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In some domains differential forms are mind-mindbogglingly expressive. E.g. Maxwell's equations boil down to [1]: dF = 0, d*F = J You literally could not use
by openrisk 2y ago
In some domains differential forms are mind-mindbogglingly expressive. E.g. Maxwell's equations boil down to [1]:
dF = 0, d*F = J
You literally could not use fewer letters. But the corresponding visualizations are not particularly easy or illuminating.
[1] https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
- ziofill 2y agoSure, but that is because there's a lot of context that is tucked away inside the formalism and corresponding syntax.
- tobinfricke 2y agoThe same could be said when they are written using Gibbs notation with div and curl. Surprisingly, when Einstein wrote his paper on special relativity, Gibbs vector notation had not yet been widely adopted, and Einstein himself wrote out Maxwell's equations in component form.
- openrisk 2y agoIn part yes, but these symbols are not contrived aliases that merely wrap the underlying complexity. They are well defined geometrical onjects individually, can be used separately in other contexts etc. That such a frugal abstraction is possible is by no means guaranteed. Other physical systems are not as elegantly expressed using differential forms.