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I stumbled across an intuition for the Laplace operator that suggests it can be considered "the difference between the value of a function at a point and the average value at "neighboring" points." As a new student, I really benefited from this interpretation - the steady-state heat equation made more sense. I found I also better understood the graph Laplacian thanks to this intuition.

This intuition makes me want to learn other perspectives using physics that build an intuition for mathematics, specifically in the fields of (1) analysis (2) discrete mathematics. Are there any books that emphasizes the intuition of these two branches (analysis, discrete mathematics) through analogies or applications in physics, specifically regarding mathematical operators used to model/describe the physical phenomena?

batlike
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  • Books by Daniel Fleisch are of that kind. https://www.danfleisch.com/ – vyali Apr 18 '23 at 16:19
  • Here's a nice, intuitive explanation of Legendre transformation by Wouter on PSE. https://physics.stackexchange.com/a/69374/363071 – vyali Apr 18 '23 at 16:23
  • The way you can build "physical intuition" is by going into the lab once in a while and by working through a lot of problems. OTOH, mathematical intuition seems to be a completely different beast, that is, in my observation of mathematicians discussing their work at the black board, more along the lines of "feeling ones way" through complex algebraic structures. In the most general case that is probably the use of commutative diagrams. – FlatterMann Apr 18 '23 at 17:38
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    Sounds like this would still be best posed to mathematics stackexchange since your view of physical isn't necessarily that of physical science. – Triatticus Apr 18 '23 at 19:12
  • Such insights are not generalisable. Instead, we often have to hunt for the physical insights whenever we come across them. – naturallyInconsistent Apr 19 '23 at 04:34

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