1
$\begingroup$

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

Tensor

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.

$\endgroup$
2
$\begingroup$

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.

This is then made a little bit more rigorous either for skewed coordinate systems, or non-flat geometries like minkowski space. There you define a vector space with upper indices, and a covector space with lower indices. Covectors take a vector and produce a scalar, so whenever you see you the same the symbol for the upper and lower index, you know that they are applying a covector to a vector to create a scalar.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.