Thursday, November 10, 2016

calculus - The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working.
u=f(x)dudx=f(x)du=f(x)dx(1)dx=duf(x)





My question is: what on earth is going on at line (1)?!?




This has been bugging me for, like, forever! You see, when I was taught this in my undergrad I was told something along the lines of the following:



You just treat dudx like a fraction. Similarly, when you are doing the chain rule dydx=dydv×dvdx you "cancel" the dv terms. They are just like fractions. However, never, ever say this to a pure mathematician.



Now, I am a pure mathematician. And quite frankly I don't care if people think of these as fractions or not. I know that they are not fractions (but rather is the limit of the difference fractions as the difference tends to zero). But I figure I should start caring now...So, more precisely,





dudx has a meaning, but so far as I know du and dx do not have a meaning. Therefore, why can we treat dudx as a fraction when we are doing integration by substitution? What is actually going on at line (1)?


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