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Now that's very interesting! According to his wiki page, he reduced Maxwell's equations from 20 equations (in 20 variables) down to the 4 iconic equations we n
by Cyph0n 9y ago
Now that's very interesting!
According to his wiki page, he reduced Maxwell's equations from 20 equations (in 20 variables) down to the 4 iconic equations we now refer to as Maxwell's equations. Imagine having to learn EM theory with 20 equations... as if current EM theory courses aren't tough enough already!
One of his creations, the step function, is also a key part of many proofs and equations in signal processing.
- ncmncm 9y agoIt should be noted that Maxwell's quaternion formulation of the same relations has just two equations. Vector notation won out over quaternions not from elegance or power, but from practical details of working them on paper. Now that we don't, anymore, quaternions are probably the better notation. Somebody should rewrite an E-M and an optics textbook.
- catnaroek 9y agopls, we use tensor calculus and exterior algebra. More, precisely, the electromagnetic tensor is the exterior derivative of the 4-potential.
- kurthr 9y agoExactly, vector calculus is a useful learning tool, but not used in most higher level (e.g. General Rel, plasma physics, etc) or numerical (super computer) physics calculations. Raising and lowering Kartoffel symbols and their covariant derivative forms make manipulating the coordinate transforms possible (4-6 dimensional)... otherwise it would just look like pages of equations. Interestingly, mechanical engineering also deals almost entirely with tensors... and they've developed extremely intuitive visualization tools.
- mkl 9y ago> Interestingly, mechanical engineering also deals almost entirely with tensors... and they've developed extremely intuitive visualization tools. What visualisation tools are you referring to? That sounds interesting.
- kurthr 9y agoSince all 3D stress/strains are tensor quantities (even for purely linear models), MEs project them into principal axes so that you can see their angle and their peak tesnsile/shear stress. https://en.wikipedia.org/wiki/Cauchy_stress_tensor https://en.wikipedia.org/wiki/Cauchy_stress_tensor Look at Mohr's circle: https://en.wikipedia.org/wiki/Mohr%27s_circle https://en.wikipedia.org/wiki/Mohr%27s_circle Within an object the stresses and strains (and principal components) vary as you move through it. That's part of what makes predicting failures and effects of thermal expansion so complex, etc difficult even ignoring nonlinear effects and failure modes (e.g. crack propagation).
- wbl 9y agohttps://en.wikipedia.org/wiki/Christoffel_symbols https://en.wikipedia.org/wiki/Christoffel_symbols is probably what you wanted to say
- kurthr 9y agoLol... german potatoes... autocorrect? Thanks, that's actually hilarious.
- Cyph0n 9y agoIsn't a quaternion just a 3D representation of a vector in terms of the coordinate system's unit vectors? From what I recall, using such a representation makes it easier to perform transforms between coordinate systems.
- qubex 9y agoQuaternions are four-dimensional, hence the radix quater, meaning “four”.
- qubex 9y agoYeah... let's also not forget that quaterion multiplication is noncommutative (AB≠︎B A), which makes theem an absolute pain to work with. This is why it made sense to reformulate them from Maxwell’s two to ythe then-prevailing twenty. Then they got reformulated in terms of 3-vector calculus (by Heavside, whose existence and achievements I was ignorant of until just a few hours ago, courtesy of Hacker News).
- nileshtrivedi 9y agoMatrix multiplication is noncommutative too. What made quaternions so particularly painful?
- qubex 9y agoThey’re not particularly more painful than matrices (indeed you can represent quaterions as matrices, beyond their ”traditional” representation of a tuple of a scalar and a 3-vector), but there’s no point going through the pain since there’s alternative formulations of Maxwell’s laws (in terms of vectors) that are commutative and get you out of the pickle (at the expense of going from two to four equations).