# Thread: Change of basis, summation notation... help?

1. ## Change of basis, summation notation... help?

(Sorry if this is in the wrong forum, wasn't sure where to post)

So I'm taking continuum mechanics and trying to get my head around the notation etc. We're using the einstein summation notation where a repeated index implies summation over that index between 1 and 3. In my class notes on change of basis we have the following (anything in red is an index):

To change basis vectors, the new basis vectors e'i can be written as a linear combination of the old basis vectors ej (i, j = {1, 2, 3}) such that:
e'i = lij ej ......(1)
Where lij are the direction cosines. Taking dot product with ej on both sides gives:
l
ij = e'i.ej .....(2)

Also, the kronecker delta d can be written:
dij = e'i.e'j = lik ek e'j = lik ljk (using (1) and (2))
That is:
dij = lik ljk .....(3)

So that's what the class notes say. But what confused me is when I tried to write (3) out in full, summing the RHS over k, I got this:

dij = li1 lj1 + li2 lj2 + li3 lj3
Now using (2) :
dij = (e'i.e1)(e'j.e1) + (e'i.e2)(e'j.e2) + (e'i.e3)(e'j.e3)
dij = e'i.e'j + e'i.e'j + e'i.e'j
dij = 3dij

Um, yeah. 3 = 1 ...
Obviously I've gone totally wrong somewhere. Can anyone point me in the right direction? o_O;

2. Hello, Squiggles!

I'll point to where you seem to have gone wrong. I'm not sure, if your indices are upper or lower indices, but I'll just let them be lower indices in my response.

Originally Posted by Squiggles
Now using (2) :
dij = (e'i.e1)(e'j.e1) + (e'i.e2)(e'j.e2) + (e'i.e3)(e'j.e3)
dij = e'i.e'j + e'i.e'j + e'i.e'j
dij = 3dij

Um, yeah. 3 = 1 ...
Obviously I've gone totally wrong somewhere. Can anyone point me in the right direction? o_O;
You seem to think that

$\displaystyle (e'_i\cdot e_1)(e'_j\cdot e_1) = e'_i\cdot e'_j.$

How did this happen? Remember that the product between $\displaystyle e'_i\cdot e_1$ and $\displaystyle e'_j\cdot e_1$ on the left hand side is just usual multiplication of real numbers (or complex, depending on your number field).

3. -facepalm- Haha yes you're right, I just saw the two e1s in the same expression and "cancelled" them without thinking