Question on integration by subst. and du/dx as a fraction

[jwpaine]

Lets say that when doing integration by substitution, we say that u = g(x), and then du/dx = g'(x), followed by (multiplying both sides by dx) to obtain du = g'(x)dx
now my question is, why is it acceptable to handle that du/dx Leibniz's notation as a fraction, when it is really just a symbol? In sum: why is it OK to multiply both sides by dx in this situation? I don't want to take this for granted without fully understanding why we are allowed to suddenly call du/dx a fraction!

Thanks a ton!

 

[pka]

Lets say that when doing integration by substitution, we say that u = g(x), and then du/dx = g'(x), followed by (multiplying both sides by dx) to obtain du = g'(x)dx now my question is, why is it acceptable to handle that du/dx Leibniz's notation as a fraction, when it is really just a symbol? In sum: why is it OK to multiply both sides by dx in this situation? I don't want to take this for granted without fully understanding why we are allowed to suddenly call du/dx a fraction!

There is no definitive answer to this old question. Perhaps the best answer is it was a lucky mistake of history that happens to work. The seventeenth century mathematician’s use of infinitesimals had no logical grounding. They treated then as ordinary numbers so du/dx became a fraction. However even after mathematicians abounded infinitesimals, certain habits were carried over because they worked. I think that this is one of those cases.

As a side note: In last forty years a firm logical foundation has been found for infinitesimals. It also interesting to note that there people working in integration theory who advocate dropping the dx in the integral notation altogether.