What is ∫ (dy)^2?

[kingwinner]

What is ∫ (dy)^2 ?

Fact:
Let a<b.
b
∫ (dy)² = 0
a

=====================

1) First of all, I'm having trouble understanding the meaning of the above integral. I tried searching my elementary calculus textbooks, but I really couldn't find ANY integral like this. I have only seen integrals like ∫ (dy) in my calculus books and so I have no idea how they can have (dy)² which makes no sense to me...

2) Why is the above fact true? How can we rigorously prove it?


Hopefully someone can clarify this.
Thank you!

 

[Bacle2]

I guess a,b are real numbers. Where did you see this integral?

 

[kingwinner]

Yes, a and b are real numbers, a<b.

I saw this integral in my first study of stochastic calculus and brownian motions.
It says...from ordinary integration, we have
b
∫ (dy)2 = 0
a

By ordinary, I'm assuming it's referring to standard elementary calculus, but I just can't recall anything similar to integrals like this in my calculus courses. Does such a thing exist in math? What is the precise definition of the above integral?

Thanks!

 

[Delta2]

One work around on this is to consider the fact that dy is independent of y thus it is like a constant when integrating with respect to y. So it is $$\int (dy)^2=\int cdy$$

Integrating cdy from a to b will give c(b-a)=dy(b-a). But dy is an infinitesimally small quantity therefore dy(b-a)=0.

 

[kingwinner]

dy is an infinitesimally small quantity
But then what does squaring an infinitesimally small quantity (i.e. (dy)²) mean? Squaring a differential just doesn't make much sense to me...

Also, I believe integral is defined as the limit of some kind of sum.
Is there an analogous sum being used to define the integral
b
∫ (dy)² ?
a

 

[Delta2]

dy is something like (b-a)/n and the integral is defined as the limit of a sum.
$$\int_{a}^{b}(dy)^2=lim_{n→∞}\sum _{i=1}^{n}(\frac{b-a}{n})^2$$

You can see that this limit is zero as n goes to infinity. But all i did here is not rigorously mathematically defined, i mean in math textbooks there is no official definition for the integral of (dy)^2 but an attempt to define such a thing in the most straightforward way is via the limit of sum i gave above.

 

[Bacle2]

I don't mean to be pedantic about this, by bringing-in the big machinery, but I can't
come up at this moment with something simpler; tho I will try: using
Green's theorem, one integrates 1-forms over surfaces, so there seems to be
a dimension problem. Maybe it is understood/implied that when using ∫ and (dy)², that we are actually using a double integral, and using dy repeatedly as the dummy variable
so that ∫(dy)² can maybe be understood as the integral over ℝ²
of the function f==1 along the x-axis , i.e., a line has measure zero as a subset of ℝ² . Or, more simply, a line has area=0 .

Maybe someone else can double-check.

 

[Bacle2]

Have you considered Ito's Lemma:

http://en.wikipedia.org/wiki/Itō's_lemma

and/or related?