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Cake day: June 12th, 2023

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  • Kogasa@programming.devtoMemes@lemmy.mlMath
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    19 days ago

    Stokes’ theorem. Almost the same thing as the high school one. It generalizes the fundamental theorem of calculus to arbitrary smooth manifolds. In the case that M is the interval [a, x] and ω is the differential 1-form f(t)dt on M, one has dω = f’(t)dt and ∂M is the oriented tuple {+x, -a}. Integrating f(t)dt over a finite set of oriented points is the same as evaluating at each point and summing, with negatively-oriented points getting a negative sign. Then Stokes’ theorem as written says that f(x) - f(a) = integral from a to x of f’(t) dt.



  • Yup, you’ll notice the only thing distinguishing C from R^(2) is that multiplication. That one definition has extremely broad implications.

    For fun, another definition is in terms of 2x2 matrices with real entries. The identity matrix

    1 0
    0 1
    

    is identified with the real number 1, and the matrix

    0 1
    -1 0
    

    is identified with i. Given this setup, the normal definitions of matrix addition and multiplication define the complex numbers.