Index notation is driving me CRAZY

[AxiomOfChoice]

I've already posted a couple of index notation questions on here and I've gotten very helpful responses. So I thought I'd try my lucky again, though I'm a little more stumped on this question than I was on the others...

Let \vec{x} be the position vector and \vec{r} the radial unit vector. In index notation, I find myself confronted with simplifying

<br /> x_k \partial_i r_k.<br />

I desperately want this to be zero, but I can't figure out why it should be. (I can't really figure out what this represents at all, as a matter of fact.) Isn't \partial_i r_k a 2 by 2 matrix? And if so, how would multiplying with x_k make everything go away? Again, maybe it doesn't.

Can anyone help?

 

[matt grime]

Edit: of course it helps if I (read and hence) write the correct thing; here's what I should have written.

You can just write the sum out (doing this is what helped me from going from "summation convention, that's black magic" to "summation convention is really useful and easy".

\sum_k x_k\partial_i r_k

Now, what exactly is \partial_i?

 

[matt grime]

Isn't \partial_i r_k a 2 by 2 matrix?

No, it is what happens when you apply \partial_i to r_k.

I'm assuming that \partial_i means partial derivative with respect to something (x_i perhaps, or r_i).

Passing back to matrices, remember that if M is a matrix we write M_{ij} for the i,j'th entry of a matrix (say a real number). M_{ij} is _not_ a matrix. It is an element of a matrix.

 

[AxiomOfChoice]

I think I've shown that x_k \partial_i r_k is identically zero: Rewrite x_k \partial_i r_k as

<br /> x_k \partial_i (\frac{x_k}{r}),<br />

where as usual r = |\vec{r}|. We can apply the chain rule to get

<br /> x_k \partial_i (\frac{x_k}{r}) = x_k x_k \partial_i(\frac{1}{r}) + \frac{x_k}{r} \partial_i x_k.<br />

Things become easier now. The second term on the RHS is just

\frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}.

Now, \partial_i (1/r) is the same as \nabla (1/r), and a quick calculation gives that \nabla (1/r) = -x_k/r^3 (in index notation). Since x_k x_k = r^2, the first term on the RHS is, using index notation,

<br /> -r^2\frac{x_k}{r^3} = -\frac{x_k}{r}.<br />

Then

<br /> x_k \partial_i r_k = -\frac{x_k}{r} + \frac{x_k}{r} = 0.<br />

I might have made a mistake, but I think this is right. Let me know if you disagree with my argument.

 

[matt grime]

There is one clear mistake in your reasoning that your last line of tex tells you to look for: the LHS of that last line is a sum. The central term is not a sum but has k, the summation index, in it.Here is one thing that is wrong: you write

<br /> \frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}<br />

The thing on the left is summed over k. The thing on the right has a k in it. It should have an i in it, not a k. You do something similar when you say that the i'th component of del of 1/r is x_k. Let me quote the thing in the next reply.

 

[matt grime]

Now, \partial_i (1/r) is the same as \nabla (1/r)

No it's not. the former is a component, the latter is a vector. See above comment about matrices and elements of matrices. It is important to know what the things you're manipulating are, and why you can do what you do to them.

and a quick calculation gives that \nabla (1/r) = -x_k/r^3 (in index notation).

Here you use k when you *must* use i, otherwise, if it weren't for this next substitution:

Since x_k x_k = r^2, the first term on the RHS is, using index notation,

<br /> -r^2\frac{x_k}{r^3} = -\frac{x_k}{r}.<br />

you'd've ended up with more than 2 k's in something.

 

[AxiomOfChoice]

Edit: of course it helps if I (read and hence) write the correct thing; here's what I should have written.

You can just write the sum out (doing this is what helped me from going from "summation convention, that's black magic" to "summation convention is really useful and easy".

\sum_k x_k\partial_i r_k

Now, what exactly is \partial_i?

Matt, thanks for your help. To answer your question:

<br /> \partial_i = \frac{\partial}{\partial x_i}.<br />

Hopefully that clears things up.

 

[AxiomOfChoice]

No it's not. the former is a component, the latter is a vector. See above comment about matrices and elements of matrices. It is important to know what the things you're manipulating are, and why you can do what you do to them.



Here you use k when you *must* use i, otherwise, if it weren't for this next substitution:




you'd've ended up with more than 2 k's in something.

Matt:

I see your point. I think what I should have in my final expression is i's, not k's. If I follow this through, I still get 0 as my final answer.

I will try to post again in a second with my corrections.

 

[AxiomOfChoice]

Things become easier now. The second term on the RHS is just
<br /> \frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}<br />.

This should be corrected as follows:

<br /> \frac{x_k}{r}\delta_{ik} = \frac{x_i}{r}<br />.

Now, \partial_i (1/r) is the same as \nabla (1/r), and a quick calculation gives that \nabla (1/r) = -x_k/r^3 (in index notation). Since x_k x_k = r^2, the first term on the RHS is, using index notation,

<br /> -r^2\frac{x_k}{r^3} = -\frac{x_k}{r}.<br />

Then

<br /> x_k \partial_i r_k = -\frac{x_k}{r} + \frac{x_k}{r} = 0.<br />

This should be corrected as follows:

A quick calculation gives

<br /> \partial_i (1/r) = -\frac{x_i}{r^3}<br />

Hence

<br /> x_k x_k\partial_i(1/r) = r^2(-x_i/r^3) = -\frac{x_i}{r}.<br />

This is because x_k x_k is the same as \vec{r} \cdot \vec{r} = r^2. It follows that

<br /> x_k\partial_i r_k = \frac{x_i}{r} - \frac{x_i}{r} = 0.<br />

(Oh God, PLEASE let that be right.)

 

[matt grime]

Yes, I that's it.

 

[Will_taylor22]

I'm bumping this because it's related to the problem I'm having. currently though, I'm just trying to understand what's been written and it would be great if you could help me out. here's a quote from what is above:




We can apply the chain rule to get

<br /> x_k \partial_i (\frac{x_k}{r}) = x_k x_k \partial_i(\frac{1}{r}) + \frac{x_k}{r} \partial_i x_k.<br />

Things become easier now. The second term on the RHS is just

\frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}.


how does that work?

\frac{x_k}{r} \partial_i x_k. = \frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}.

I don't understand what you've done there, how that's equal.

 

[matt grime]

What don't you understand? (note you've copied part that was wrong.)

This:
<br /> \frac{x_k}{r}\delta_{ik} = \frac{x_k}{r}<br />

is not correct. Note again that on the left the k is a summed index. So it cannot appear on the right on its own. We are also trying to work out the i'th component of something so i should appear on the right. It should read

<br /> \frac{x_k}{r}\delta_{ik} = \frac{x_i}{r}<br />

Is that the problem solved?

 

[Will_taylor22]

yes, thankyou.