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I have recently been studying and using the index notation in physics, but I have a specific question, which is not very clear to me.

Say we have the radial vector $\textbf{r}$ and the usual Del-operator, $\boldsymbol{\nabla}$. I know that in three dimensions we have $\partial_i r_i=\sum_{i=1}^{3}\partial_i r_i = \boldsymbol{\nabla} \boldsymbol{\cdot} \textbf{r} = 3$ which of course yield the same result whether we use cartesian or spherical coordinates. I understand that two equal indices imply summation (Einstein notation).

Using index notation we may experience $\partial_n r_k = \delta_{nk}$. At first I believed we should have $\partial_n r_k=3\delta_{nk}$ because, again, two equal indices imply summation. My current understanding is that we implicitly look at the $\textit{element}$ of the vector, so $\partial_n r_k = \left[\partial_n r_k\right]_i = \delta_{nk}$ where subscript $i$ implies index. If this is the case then summation is $\textit{only}$ implicit if we have an expression where the product of two numbers is expressed by the same index (and not if the two numbers only yield a non-zero output in the special case of equal indices [expressed by Kronecker-Delta]).

Is this interpretation correct, and how do we, in general, know if we look at the element of a vector when using index notation?

Rasmus Andersen
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1 Answers1

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Some rules of index/summation/Einstein notation:

  • An index may not appear more than twice in a term.
  • An index appearing once is a free suffix.
  • An index appearing twice is a dummy suffix.

Summation only occurs over dummy suffices. So, $$\partial_i r_i = \sum_{i=1}^3 \partial_i r_i$$ since $i$ appears twice. However, there is no summation in $$\partial_n r_k.$$ This is just an expression that depends on two indices. It's just like a matrix: $A_{ij}$ is the $(i,j)$-th element of the matrix $\mathbf A$. The same index must appear twice for summation.

kipf
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