Index notation tensors quick question

[binbagsss]

My text has:

$\frac{\partial x^{a}}{\partial x^{p}}V^{p}-\frac{\partial x^{a}}{\partial x^{r}}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}=\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}$

Looking at the 1st and 2nd terms, I see that $p$ and $r$ are dummy indices, so we can just rename them. But, surely this affects the 3rd term - e.g- say I name $p=r$ then they cancel, but I would have 4 r's in the 3rd term - which is not allowed. you can only have an index repeated twice in a single term right?

I'm not sure how to manipulate the indices to get this equality.

Thanks for your help. (this won't look like anything well-known, I've taken out the irrelevant terms that do not contain any of the indices above).

 

[stevendaryl]

My text has:

$\frac{\partial x^{a}}{\partial x^{p}}V^{p}-\frac{\partial x^{a}}{\partial x^{r}}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}=\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}$

Looking at the 1st and 2nd terms, I see that $p$ and $r$ are dummy indices, so we can just rename them. But, surely this affects the 3rd term - e.g- say I name $p=r$ then they cancel, but I would have 4 r's in the 3rd term - which is not allowed. you can only have an index repeated twice in a single term right?

I'm not sure how to manipulate the indices to get this equality.

I'm not sure what manipulation you are talking about. As you noted, the first term is the negative of the second term, which you can see by renaming dummy indices. So they cancel, leaving just the 3rd and 4th terms.

 

[binbagsss]

I'm not sure what manipulation you are talking about. As you noted, the first term is the negative of the second term, which you can see by renaming dummy indices. So they cancel, leaving just the 3rd and 4th terms.

But..surely this affects the 3rd term - e.g- say I name p=r then they cancel, but I would have 4 r's in the 3rd term - which is not allowed. you can only have an index repeated twice in a single term right?

 

[George Jones]

If you relabel a free index in one term, you have to relabel this index in all the terms in which it appears. If you relabel a pair of dummy indices index in one term, you do not have have to relabel dummy indices in any terms. In fact, you are prohibited from relabeling if relabeling results in something illegal.

 

[binbagsss]

If you relabel a free index in one term, you have to relabel this index in all the terms in which it appears. If you relabel a pair of dummy indices index in one term, you do not have have to relabel dummy indices in any terms. In fact, you are prohibited from relabeling if relabeling results in something illegal.

I see. thanks.