grasp the equation immediately with help of the diagrams, and translate that to an internal abstraction representing the world. For me the equations themselves are a much more inert thing to approach and take much more effort to untangle. We all think differently. Especially so the more abstract the thought is it seems. As we approach each individual's limit of abstraction. Then we start to diverge further and further in how we think. I personally am a very visual thinker, and find Einstein's descriptions of thoughts to match my experience of cognition: “Words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined…but taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.” Einstein was smarter than me though. I'm a computer scientist. I know some mathematicians, and was talking about matters of cognition with them. One of the mathematicians – a mathematician's mathematician who is easily able to learn math from maths books – told me that his style of thinking is effectively the same as is printed the books. My mind was blown. For me, reading the same books involved "translating" into half-abstract visual/sensory elements. Once translated into my native representation of cognition the acquisition is complete and I can think in the new visual-sensory-abstract symbols. My friend on the other hand basically thinks in the words and math symbols found in the books.

bollu 7y ago

While this is true, note that this special relationship between gradient, divergence, and curl only exists in 3D, due to the fact that 2 + 1 = 3, and the presence of the [hodge star](https://en.wikipedia.org/wiki/Hodge_star_operator https://en.wikipedia.org/wiki/Hodge_star_operator) operator that allows us to convert "areas" into "normal vectors to areas". A curl is really a "infinitesimal area", which we then choose to interpret as "a normal vector". This is captured by the [De Rham cohomology](https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/De_Rham_cohomology), whose calculations are sometimes simplified by using the [Penrose notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation https://en.wikipedia.org/wiki/Penrose_graphical_notation) as outlined above. Having used the notation, I don't think it helps for vector calculus. And I 100% agree with you that we should teach the intuition for div, grad, and curl first, but I wanted to bring to the attention of anyone who's never seen the "fully general" form of these objects as to what it looks like.

kmill 7y ago

The paper doesn't really explain everything you need to be able to read the notation, but, being a tensor diagram aficionado, I have to say I found the way they encoded divergence and curl very exciting: it is all consistent and helpful for doing calculations. The loop-around-a-tensor notation is calculating the covariant derivative. For example, if you have a vector field, the covariant derivative is a "matrix field," where at every point in space you have a matrix that can take in a vector there and spit out the component-wise directional derivative. The divergence graphical notation (connecting the two wires together) is equivalent to taking a trace of this, meaning you create a scalar field where at every point you take the trace of the matrix from the covariant derivative. I guess all I'm saying is that the notation for divergence and curl are consequences to a couple of building blocks that have more generality. I'll also add that, like any language, it takes some getting used to. > Why not just represent divergence and curl for what they are? I'm curious how you would represent them for what they are. Maybe you have something in mind that would make calculations easier?

AnthonBerg 7y ago

I have come to believe the theory that people have profoundly different styles of cognition :) Alternate representations are incredibly useful to get as many styles of cognition on board as possible. I found I could grasp the equation immediately with help of the diagrams, and translate that to an internal abstraction representing the world. For me the equations themselves are a much more inert thing to approach and take much more effort to untangle. We all think differently. Especially so the more abstract the thought is it seems. As we approach each individual's limit of abstraction. Then we start to diverge further and further in how we think. I personally am a very visual thinker, and find Einstein's descriptions of thoughts to match my experience of cognition: “Words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined…but taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.” Einstein was smarter than me though. I'm a computer scientist. I know some mathematicians, and was talking about matters of cognition with them. One of the mathematicians – a mathematician's mathematician who is easily able to learn math from maths books – told me that his style of thinking is effectively the same as is printed the books. My mind was blown. For me, reading the same books involved "translating" into half-abstract visual/sensory elements. Once translated into my native representation of cognition the acquisition is complete and I can think in the new visual-sensory-abstract symbols. My friend on the other hand basically thinks in the words and math symbols found in the books.