Solving a Problem with Integrals in R3

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In summary, The conversation is about a problem with a complex integral involving multiple variables and functions. The speaker is attempting to simplify the integral by removing delta functions and integrating over a smaller range. They also mention a substitution for one of the variables, which would result in two separate integrals.
  • #1
cyc454
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Hi! I've got a problem with an integral. Let's assume we've got something like this:

R3d3x1R3d3x2R3d3x3R3d3x4P(|x1|)P(|x3|)δ(x1+x2)δ(x3+x4)W(|x1+x2|)W(|x3+x4|)


xi is a vector
The "δ" is the Dirac delta.
P(|x|i) & W(|xi+xj|) are some functions
I would like to make it looks a bit simpler---I mean get rid of deltas and two integrals. How can I make it?
Thanks for help and sorry for spelling mistakes!
 
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  • #2
What have you attempted? Do you understand the properties of the delta function?
 
  • #3
if the xi=-xj then δ ≠0.

R3δ(x)d3x should be equal 1. Well, actually it should looks

like this:

-∞δ(x)dx=1

but it is the same I thing.. This is all I know.
 
  • #4
Okay, also note that [itex] \int \cdots \int f(\vec{x}) \delta(\vec{x} - \vec{x}_o) d^Nx = f(\vec{x}_o)[/itex]. This can allow you to fix some variables.

My next question is, are we integrating from [itex]-\infty \rightarrow \infty [/itex]? If the variable being integrated is not within the bounds, we can simplify things greatly.

I must say, it has been awhile since I have done integrals of this form.
 
  • #5
Well, we are integrating it over the entire R3..
I don't get it. There isn't any function depending on x. there is only P and W that depend on |x| or |xi+xj|

PS I can't put P and W before the integrals, can I?
PPS One more thing. There is a integral:
∫d3x1
and let's assume x1=x2+x3 so the d3x1=d3x2+d3x3. So after substitution
∫d3x1=∫d3x2+∫d3x3? is it correct?
 
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1. What is the purpose of using integrals in R3 to solve a problem?

Integrals in R3 are used to find the volume of 3-dimensional objects or the area of surfaces in 3-dimensional space. They can also be used to solve problems involving mass, force, and other physical quantities.

2. How do you set up an integral in R3 to solve a problem?

To set up an integral in R3, you first need to define the limits of integration for each variable (x, y, and z) and choose the appropriate coordinate system. Then, you can write the integrand function in terms of the variables and use the appropriate integration rules to solve the integral.

3. What are some common applications of solving problems with integrals in R3?

Some common applications of solving problems with integrals in R3 include calculating the volume of irregularly shaped objects, finding the center of mass of a 3-dimensional object, and determining the work done by a force on an object in 3-dimensional space.

4. Can integrals in R3 be solved using software programs?

Yes, there are many software programs available that can solve integrals in R3, such as Mathematica, Maple, and MATLAB. These programs use numerical methods to approximate the solution or can find exact solutions for simpler integrals.

5. Are there any tips for solving problems with integrals in R3?

Some tips for solving problems with integrals in R3 include carefully choosing the coordinate system to simplify the integrand, breaking down the integral into smaller parts when possible, and checking your answer by using alternative methods or approximations.

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