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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: 5, Funny) by maxwell demon on Saturday March 28 2020, @07:16AM (6 children)
Not to mention that he also invented me!
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2, Insightful) by Anonymous Coward on Saturday March 28 2020, @08:34AM (5 children)
I want my free energy check, you demon!
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @06:01PM (3 children)
You can have one of these,
http://massmind.org/images/com/visi/www/http/~darus/hilsch/hilsch.html [massmind.org]
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @06:03PM (2 children)
and the rest of the pull quote,
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @07:41PM
I had a colleague in grad school doing research in Hilsch tubes. They are loud as hell as you have high pressure gas blowing through a tube.
(Score: 1) by khallow on Sunday March 29 2020, @02:55AM
(Score: 2) by maxwell demon on Sunday March 29 2020, @07:07AM
You are aware that the service of those who do no actual work but constantly make decisions is always the most expensive? I don't think you could afford my services!
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by pkrasimirov on Saturday March 28 2020, @12:28PM (1 child)
Very well explained. Thank you for sharing!
(Score: 3, Informative) by FatPhil on Saturday March 28 2020, @03:00PM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @12:43PM
Back in those innocent days, you didn't have to be uber-uber specialists.
(Score: 1, Informative) by Anonymous Coward on Saturday March 28 2020, @01:18PM (3 children)
I didn't watch the video because of time constraints, but if it wasn't mentioned there, I will mention it here: the form of Maxwell's Equations which are quite compact and simple (using vectors) that we all use is due to Oliver Heavyside.
He is someone who doesn't get nearly the recognition he deserves. I think only electrical engineers know who he is because he was one of them.
(Score: 2) by opinionated_science on Saturday March 28 2020, @02:41PM
quite, Heaviside (Step Function) and LaPlace (Transform) are common tools in engineering space...
I have visited the grave of the latter (LaPlace) in Paris...
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @04:18PM
Did they have the vector calculus notations during Maxwell's time? I know it's not just the notation convention.
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @05:52PM
"Heaviside worked to eliminate the potentials (electric potential and magnetic potential) that Maxwell had used as the central concepts in his equations; this effort was somewhat controversial, though it was understood by 1884 that the potentials must propagate at the speed of light like the fields, unlike the concept of instantaneous action-at-a-distance like the then conception of gravitational potential." -- The term Maxwell's equations [wikipedia.org]
Nikola Tesla promised us that one day our machinery would be powered by the very wheelwork of nature [abc.net.au], which I interpret as meaning that Tesla grokked Maxwell's original equations, and recognized the mistake in Heaviside's restatements.
(Score: 0) by Anonymous Coward on Saturday March 28 2020, @02:21PM (14 children)
pshaw! a fancy symbol nobody understands (curl) and two constants and voila! lightspeed is a constant too.
then again, duh, the pointer to go faster is right there in them equations: the two "other" constants and how we define/measure them.
anyways, the whole fun (AND USEFULLNESS) probably would start if some moron would just explain that shitty symbol (curl) already.
talking about maxwells equation without first explaining the curl operator is like hypeing the usefullness of whole numbers on the vegetable market but totally ignoring addition and subtraction.
also, methinks the whole thing would be much easier explained with quaternions (yeah! DOOM) but i guess gimble lock is a thing that the "dream police" wants implemented rigorously.
(Score: 2) by FatPhil on Saturday March 28 2020, @02:57PM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(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