Interpretation of the integration domain symbol

In summary, the conversation discusses the interpretation of the domain of integration symbol in physics and whether it is an invariant geometric object or a representation of the values taken by a given set of variables. It is concluded that the domain of integration is a coordinates independent object, but explicit coordinates are needed to perform the integration. The question is deemed to be pertinent and the answer is obvious, closing the discussion.
  • #1
Barnak
63
0
I'm having an interpretation problem with the notation used in physics, under the integration sign.

What is the proper interpretation of the domain of integration symbol, on the integration sign ?

To be more precise, consider a function [itex]F(x)[/itex] of one or several variables. Its integral on a domain [itex]D[/itex] is simply

[itex]I = \int_D F(x) \, dx[/itex].

Now, we procede to a change of variables : [itex]x \rightarrow x' = h(x)[/itex], so the function is now a new function of the new variables : [itex]F(x) \rightarrow G(x')[/itex]. Of course, the integral above gives the same number, but the limits of integration have to change to adapt to the new variables. We can write :

[itex]I \equiv \int_{D'} G(x') \, dx'[/itex].

My question is this : is the symbol [itex]D[/itex] on the integration sign actually an invariant, so [itex]D' \equiv D[/itex] ? Or is it a representation of the values that the variables are taking under the integral (so [itex]D' \ne D[/itex]) ?

In other words : do the domain of integration [itex]D[/itex] change with the coordinates transformation ?
 
Mathematics news on Phys.org
  • #2
In general, D and D' will represent different sets of numbers, so they are different subsets of the real number line.
 
  • #3
arildno said:
In general, D and D' will represent different sets of numbers, so they are different subsets of the real number line.

So you're saying that [itex]D[/itex] isn't invariant : it's a representation of the values taken by a given set of variables (coordinates) ?

I still have doubts on this. For example, when we define a surface integral like this :

[itex]I = \int_{\mathcal{S}} {\bf A} \cdot d{\bf S}[/itex],

the symbol [itex]\mathcal{S}[/itex] is both the domain of integration of the integral and the absolute surface in 3D space on which the integration is performed. So this implies that the domain [itex]D[/itex] is invariant. This is what puzzles me !
 
  • #4
Why do you think that a domain of integration must be invariant if you change the variable ?
Obviously not ! The domain of integration depends on the way that the function is defined.
For example, consider the function F(x)=a if -1<x<1 and F(x)=b elsewhere.
Let x=x'/2 so, F(x)=F(x'/2)=G(x') then G(x')=a if -2<x'<2 and G(x')=b elswhere.
Integrating F(x) on D=[-1,1] is the same as integrating G(x') on D'=[-2,2]
If you integrate G(x') on [-1,1] instead of [-2,2], you will obtain a different result.

Similary in 3D.
A surface (S) can be defined on many ways, i.e. with many different equations, by change of variables, or by change of coordinates (Cartesian, polar, ...) or by change of axes system. For each one of these definitions of the same surface with same border, the equations which characterize the border are different. If we integrate on (S), of course we have to consider the convenient related equation of the border, not another one, hense not always the same equation (but the border is the same on geometical viewpoint).
 
Last edited:
  • #5
It is "invariant" in the sense that it refers, geometrically, to a specific set of points. When you change variables, you are changing "coordinates" and how those points are given in the new coordinates will change.
 
  • #6
But then how should we indicate the domain below the integral sign ? As an invariant geometric object ([itex]D' \equiv D[/itex]), or as a coordinates relative set ([itex]D' \ne D[/itex]) ?

Of course, the integration limits are changing with the coordinates change (obvious !), but those limits aren't the same as indicating "[itex]D[/itex]" below the integration sign.

the question can be asked differently : What is an integration domain ? Is it an invariant geometic object on which the integration is performed, or is it the set of values of the integration variables ? I suspect it is the first one.
 
Last edited:
  • #7
Barnak, could you tell us what you mean by "invariant geometrical object"?
 
  • #8
pwsnafu said:
Barnak, could you tell us what you mean by "invariant geometrical object"?

Well, I badly expressed myself (sorry, English isn't my primary language).

I mean an "absolute" or coordinates independant object, like a surface in 3D space. Or a region [itex]D[/itex] in space.

To me, it is almost clear (? not sure yet) that the integration domain [itex]D[/itex] is a coordinates independant thing. Of course, to explicitely perform the integration, we need some coordinates and specify the limits on the integral sign. But there should be a difference between the following two notations :

[itex]\int_{D} F \, d^3 x = \iiint_{-\infty}^{+\infty} F(x,y,z) \, dx \, dy \, dz.[/itex]
 
  • #9
Hi !

The explanation of HallsofIvy (post #5) is clear and suffisant. This should close the discussion. The question was pertinent and the answer is obvious.
It seems to me that it would therefore be misplaced to discuss ad infinitum the gender of angels.
 

What does the integration domain symbol represent?

The integration domain symbol, which is ∫, represents the integral of a function over a specific domain. It is used in calculus to evaluate the area under a curve or to find the volume of a solid.

What is the difference between a definite and indefinite integral?

A definite integral, which has a specified integration domain, gives a numerical value as the result. An indefinite integral, on the other hand, does not have a specified integration domain and instead gives a function as the result.

How is the integration domain symbol used in multivariable calculus?

In multivariable calculus, the integration domain symbol represents integration over multiple dimensions. It is used to calculate the volume under a curved surface or to find the mass of an object with varying density.

What are the limits of integration in the integration domain symbol?

The limits of integration, also known as the bounds, define the range over which the integral is evaluated. They can be constants, variables, or functions, and are written as the lower and upper limits of the integral sign.

Can the integration domain symbol be used to solve real-world problems?

Yes, the integration domain symbol is commonly used in physics, engineering, economics, and other fields to solve real-world problems. It allows for the calculation of areas, volumes, and other quantities that can be represented by integrals.

Similar threads

Replies
4
Views
825
  • General Math
Replies
1
Views
678
Replies
4
Views
283
  • General Math
Replies
10
Views
1K
Replies
8
Views
4K
  • Calculus
Replies
29
Views
513
Replies
2
Views
178
Replies
11
Views
11K
Replies
1
Views
796
Back
Top