Proving the relation using multivariable calculus

[jhartc90]

1. The problem


I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral).

$$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec r)\ \cdot \nabla f(\vec r)\,d\vec r$$

Homework Equations

The Attempt at a Solution

If I use integration by parts, we have:

$$uv - \int(v\ du)$$

Does anyone see the way in which to prove this relation?

I can start by saying:

$$u=f(r)$$
$$du = \nabla fdr$$
$$dv = \nabla \cdot A(r)dr$$
$$v = A(r)$$

Does this look correct?

I am not sure if I am doing integration by parts correctly, because I am not sure how to handle gradient/divergence in this manner. Is there anyone out there who can provide some insight?
m statement, all variables and given/known data
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[Orodruin]

Hint: In cartesian coordinates, ##\vec\nabla = \sum_i \vec e_i \frac{\partial}{\partial x_i}##

 

[jhartc90]

I am sorry, but I am not entirely sure what do with your hint? Why would I need to introduce summations?

 

[jhartc90]

Hint: In cartesian coordinates, ##\vec\nabla = \sum_i \vec e_i \frac{\partial}{\partial x_i}##

I am sorry, but I am not entirely sure what do with your hint? Why would I need to introduce summations?

 

[Orodruin]

I suggest you use it and tackle one term at a time. That the ##\vec\nabla## operator can be written in this way in cartesian coordinates should be familiar.

 

[jhartc90]

I suggest you use it and tackle one term at a time. That the ##\vec\nabla## operator can be written in this way in cartesian coordinates should be familiar.

You are referring to the del symbol, but in the context of this problem, the integral on the left side is the dot product. Are you aware of this?

 

[Orodruin]

You are referring to the del symbol, but in the context of this problem, the integral on the left side is the dot product. Are you aware of this?

Perfectly aware, this changes nothing. Are you aware of the following?
∇f = grad(f)
∇⋅A = div(A)

 

[jhartc90]

Perfectly aware, this changes nothing. Are you aware of the following?
∇f = grad(f)
∇⋅A = div(A)

I am aware, so are you saying to integrate the right side instead of the left? And have a triple integral?

 

[jhartc90]

Do you agree with the integration by parts that I did? Does that work for the left side, and you are suggesting a way to solve the right side?

 

[Orodruin]

You can do it in any direction you like. Integration by parts would work in both directions. Why don't you try writing out one of the sides using the expansion of ∇ and show us what you get? (If you know the Einstein summation convention I would suggest using it for brevity.)

 

[jhartc90]

You can do it in any direction you like. Integration by parts would work in both directions. Why don't you try writing out one of the sides using the expansion of ∇ and show us what you get? (If you know the Einstein summation convention I would suggest using it for brevity.)

Can you explain to me why the integration by parts as I show in my original post is not the corrrect way to do it? I believe you, I just don't understand why that method is wrong. I am not sure what the Einstein summation method is.

If I were to right out the right side using the convention you said, it would just be the integral over 3 summations wouldn't it? The d//dx*e_x component, and similarly for y and z. then dr = dxdydz so i guess i could split into 3 separate integrals? One for dx, dy and dz?

 

[vela]

The suggestion is to write the integrals in terms of stuff you should already know instead of guessing at how integration by part works with divergence and gradient. For example,
$$\int f(\vec{r})(\nabla \cdot \vec{A})\,d\vec{r} = \int f(\vec{r})\left[\frac{\partial}{\partial x} A_x + \frac{\partial}{\partial y} A_y + \frac{\partial}{\partial z} A_z \right]\,d\vec{r}.$$ You should know how to apply integration by parts to
$$\iiint f(\vec{r})\frac{\partial A_x}{\partial x}dx\,dy\,dz$$ as well as the other terms. What do you get when you do that?

 

[jhartc90]

Let's see here:

$$

 

[jhartc90]

$$\int(f(r)\frac{dAx}{dx}dx)$$

So $$dv =\frac{dA_x}{dx}$$
$$v = A_x$$

$$u = f(r)dx$$

Am I starting right?

 

[jhartc90]

Not sure if that is the right start

 

[vela]

No, it's not correct because of sloppy notation. Writing ##dv = A_x'## doesn't make sense. It should be ##dv = A_x' \,dx##. Similarly, writing ##u=f(r)\,dx## doesn't make sense either.

Do you know how to do integration by parts? The fact that you seem to be guessing here suggests you don't really understand how to do it. If you're rusty, reviewing it would be a good idea.

 

[jhartc90]

No, it's not correct because of sloppy notation. Writing ##dv = A_x'## doesn't make sense. It should be ##dv = A_x' \,dx##. Similarly, writing ##u=f(r)\,dx## doesn't make sense either.

Do you know how to do integration by parts? The fact that you seem to be guessing here suggests you don't really understand how to do it. If you're rusty, reviewing it would be a good idea.

I do generally know how to integration by parts. I think I am confused by gradient and divergence function in here.

So, in that case, $$v=A_x$$

and

$$u=f(r)$$

And then

$$du=f'(r)dr$$

And then $$uv-\int(vdu)$$

And multiply by y and z components?

 

[vela]

What is f'(r) supposed to mean? What are you differentiating with respect to? What is dr?

 

[jhartc90]

What is f'(r) supposed to mean? What are you differentiating with respect to? What is dr?

f(r) is in the original problem.
You even wrote it in the problem before saying I needed to do integration by parts. I was differentiating wrt r. This is why I am confused, I would think it would be with respect to x.

Since this is r_vector, do i just do x-component, where f(r)_x = xi?
So f(x) =x

so u =x
du =dx

Am I on the right track?

