Integration by parts (multivariable)

In summary, the conversation discusses finding the formula for integration by parts for multi-variable integrands and determining the appropriate substitutions to show the steps. It is recommended to look for a specific example to better understand the concept.
  • #1
Pacopag
197
4

Homework Statement


I can find on Wikipedia the "formula" for integration by parts for the case where there is a multi-variable integrand, but I would like to know what substitutions to make in order to show my steps.


Homework Equations


For multiple variables we have
[tex]\int_{\Omega}{{\partial u}\over{\partial x_i}}vdx=-\int_{\Omega}{{\partial v}\over{\partial x_i}}udx[/tex].
assuming that we can drop the surface term for physical reasons. Here, u and v are functions of several variables, say {[tex]x_1, x_2, ...x_n[/tex]}
First of all, should the [tex]dx[/tex] be a [tex]dx_i[/tex] ??
Now, my real question is; what substitutions do I make in order to show this?


The Attempt at a Solution


I feel like the generalization from 1D to the above higher-D version should be obvious, but it just isn't to me. I guess what is bothering me is that e.g.
[tex]du = \sum {{\partial u}\over{\partial x_i}}dx_i[/tex]. And I can't get this to fit into a derivation of the equation in 2. (above)
 
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  • #2
Just as in one variable calculus, what you choose to be u and what you choose to be dv depends heavily on the particular function you are trying to integrate! Do you have a particular example to show us?
 
  • #3
I don't have particular functions to work with. I have a sum like
[tex]\sum_{i} \int dx_{1}...dx_{n}dp_{1}...dp_{n} A t \left( {\partial H \over {\partial x_{i}}}{\partial \rho \over \partial p_{i}}-{\partial H \over {\partial p_{i}}}{\partial \rho \over \partial x_{i}} \right)[/tex],
and I am just trying to bring the derivatives from [tex]\rho[/tex] over to the A. Here, [tex]\rho[/tex] and A are functions of the x_i and p_i's, and t is time, i.e an independent variable.

What would be nice is a derivation of the integration by parts formula in part 2. of my original post
 
Last edited:

Related to Integration by parts (multivariable)

1. What is integration by parts?

Integration by parts is a method used in calculus to evaluate integrals of functions that are the product of two other functions. It involves using the product rule of differentiation to rewrite the integral in a different form that is easier to solve.

2. How does integration by parts work?

To use integration by parts, you must first identify which function in the integral is the "u" term and which is the "dv" term. Then, you apply the following formula: ∫udv = uv - ∫vdu. This allows you to rewrite the integral in a way that simplifies the process of solving it.

3. When should I use integration by parts?

Integration by parts is useful when the integral you are trying to solve involves a product of functions that cannot be easily integrated using other methods, such as substitution or partial fractions. It is also helpful when the integral involves trigonometric functions.

4. What are the steps for integration by parts?

The steps for integration by parts are as follows:

  1. Identify the "u" and "dv" terms in the integral.
  2. Apply the formula ∫udv = uv - ∫vdu to rewrite the integral in a simpler form.
  3. Solve the new integral using other integration methods.
  4. Substitute the original "u" and "dv" terms back into the final solution.

5. Are there any tips for using integration by parts?

One tip for using integration by parts is to choose "u" to be the most complicated function in the integral, as this will often lead to a simpler integral to solve. It is also helpful to keep track of your "u" and "dv" choices by using a table or writing them out clearly. Additionally, if the integral involves trigonometric functions, try choosing "u" to be the trigonometric function that is being differentiated.

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