Finding Einstein notation version of a given flow equation

[geojenny]

Homework Statement

This is from lecture, not a homework problem per se. But I need assistance.

The problem was to write this form of a flow equation in Einstein's notation:

Homework Equations

\frac{\partial }{ \partial x_1}(K_1 \frac{\partial h}{\partial x_1}) + \frac{\partial }{ \partial x_2}(K_2 \frac{\partial h}{\partial x_2})+ \frac{\partial }{ \partial x_3}(K_3 \frac{\partial h}{\partial x_3}) = 0

Where K_1, K_2, K_3 are from a diagonalized form of the K tensor.

The Attempt at a Solution

The given solution is \frac {\partial} {\partial x_i} (K_i \frac {\partial h} { \partial x_i}) = 0, where i = 1,..,3

Which is supposed to be related to
\frac {\partial} {\partial x_i} (K_{ij} \frac {\partial h } { \partial x_j }) = 0 when K_{ij} = 0 if i ≠ j

I'm confused because there's an index repeated twice (appears three times), which I learned is bad in a different class. Is there a better way to write this equation?

 

[geojenny]

Does anybody have any idea? This isn't a homework question, and my engineering professor said to simply replace all three subscripts with an i to get Einstein's summation convention, and a repeated subscript implies summation.

However, I'm pretty sure this is incorrect because I have independently learned that the subscript can only be repeated once (appear twice). Is this really a correct way to express the equation in Einstein's notation?

 

[chiro]

Hey geojenny.

I'm not exactly sure where the confusion is: the reduction given is the minimally reduced form of the expression since the summation can be simplified using only the one index (which is why the minimum form has only that one index).

If you could not reduce it down to something further and needed an extra dummy variable, then you simply use another dummy variable: that is pretty much it. If you can collapse tensors then this just means that you can simplify the actual number of indices.

You might want to consider that for K_ij see what happens we multiply that tensor by the kronecker delta tensor d_ij (which in matrix form is just an identity matrix) and when you expand everything out, see what terms disappear (from multiplication by 0) and see what terms don't and then look at the form to see if you can find the minimal representation (as a summation) in the Einstein convention form.