3 ms·

pls, we use tensor calculus and exterior algebra. More, precisely, the electromagnetic tensor is the exterior derivative of the 4-potential.

kurthr 9y ago
Exactly, 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 ago
    Since 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 ago
  https://en.wikipedia.org/wiki/Christoffel_symbols https://en.wikipedia.org/wiki/Christoffel_symbols is probably what you wanted to say
  - kurthr 9y ago
    Lol... german potatoes... autocorrect? Thanks, that's actually hilarious.

kurthr 9y ago

Exactly, 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 ago
  Since 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 ago
https://en.wikipedia.org/wiki/Christoffel_symbols https://en.wikipedia.org/wiki/Christoffel_symbols is probably what you wanted to say
- kurthr 9y ago
  Lol... german potatoes... autocorrect? Thanks, that's actually hilarious.