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Arthur T Knackerbracket has found the following story:
We are unabashed fans of [The History Guy’s] YouTube channel, although his history videos aren’t always about technology, and even when they are, they don’t always dig into the depths that we’d like to see. That’s understandable since the channel is a general interest channel. However, for this piece on James Clerk Maxwell, he brought in [Arvin Ash] to handle the science side. While [The History Guy] talked about Maxwell’s life and contributions, [Arvin] has a complimentary video covering the math behind the equations. [...]
Deriving Maxwell’s equations is a math nightmare, but [Arvin] doesn’t do that. He uses some amazing graphics to explain how the equations relate electricity and magnetism. A great deal of our modern world — especially related to any sort of radio technology — builds on these four concise equations.
One thing we didn’t realize is how wide-ranging Maxwell’s interest were. He contributed to astronomy by explaining Saturn’s rings, derived statistical laws about gasses, and worked on color vision, including creating the first light-fast color photograph. He also contributed to thermodynamics, control theory, and optics. Those were the days!
(Score: 1) by khallow on Sunday March 29 2020, @03:18AM (12 children)
It's just the same math no matter how you label it. You have two basic three dimensional vector bilinear operators, inner product and cross product (both which appear as components of quaternion multiplication). When you replace the first vector with a linear differential operator (of form (d/dx1, d/dx2, d/dx3)) acting on the second vector, you get divergence and curl respectively. These make more sense from a differential form [wikipedia.org] perspective. A k-form is basically a mathematical object that can be integrated over a k-dimension subspace (where in 3-dimensional space, k can be 0, 1, 2, or 3). 1-forms are analogous to differentiable vectors, 0-forms are just a differentiable function over the space and in 3 dimensions 2-form and 3-forms are duals which act as differentiable vectors and functions respectively. Gradient is a differential operator that takes a 0-form to a 1-form. Divergence takes a 2-form to a 3-form. And cross product takes a 1-form to a 2-form. You can effectively treat this as one single operator on all forms 0 through 3 with 3-forms zeroed out (similarly, there is a dual operator that goes in the opposite direction). It has the special property that applying it twice returns zero (same for the dual). Every form is zeroed out. This covers the usual identities: the curl of a gradient is zero or the divergence of a curl is zero.
(Score: 1) by khallow on Sunday March 29 2020, @03:23AM (11 children)
FTFM.
(Score: 2) by hendrikboom on Saturday April 04 2020, @01:08PM (10 children)
Correct. Cross-product takes *two* one-forms to a two-form.
To convert that two-form to a one-form, thereby getting it into the same space as the original one-forms, you need an inner product.
(Score: 1) by khallow on Saturday April 04 2020, @02:47PM (8 children)
Actually, it's curl again in dual form. Composing the dual curl on the curl yields the Laplacian of the original one form. There's a dual operator (which exists whenever the space is orientable - no Mobius strip-like switching of handedness), usually labeled "*" (with ** = identity), that maps 1-1 and onto k-forms to (3-k)-forms (for the 3-dimensional space case). That means among other things that one- and two-forms actually have the same dimension of 3. Taking "curl" to be the original curl, then the dual curl has form *curl*, and the resulting Laplacian (which ends up the sum of the second derivatives of each coordinate and takes k-forms to k-forms) on a one-form is (*curl*)curl. For the overall differential operator "d" (recall that dd = 0) that acts on all k-forms, the Laplacian is the sum of the two different ways one can compose d and its dual *d* (both which act on all k-forms) as (*d*)d + d(*d*).
(Score: 2) by hendrikboom on Sunday April 05 2020, @01:12AM (7 children)
I am completely baffled why they didn't teach differential forms by third-year honours math or physics when I went to university. So much simpler than all these vector operations. And they work in other-dimensional spaces.
(Score: 1) by khallow on Sunday April 05 2020, @01:44AM
(Score: 1) by khallow on Sunday April 05 2020, @01:48AM (5 children)
(Score: 2) by hendrikboom on Sunday April 05 2020, @02:25AM (4 children)
Lovely book.
(Score: 2) by hendrikboom on Sunday April 05 2020, @02:27AM (3 children)
Misner, Thorne and Wheeler have another introduction to differential forms in their massive tome "Gravitation". Of course there they use it on 3+1-dimensional spacetime.
(Score: 1) by khallow on Sunday April 05 2020, @02:55AM (2 children)
(Score: 2) by hendrikboom on Sunday April 05 2020, @02:33PM (1 child)
But they figure out how to draw pictures of differential forms! Something I haven't seen elsewhere. Useful for those who think visually.
(Score: 1) by khallow on Sunday April 05 2020, @03:11PM
(Score: 1) by khallow on Saturday April 04 2020, @02:51PM