31–40 of 95 posts
Earlier quoted context omitted.
Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations: d F=J dF=0 Geometric algebra produces two equations too, by the way, not one.
You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to d dA = J.
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-th…
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
Earlier quoted context omitted.
Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations: d F=J dF=0 Geometric algebra produces two equations too, by the way, not one.
You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to d dA = J.
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-th…
Would really like to have that free time to delve and be able to suggest the alternative formulation myself.
Earlier quoted context omitted.
Probably true! That said, Wikipedia's mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the "reference of least resistance."
I'm at least ambivalent about a general purpose encyclopedia having whole categories of articles that are mostly only accessible to specialists. In an ideal world, there would probably be a companion "Mathepedia" or something along those lines with a different target audience. But, of course, Wikipedia doesn't really have the concept of a specific reader persona like a conventionally edited reference does.
FWIW, there's Simple Wikipedia (https://simple.wikipedia.org), but it is, quite frankly, terrible for learning about anything of a mathematical nature beyond high-school algebra.
Perhaps Wikipedia should add a "Learn About This" section to math, physics, engineering, and hard-science pages?
Earlier quoted context omitted.
Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.
Well, it's not Wikipedia's fault if people producing intuitive content are not contributing to it. It's not like editors go out of their way to remove nice explanations.