[HN Gopher] Geometrically understanding calculus of inverse func...
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       Geometrically understanding calculus of inverse functions (2023)
        
       Author : tobytylam
       Score  : 92 points
       Date   : 2025-05-05 15:01 UTC (7 hours ago)
        
 (HTM) web link (tobylam.xyz)
 (TXT) w3m dump (tobylam.xyz)
        
       | crdrost wrote:
       | Upvoted for the cute proof- without-words geometrical diagram of
       | the Legendre transform, but the fact that you defined the inverse
       | map as (x, y) to (\hat y, \hat x) I found impossible to keep my
       | head straight. Probably it's easy if I slow down and stop
       | skimming the article.
       | 
       | IMO the easier derivation -- may just be personal tastes as
       | someone more on the engineering side -- is just another
       | integration by parts.
       | 
       | So with f(g(x)) = g(f(x)) = x, define y = g(x) for U-substitution
       | with x = f(y):                   [?] g(x) dx = [?] g(f(y)) f'(y)
       | dy                   = [?] y f'(y) dy                   = y f(y)
       | - [?] f(y) dy                   = g(x) x - F(g(x)) + C
       | 
       | The more interesting thing is that this is a really basic
       | integration by parts which means that this diagram of yours that
       | I like, is more universal than it appears at first? I'd have to
       | think about that a bit more, how you can maybe graphically teach
       | integration by parts that way, is there always a u substitution
       | so that you can get u f(u) or so and get this nice pretty
       | rectangle in a rectangle... hmm.
        
         | messe wrote:
         | Yes, there's a similar diagram on the Wikipedia page for
         | Integration by Parts.
        
           | gowld wrote:
           | And it links to the more specific page on Integral of inverse
           | functions:
           | 
           | https://en.wikipedia.org/wiki/Integral_of_inverse_functions
        
       | gitroom wrote:
       | haha this stuff twists my brain up but i love the visual angle.
       | makes me wonder if breaking things down more like this really
       | changes the way i remember them long-term. stuff like diagrams
       | actually help or am i just tricking myself?
        
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       (page generated 2025-05-05 23:00 UTC)