Different between integrals and Einstein summation?

[ktpr2]

From http://mathworld.wolfram.com i see that the integral notation was "the symbol was invented by Leibniz and chosen to be a stylized script "S" to stand for 'summation'. "

So from that i figure integrals are just summations. So what's the difference from Einstein Summation, where "repeated indices are implicitly summed over."

 

[mathman]

These two concepts have nothing to do with each other.

Integration can be viewed as the limit of sums (for example Riemann sums where the interval is divided into small intervals and the sum is over approx hieght and width - the limiting process is carried out by making the intervals smaller and smaller).

The summation convention has to do with differential geometry where you are summing over a small number of terms - in relativity, usually 4 (four dimensional space-time). The expressions involved are tensors and vectors. As you observed, there is a convention to omit the summation sign for repeated indices.

 

[ktpr2]

Hmmmm. So both notations carry out summation and both notations can be carried out to infinity, with any function (including one for Einstein summation that worked with smaller and smaller intervals, just like integration) ... so the only difference is that integration goes "beyond" infinity and takes the limit, while Einstein Summation does not?

 

[matt grime]

Nope, you're reading to much into mere notation. Inifinite sums are only defined under very strict circumstances, and predominately summation convention is usef for finitely indexed sums, but all in all it is neither here nor there.

summation convention is just a very useful tool to let us manipulate things such as vectors and and so on:

$$\delta_{ij}\epsilon_{ijk}=0$$
and if i recall correctly

$$\epsilon_{ijk}\epsilon_{ipq}=\delta_{jp}\delta_{kq} - \delta_{jq}\delta_{kp}$$

Riemann integrals are, when they exist and agree, the limits of certain sums.

 

[ktpr2]

hmm okay that seems to jive with the end of my last post.

 

[DeadWolfe]

No it doesn't.

 

[ahrkron]

You'd probably need to study some linear algebra to understand the usefulness of Einstein's convention. It is just a practical way of saving some ink and effort while writing.

On the other hand, you can think of integration as a "process" by which you find the area under a curve, or the total of some quantity over a path. It has to do with processes, while Einsteins convention simply helps writing things down.

 

[matt grime]

Linear algebra alos shows the problem with smmation convention: suppose T is a linear map and e_i is a basis of eigenvectors with eigen values :

Te_i= k_ie_i

without summation convention, obviously

 

[ahrkron]

Agreed.

On a minor note, in that case you can see that the summation convention should not be applied because there is an index i on the left hand side.