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In Physics and Mathematics it is widely accepted that having an efficient notation is of paramount importance when thinking about a given problem. A good exampl
by ThePhysicist 11y ago
In Physics and Mathematics it is widely accepted that having an efficient notation is of paramount importance when thinking about a given problem. A good example for this are Maxwell's equations:
https://en.wikipedia.org/wiki/Maxwell%27s_equations https://en.wikipedia.org/wiki/Maxwell%27s_equations
Using the vector differential operator notation those equations neatly fit on a napkin, whereas Maxwell's original equations (written in Integral form) would cover several pages. In fact, looking at his original manuscript always fills me with awe for the man as he had much less sophisticated tools at hand for his job but still managed to discover the beauty and simplicity of electromagnetism with them, which would also provide the foundation of special relativity.
- beagle3 11y agoIndeed. Most texts stop at the 4-equation notation that has both electric (E) and magnetic (B) fields, but with a small extension it becomes a single equation, which is lorentz-invariant: https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism#Maxwell.27s_equations_in_the_Lorenz_gauge https://en.wikipedia.org/wiki/Covariant_formulation_of_class... - I regard it as a much more fundamental and succinct description of how EM fields work.
- effie 11y agoThe covariant notation is concise and to some even elegant, but it is also less useful in practice. One often has to deal either with electric or magnetic field specifically as opposed to EM field tensor, and even if one does not need that, it is often possible and useful to work with only electric (or magnetic) component in 3D vector notation.