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In my book for differential forms it is argued that using the external derivative on function $f(x, y, z) = x$ the result is:

$$ df = dx = f_x (x, y, z) e_1' = e_1' $$

Where $e_1'$ is the first element for the basis of spaces, so it is the function $e'_1(x, y, z) =x$. This justifies the notation $e_1' = dx, e_2' = dy, e_3' = dz$.

My question is if this notation has something to see with the $dx$ symbol that appears in the integral, so we we could write $\int f(x) dx = \int f(x) e_1'$.

Rodrigo Serna Pérez
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  • Yes it does! its precisely the same dx. You integrate forms and $f(x)dx$ is a form! – user57 Jan 13 '24 at 19:35
  • Then I can't see the relation between $dx = e_1'$ and $Delta x$ that is used to define the Riemman integral. – Rodrigo Serna Pérez Jan 13 '24 at 21:54
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    That relation was given to differential forms by Elie Cartan. This answer sums it up in a funny yet very enlightening way and gives a link to further material such as a video of a lecture by Ted Shifrin. – Kurt G. Jan 14 '24 at 10:18

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