# Index notation tensors quick question

## Main Question or Discussion Point

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.

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

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stevendaryl
Staff Emeritus
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.

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
Staff Emeritus