I have the following operator, where $\rho$ is a scalar and $u$ is a vector: $$ \nabla (\rho u) - (\nabla \rho)u - u(\nabla \rho) $$ My book writes this in index notation as $$ \partial_\alpha(\rho u_\beta) - u_\beta\partial_\alpha\rho-u_\alpha\partial_\beta\rho $$ My question is, why does the book use the indices $\alpha$ and $\beta$ again for the second and third term? Shouldn't they have their own set of indices?
What you are really doing when introducing indices is replacing $\nabla$ by $e_α\partial_α$ (using Einstein summation) and $u$ by $u_βe_β$, $e_α$ being the canonical basis vectors. Then
$$∇(ρu)=∂_α(ρu_β)\cdot e_α\otimes e_β$$
and in the full expression you combine the coefficients of the same basis vector $e_α⊗e_β$ of the tensor product in all three terms.