# Einstein Summation

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1. Apr 7, 2016

### Raptor112

1. The problem statement, all variables and given/known data
My question is regarding a single step in a solution to a given problem. The step begins at:

$\large \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{y^p} \frac{\partial x^j}{\partial y^q} - \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{\partial y^q} \frac{x^j}{ \partial y^p}$

The solution says that the second term has the dummy i and j indices which can be switched in order to factorise which gives us:

$\large \big (\frac{\partial \alpha _j}{\partial x ^i} - \frac{\partial \alpha _i}{\partial x ^j} \big ) \frac{\partial x^i}{\partial y^p} \frac{\partial x^j}{\partial y^q}$

1. To clarify, the Einstein summation means that changing the index i in the second term has no implication for the index i in the first term?

2. If the dummies i and j were switched from the first term then I would result in the negative answer. So how does one know beforehand which term for whcih the indices have to be switched?

2. Apr 7, 2016

### George Jones

Staff Emeritus
In the second term, does replacing index i by index m and replacing index j by index n change anything?

3. Apr 7, 2016

### Raptor112

Well no because the summation will still be the same. But that makes it diffucult to see the factorization that is needed.

4. Apr 7, 2016

### George Jones

Staff Emeritus
So, now the second term is

$$\frac{\partial \alpha _n}{\partial x ^m} \frac{\partial x^m}{\partial y^q} \frac{\partial x^n}{ \partial y^p}.$$

Does replacing index m by index j and replacing index n by index i change anything?

5. Apr 8, 2016

### Raptor112

But I still don't see how this answers by second question?

6. Apr 8, 2016

### PeroK

I might suggest a radical approach of putting the sigmas back in so you can see what's going on. The summation convention is only a shorthand, after all.

7. Apr 9, 2016

### George Jones

Staff Emeritus
If you type in the expression that results when this is done, we can discuss the expression.

Also, this might help you to understand things. I sometimes find that my understanding changes when I actually write an expression down. Unfortunately, it can go both ways! Sometime, I don't understand an expression when I visualize it in my mind, but things become clear when I write the expression down. Other times, I think that I understand something, but I when I write it down, my "understanding" fades.

I only recommend doing this as a short-term aid for learning how to deal with expressions that omit the Sigmas.

8. Apr 9, 2016

### Fred Wright

Your question 1 is true; the Einstein summation convention only applies to terms connected by multiplication . For question 2 there is no way to know before hand and your reasoning must lead to the conclusion that the term in braces equals zero, i.e. ∂αj/∂xi = ∂αi/∂xj