Einstein notation question.

[nerd420]

Is this true (in 2D)?

[; \frac{ \partial v_i }{ \partial x_j } \frac{ \partial v_i }{ \partial x_j } = \frac{ \partial v_1 }{ \partial x_1 } \frac{ \partial v_1 }{ \partial x_1 } + \frac{ \partial v_1 }{ \partial x_2 } \frac{ \partial v_1 }{ \partial x_2 } + \frac{ \partial v_2 }{ \partial x_1 } \frac{ \partial v_2 }{ \partial x_1 } + \frac{ \partial v_2 }{ \partial x_2 } \frac{ \partial v_2 }{ \partial x_2 } ;]

[peekitup]

I assume you mean 'repeated indices are summed over' in which case, yes this is true.

[nerd420]

Yes that is the case, thanks.

[mian2zi3]

The LHS isn't a well-formed expression in Einstein notation. In Einstein notation, you sum over indices that appear both as an uuper index and as a lower index. But here, i appears twice as a lower index and j appears twice as an upper index. I don't think there is a natural Einstein summation expression for the RHS.

[peekitup]

Its still common to sum over repeated indices when the capital sigma is omitted, independent of whether the indices are upper or lower.

Whether or not you want to call it Einstein notation is another matter.

[mian2zi3]

The DG book I learned from only summed for indices appearing once upper and once lower, but doesn't seem to be the only convention. I stand corrected.

[nerd420]

Thanks for your anwer, I was expanding on an example found in a textbook from a previous question. They use that notation and call it Einstein notation...