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> It seems that this is written out of a perspective of some missing knowledge. Well, the author talks about what "we learned in school", not university, so th

> It seems that this is written out of a perspective of some missing knowledge. Well, the author talks about what "we learned in school", not university, so that checks out but only because you two have different audiences in mind.

azalemeth 5y ago
Indeed. I learnt about vector dot and cross products, basic linear algebra (including diagonalisation, simple Markov chains and similar), partial differentiation, grad div and curl, the volume of a parallel piped and "all that Jazz" in high school, as a 17-18 year old. I learnt about the divergence theorem, Stokes's theorem, multivariate integration, integrating factors and higher order ODEs and simple PDEs, Fourier transforms and other integral transformations and similar in the first year of university (studying Physics). This is not uncommon in the UK – but it is also not common either, and depends on exactly what A-level modules you did. My understanding is that it's quite rare to do exactly this in high school in the united states – but there, I think limits are taught much more heavily. I think having a clear, short statement of having "assumed knowledge" somewhere probably helps avoid these issues. (I thought the article was excellent, and beautifully illustrated!)
- deleted 5y ago
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- spacedcowboy 5y ago
  I'll second this - my experience is pretty much the same (though I did some basic Fourier transforms at school), studying Physics at college. I did 3 Maths A levels along with Physics and Chemistry (and "General Studies" :) however, so I too may be atypical.
learc83 5y ago
In the US, it’s common to refer to university as school.
- vanderZwan 5y ago
  Really? I was not aware. Do we know if the author is from the US though?
  - jacobolus 5y ago
    The author went to MIT, worked for NASA, and now lives in San Francisco https://twitter.com/mferraro89 https://twitter.com/mferraro89 It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students). But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc. Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject. Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935
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azalemeth 5y ago

Indeed. I learnt about vector dot and cross products, basic linear algebra (including diagonalisation, simple Markov chains and similar), partial differentiation, grad div and curl, the volume of a parallel piped and "all that Jazz" in high school, as a 17-18 year old. I learnt about the divergence theorem, Stokes's theorem, multivariate integration, integrating factors and higher order ODEs and simple PDEs, Fourier transforms and other integral transformations and similar in the first year of university (studying Physics). This is not uncommon in the UK – but it is also not common either, and depends on exactly what A-level modules you did. My understanding is that it's quite rare to do exactly this in high school in the united states – but there, I think limits are taught much more heavily. I think having a clear, short statement of having "assumed knowledge" somewhere probably helps avoid these issues. (I thought the article was excellent, and beautifully illustrated!)

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spacedcowboy 5y ago
I'll second this - my experience is pretty much the same (though I did some basic Fourier transforms at school), studying Physics at college. I did 3 Maths A levels along with Physics and Chemistry (and "General Studies" :) however, so I too may be atypical.

learc83 5y ago

In the US, it’s common to refer to university as school.

vanderZwan 5y ago
Really? I was not aware. Do we know if the author is from the US though?
- jacobolus 5y ago
  The author went to MIT, worked for NASA, and now lives in San Francisco https://twitter.com/mferraro89 https://twitter.com/mferraro89 It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students). But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc. Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject. Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935
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