Assuming boundary conditions when integrating by parts

In summary: In the first example, the boundary conditions are assumed to be such that the terms ##\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec x-\vec x')## and ##- \frac{i}{2} \nabla'\phi(\vec x', t) \delta^3(\vec x-\vec x')## vanish, leaving only the term ##\frac{i}{2} \int d^3 x' \nabla'^2 \phi(\vec x', t) \delta^3(\vec x-\vec x')##. In the second example, the boundary conditions are assumed to be such that the terms ##\partial^{m
  • #1
JD_PM
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TL;DR Summary
I want to understand what 'assuming boundary conditions' means in the context of integration by parts technique. I've stumbled upon the same issue twice and I want to understand it once and for all.
Let's present two examples

$$-\frac 1 2 \int d^3x'\big (-i \phi(x', t)\nabla^2\delta^3(x-x') \big )$$

Explicit evaluation of this integral yields

$$-\frac 1 2 \int d^3x'\big (-i \phi( \vec x', t)\nabla'^2\delta^3(\vec x-\vec x') \big ) =\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec x-\vec x') -\frac{i}{2} \int d^3 x' \nabla' \phi(\vec x', t) \nabla' \delta^3(\vec x-\vec x')$$
$$=\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec x-\vec x') - \frac{i}{2} \nabla'\phi(\vec x', t) \delta^3(\vec x-\vec x') + \frac{i}{2} \int d^3 x' \nabla'^2 \phi(\vec x', t) \delta^3(\vec x-\vec x')$$

Now ##\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec x-\vec x'), - \frac{i}{2} \nabla'\phi(\vec x', t) \delta^3(\vec x-\vec x')## terms vanish 'assuming boundary conditions'. I do not quite get this assertion and I'd like to discuss it in further detail.

The other example is this:

$$\int_{\mathbb{R}^{n}} dx \ \partial^{m+1}f(x) \ \varphi (x) = \partial^{m}f(x) \ \varphi (x)+(-1)\int_{\mathbb{R}^{n}} dx \ \partial^{m}f(x) \ \partial \varphi (x)$$

Applying integration by parts ##m## times one gets

$$\int_{\mathbb{R}^{n}} dx \ \partial^{m+1}f(x) \ \varphi (x) =$$

$$=\partial^{m}f(x) \ \varphi (x)-\partial^{m-1}f(x) \ \partial \varphi (x) +...+(-1)^{|m+1|}\int_{\mathbb{R}^{n}} dx \ f(x) \ \partial^{m+1} \varphi (x)$$

Now ##\partial^{m}f(x) \ \varphi (x),-\partial^{m-1}f(x) \ \partial \varphi (x), +...## terms vanish due to the same reason.

This originated from #15 on here.

Thanks :smile:
 
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  • #2
It means exactly what it says. Here is an example.

##\int_a^bf(x)g'(x)dx=f(x)g(x)|_a^b-\int_a^bf'(x)g(x)dx=f(b)g(b)-f(a)g(a)-\int_a^bf'(x)g(x)dx##

if at least on of the functions vanishes on the boundary, the first two terms in the last expression will be zero.
 
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Related to Assuming boundary conditions when integrating by parts

1. What are boundary conditions?

Boundary conditions refer to the specific values or constraints placed on a mathematical function at the boundaries of the domain. These conditions are necessary for determining a unique solution to a problem.

2. Why is it important to assume boundary conditions when integrating by parts?

Assuming boundary conditions when integrating by parts helps to ensure that the resulting solution is valid and accurate. Without these conditions, the solution may not accurately reflect the behavior of the function at the boundaries of the domain.

3. How do I determine the appropriate boundary conditions for a given problem?

The appropriate boundary conditions for a problem can be determined by considering the physical or practical context of the problem. These conditions should reflect the behavior or constraints of the system being studied.

4. Can boundary conditions change during the integration by parts process?

No, boundary conditions should remain constant throughout the integration by parts process. Changing the boundary conditions can lead to an incorrect solution.

5. Are there any common mistakes to avoid when assuming boundary conditions for integration by parts?

Yes, some common mistakes to avoid include assuming incorrect boundary conditions, not including all necessary boundary conditions, and not properly considering the physical or practical context of the problem.

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