# What to do with an extra index in the definition of a tensor?

I came across this definition of a tensor while reading some vector calculus literature

This definition contains the index $\ell$ in the last term, however the tensor itself only depends on $j$ and $k$. What am I supposed to do with this extra index? Do I sum over it?

Note: The term $\frac{\partial \phi_k}{\partial x_j}$ is irrelevant here; it is just another long expression in terms of the position vectors, although it also contains an $\ell$ within it.

Yes, generally these sorts of expressions follow the Einstein summation convention. This says that whenever you see an index repeated in a multiplication expression, it means to implicitly sum over that index. So $(x_a - X_a)$ is a vector subtraction, but $x_a X_a$ is an inner product.