Quick question on Einstein Notation

In summary: I just wish someone would write one of these books that wasn't so dry and boring.In summary, the conversation discusses the use of Einstein notation and its application in tensor calculus. The author struggles to understand how to interpret the right hand side of an equation in Einstein-summation form and seeks help in proving a given equality. Recommendations for better resources on the subject are also mentioned.
  • #1
Autodidact
3
0
Hello all,

I have a quick question on Einstein notation. I'll write the tensors as a capital letter and the covariant indices as lower case letters (and not use anything that has contravariant indices). I'll also use != for not equal or not congruent to.

In Schaum's outline of tensor calculus, the author stipulates (page 3):

Aij (Xi + Yj) != AijXi + AijYj

but I don't understand how to interpret the right hand side, or to figure out exactly what it does equal in an Einstein-summation form without the parenthetical expression, without choosing some "n" and simply expanding it out entirely.

This becomes relevant because problem 1.28 part c on page 7 (which sets Ei = 1 for all i) asks for a proof of:

Aij (Xi + Xj) = 2 Aij Ei Xj (with Aij symmetric)

I was doing this by expanding the right hand side, but apparently I have not been able to figure out how to do that properly as I have been unable to demonstrate this equality.

I appreciate any insight!

Thanks.
 
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  • #2
Version with proper math

(This is a re-post now that I see that I can use LaTeX in the message.)

Hello all,

I have a quick question on Einstein notation.

In Schaum's outline of tensor calculus, the author stipulates (page 3):

[tex]a_{ij}(x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j[/tex]

but I don't understand how to interpret the right hand side, or to figure out exactly what it does equal in an Einstein-summation form without the parenthetical expression, without choosing some "n" and simply expanding it out entirely.

This becomes relevant because problem 1.28 part c on page 7 asks for a proof of:

Given [tex]a_{ij}[/tex] symmetric and [tex]\varepsilon_i = 1[/tex] for all [tex]i[/tex], prove:

[tex]a_{ij} (x_i + x_j) = 2 a_{ij} \varepsilon_i x_j[/tex]

I was doing this by expanding the right hand side, but apparently I have not been able to figure out how to do that properly as I have been unable to demonstrate this equality.

I appreciate any insight!

Thanks.
 
  • #3
You meant,expand the LHS,right...?

Daniel.
 
  • #4
I did mean the RHS...

dextercioby said:
You meant,expand the LHS,right...?

I actually did mean the RHS (the side with the parenthetical expression) but you are correct in that I could expand either side. The problem is that I don't understand the RHS while the LHS makes perfect sense to me.

Thanks!
 
  • #5
Since [itex]\varepsilon_{i} [/itex] is 1,no matter the value of the subscript "i" and is involved in a multiplication,what is the soul purpose for its presence in the RHS...?

Daniel.
 
  • #6
Autodidact said:
In Schaum's outline of tensor calculus, the author stipulates (page 3):

[tex]a_{ij}(x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j[/tex]

Always keep in mind that all Einstein notation is, is a convention as to when one doesn't have to write the summation signs, just for convenience. So, when in doubt, add the sigmas. The above expression becomes obvious when you do this:

[itex]\sum_{i,j} \left(a_{ij}(x_i + y_j) \right) \not= \sum_{i} (a_{ij}x_i) + \sum_j(a_{ij}y_j)[/itex]

There is only one summation for each of the addends on the right-hand side, because in each of those expressions, there is only one repeated index.

I'll be honest with you: I've reviewed dozens of tensor analysis books and the Schuam's study book is by far NOT the best one. It's a bit heavier than most, but I would recommend Bishop and Goldberg's Tensor Analysis on Manifolds. (Note: you might interpret this as a subjective opinion, since Goldberg is my academic grandfather).
 
  • #7
Doodle Bob said:
I'll be honest with you: I've reviewed dozens of tensor analysis books and the Schuam's study book is by far NOT the best one. It's a bit heavier than most, but I would recommend Bishop and Goldberg's Tensor Analysis on Manifolds.

I agree with Doodle. I started out with the Schuam book quite frankly because it was monetarily cheap and I wound up more confused than when I started. Fortunately, a friend of mine (knowledgeable in diff geo) pointed me in the right direction: away from the physics-oriented texts and more toward the math-oriented approach to the subject. This indices business becomes nonsensical after awhile.

The Bishop text is good, as are some others that folks on here have recommended in other threads. These books take the more modern approach of viewing tensors as multilinear functionals and not just some arbitrary thing that changes in such-and-such a way with "coordinate transformation". (this begs the question: what's the point of working in coordinates anyway?)
 

1. What is Einstein Notation?

Einstein Notation, also known as Einstein Summation Convention, is a mathematical notation used to express and manipulate equations in the theory of relativity. It involves using Greek letters to represent the four dimensions of space and time, and using subscripts and superscripts to indicate which dimensions are being multiplied together or summed over.

2. Why is it called Einstein Notation?

It is named after the famous physicist Albert Einstein, who introduced this notation in his theory of general relativity. He used this notation to simplify and condense complex equations, making them easier to write and understand.

3. What are the benefits of using Einstein Notation?

Einstein Notation allows for a more compact and efficient way of writing equations, reducing the number of terms and making them easier to manipulate. It also helps to highlight and simplify the underlying mathematical structure of the equations.

4. How is Einstein Notation used in physics?

Einstein Notation is commonly used in fields such as general relativity, electromagnetism, and quantum mechanics. It is particularly useful in calculations involving vectors, tensors, and matrices, as well as in the study of spacetime and the behavior of particles.

5. Are there any rules or conventions to follow when using Einstein Notation?

Yes, there are a few rules and conventions that should be followed when using Einstein Notation. For example, the same index can only appear twice in a single term, once as a subscript and once as a superscript. The repeated index is also summed over, unless it is preceded by a delta symbol (δ). It is important to carefully keep track of the indices and their placement in order to avoid errors in calculations.

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