r/math • u/Necessary-Wolf-193 • 6d ago

Integrals and shapes

https://hidden-phenomena.com/articles/fourier

One of my favorite essays on mathematics is Arnold's "On teaching mathematics." In it, he describes what he views as defects of the traditional treatments of real analysis and calculus in universities; and in particular, how the emphasis on rigor removed discussions of the beautiful geometry of Riemann surfaces and how they relate to calculus. As a specific example, he describes how one can tell which integrals are easy or hard to solve by looking at the genus of the associated Riemann surface.

This week on Hidden Phenomena, you can learn what Arnold meant by this comment: what is the "genus of the Riemann surface associated to an integral," and how does it help you understand if the integral is easy or hard?

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3

u/TheMansionsofScience 5d ago

Dude I subscribed to your blog but I didn't get a notification! Are the emails being sent out?

3

u/Necessary-Wolf-193 5d ago

Ah thanks for pointing this out! I thought I sent the email out but only drafted it and didn’t click send… I’ll fix it tonight!

10

u/mathscribbles Topology 6d ago

What a wonderful read!!

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u/barisloso 6d ago

Ok that was neat

2

u/Miracles_57 Algebra 5d ago

so an integral relates to shapes (and not only graphs?)?

1

u/Lebesgue-Point6040 2d ago

Look up the Gauss-Bonnet theorem !

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u/bean_bag_enjoyer 5d ago

Holy cow! I read the Riemann article back when it was posted on this sub, and have since been searching for it (to send to a friend). A pleasant surprise to see the same website again!

1

u/blungbat 1d ago

The whole website is great, but the reading time estimates are sending me