 

[jhartc90]

Well, I think I made a mistake. r_vector = xi+yj+zk, though I have f(r), so I have f(xi+yj+zk)

 

[jhartc90]

What is f'(r) supposed to mean? What are you differentiating with respect to? What is dr?

And I know dr = dxdydz

 

[vela]

Integration by parts comes from the product rule: (fg)' = f'g + fg', which you rearrange to get f'g = (fg)' - fg'. When you integrate both sides, you end up with
$$\int f'g = fg - \int fg'.$$ Note that the primes all denote differentiation with respect to the same variable.

Now in the integral
$$\iiint f(\vec{r})\frac{\partial A_x}{\partial x}dx\,dy\,dz$$
the differentiation is with respect to ##x##, right?

So you start with ##u = f(\vec{r}) = f(x,y,z)## and ##dv = \frac{\partial A_x}{\partial x}dx##. This gives you ##v=A_x##. Now what should ##du## be? Don't just say f'. You have to explicitly say what variable you're differentiating with respect to.

 

[jhartc90]

I think

$$u=f(x,y,z)$$

So

$$du=f(x',y,z)dx$$

 

[jhartc90]

Integration by parts comes from the product rule: (fg)' = f'g + fg', which you rearrange to get f'g = (fg)' - fg'. When you integrate both sides, you end up with
$$\int f'g = fg - \int fg'.$$ Note that the primes all denote differentiation with respect to the same variable.

Now in the integral
$$\iiint f(\vec{r})\frac{\partial A_x}{\partial x}dx\,dy\,dz$$
the differentiation is with respect to ##x##, right?

So you start with ##u = f(\vec{r}) = f(x,y,z)## and ##dv = \frac{\partial A_x}{\partial x}dx##. This gives you ##v=A_x##. Now what should ##du## be? Don't just say f'. You have to explicitly say what variable you're differentiating with respect to.

So:

I think

$$u=f(x,y,z)$$

So

$$du=f(x',y,z)dx$$

Therefore,

$$uv-\int(vdu)$$

$$f(x,y,z)A_x-\int(A_x*f(x',y,z)dx)$$

 

[jhartc90]

How will the uv terms disappear to show the relation that we want? When I tried this without substituting r for x,y, and z, I could not figure out how to get the uv term to disappear.

I know that integration by parts is $$uv-\int(vdu)$$

and if only uv disappears, we have proved the relation. How will doing it for each component change anything? Have you figured that part out? I am stumped

 

[vela]

Right, though you should write
$$du = \frac{\partial f}{\partial x}dx.$$ So what does
$$\iint \left[\int f(\vec{r})\frac{\partial A_x}{\partial x}dx\right]dy\,dz$$ become?

How will the uv terms disappear to show the relation that we want?

You need to have boundary conditions on f and A that allow you to argue the uv term will vanish.

 

[jhartc90]

Right, though you should write
$$du = \frac{\partial f}{\partial x}dx.$$ So what does
$$\iint \left[\int f(\vec{r})\frac{\partial A_x}{\partial x}dx\right]dy\,dz$$ become?You need to have boundary conditions on f and A that allow you to argue the uv term will vanish.

What boundary conditions are you talking about? We were not given any boundary conditions.

In the integral you wrote, we would get:

$$\int\int((f(x,y,z)A_x-\int(A_x*\frac{df}{dx}*dx)dydz)$$

 

[jhartc90]

How will the dy and dz terms affect this? Wouldnt they just turn into yz?

 

[jhartc90]

can you please clarify the effect the dydz would have? It seems like they would an arbitrary yz which I do not see how this fits into the problem. Do you see this issue?

Also, what B.C. are you assuming in order to say that the uv term vanishes?

 

[jhartc90]

Right, though you should write
$$du = \frac{\partial f}{\partial x}dx.$$ So what does
$$\iint \left[\int f(\vec{r})\frac{\partial A_x}{\partial x}dx\right]dy\,dz$$ become?You need to have boundary conditions on f and A that allow you to argue the uv term will vanish.

Do you have time to finish up our conversation from yesterday?

 

[vela]

If only uv disappears, we have proved the relation. How will doing it for each component change anything? Have you figured that part out? I am stumped

For right now, let's just assume the uv term vanishes. I don't see how you can conclude you've proved the relation, especially in light of your questions.

How will the dy and dz terms affect this? Wouldnt they just turn into yz?

No because the integral of ##f(\vec{r})\partial_x A_x## is generally still a function of ##y## and ##z##.

It seems to me that you're reluctant to do any calculations because you can't see yet how it's going to work out. Sometimes you just have to try stuff and then it becomes clear. So deal with the other terms and see what you get.

 

[jhartc90]

For right now, let's just assume the uv term vanishes. I don't see how you can conclude you've proved the relation, especially in light of your questions.No because the integral of ##f(\vec{r})\partial_x A_x## is generally still a function of ##y## and ##z##.

It seems to me that you're reluctant to do any calculations because you can't see yet how it's going to work out. Sometimes you just have to try stuff and then it becomes clear. So deal with the other terms and see what you get.

You are right, I am reluctant because I do not see how its going to work out. The issue I see is that We have a triple integral here. The first one we are integrating wrt x. From there, we get the uv - int(vdu) term.

Then we have two outside integrals. I am lost on how this will affect the inside integral

 

[vela]

You're not doing any integration. You still have integrals with respect to each coordinate. Note that that's what you have on the righthand side as well.

 

[jhartc90]

In the integral you wrote, we would get:

$$\int\int((f(x,y,z)A_x-\int(A_x*\frac{df}{dx}*dx)dydz)$$

Is this the final answer for the x-component, and we just repeat with y and z? And then add them all together?

 

[Orodruin]

Yes. Note that we are assuming boundary conditions such that the first term (the one with only two integrations) vanishes.