# Differential forms and integrals

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 $$\int f(x) dx = \int f(x) e_1'$$.

• Yes it does! its precisely the same dx. You integrate forms and $f(x)dx$ is a form! Commented Jan 13 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. Commented Jan 13 at 21:54
• 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. Commented Jan 14 at 10:18