Friday, June 3, 2011

Leibniz notation of integration

Look at the standard integral notation
   ab(some expression f of running variable x)dx ..........(1)

This usually but NOT always means (f of x) multiplying dx, then integrating from a to b, even though in some contexts it looks exactly like that and you feel "no doubt it means exactly that".
In better computer graphic, it's written as \int_a^b \! f(x)\,dx \,where f(x) is our complicated expression of x .

Instead, this Leibniz notation is based on the algebraic summation of n items, each a product of (f * Δ),
\sum_{i=1}^{n} f(t_i) \Delta_i ; ..............(2)
What is expressed in the integral in (1) is that we integrate over infinitesimal divisions. These divisions divide up and cover the entire continuous range [a,b]. Therefore (1) denotes the sum in (2) as Δ goes infinitesimal. Indeed, most of the time we can treat the (...)dx as a product as in (2), but i just feel there are some special contexts /with invisible broken glass on the floor/. I can't identify exactly those contexts, but here are some hints.

People often put funny things after the /innocent-looking/ "d". like
(...expression of x) d(-x/2)
(...expression of x) dw(x)
(...expression of x)dy dx/dy

dw(x) means dw, where w is treated as a variable, even though we know that w is a function of x.

I don't know if teachers ever do these things, but students do. They muddy the water.

How about double integral
   cd(∫ab(f of x and y)dx)dy

Whenever it's confusing I feel we had better refer to the the original definition of Leibniz's notation, based on (2)

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