- #1

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## 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?

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

##\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.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).