Einstein Summation Convention

In summary: \epsilon_{ijk}=\sum_{i=0}^3a_ib_j(\mathbf{e_k}-\mathbf{e_1})\mathbf{e_2}-\sum_{i=0}^3a_ib_j(\mathbf{e_k}+\mathbf{e_1})-\sum_{i=0}^3a_1b_3\mathbf{e_2}+\sum_{i=0}^3a_2b_1\mathbf{e_3}\end{align*}...the i-th component of the cross product.
  • #1
bon
559
0

Homework Statement



Ok so I'm meant to answer:

To what scalar or vector quantities do the following expressions in suffix notation correspond? (expand and sum where possible).

1) aibjci
2) aibjcjdi
3) dijaiaj
4) dijdij
5) eijkaibk
6)eijkdij


Homework Equations





The Attempt at a Solution



Okay so I'm pretty stuck to say the least..any help would be great..Here are some thoughts:

1)doesnt this just mean we sum that components of a and c and b can take any value? I'm confused..does this mean the result is a vector.?
2)so we sum a and d and b and c..but how do i simplify this?
3)ok so this one i think i know...isn't it just ajaj so sum the components of a? a.a?
4)doesnt this just equal dij?
5) Not sure
6) Again,not sure :(

thanks
 
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  • #2
Does the d_ij represent the Kronecker delta? Are a, b, c and d numbers or can they also be operators. If they are just numbers remember that order doesn't matter. How would you write down the dot product in index notation, what about the cross product?

As for 4). Your answer is not right. Keep in mind that you're summing over both i and j so when i != j you get 0, which doesn't contribute to the sum. Therefore the only entries that contribute to the sum are the entries where i=j. Try to write it out by letting i and j run from 0 to 1 and note that your answer can't be right.
 
Last edited:
  • #3
yes d_ij is k delta.

i think abc are just numbers..

dot product would be a_ib_i
cross would be e_ijk a_i b_j e_k?

where the fist e is an epsillon..

but i don't see how to do the problems..:S
 
  • #4
so is it d_ii?
 
  • #5
Yes d_ii is correct. However note that d_ii has repeated indices so we are summing all it's diagonal entries together. Can you express d_ii as a number?

Your dot and cross products are correct. So for 1) we get [itex]a_i b_j c_i=a_ic_ib_j=(\mathbf{a} \cdot \mathbf{c})b_j[/itex]. So what kind of object do we get in the end?
 
  • #6
Aha..so is d_ii = 3?

And so for 1) do we get a vector in the end? so should i write (a.c) (b1, b2, b3)?

Thanks a lot for your help..starting to make more sense..

So for 2) is it just a number, namely (a.d)(b.c)?

3)a.a?
4)3
5)still not sure?
6) ditto..

thanks again
 
  • #7
Also in another question, I am asked to find the div and curl of this expression, where r = (x,y,z) and a and b are fixed vectors..

(a.r)b

so i took the div, which i found i could rearrange to be (a.r)(del.b) but can i simplify further? what does it mean when it says a and b are fixed vectors? does it mean that all components are constants, in which case, del.b = 0? thanks :)
 
  • #8
bon said:
cross would be e_ijk a_i b_j e_k?

Cyosis said:
Your dot and cross products are correct.

This cross product is incorrect. Remember that the cross product of two vectors is itself a vector, and so much have a free index. So, if [itex]{\bm a}\times{\bm b}={\bm c}[/itex], then [itex]c_i=\epsilon_{ijk}a_jb_k[/itex]
 
  • #9
I think he meant that e_k is a basis vector, and in that case it's correct.

Bon, I'm not sure what you find confusing about 1. It's really trivial if you think about what the notation means. For example, what does bj mean? Perhaps it would help if you write aibjci with a summation sigma and some parentheses?
 
  • #10
Fredrik said:
I think he meant that e_k is a basis vector, and in that case it's correct.

Oh, well then that's incredibly confusing to use the same symbols to mean different things!
 
  • #11
Cristo what you have written is the i-th component of the cross product. What bon has written down is the full vector with e being the unit vector.

Example:
[tex]
\epsilon_{ijk}a_ib_j\mathbf{e_k}=a_2b_3\mathbf{e_1}-a_3b_2\mathbf{e_1}+a_3b_1\mathbf{e_2}-a_1b_3\mathbf{e_2}+a_1b_2\mathbf{e_3}-a_2b_1\mathbf{e_3}=(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-b_2a_1)
[/tex]
 
  • #12
Cyosis said:
Cristo what you have written is the i-th component of the cross product. What bon has written down is the full vector with e being the unit vector.

Example:
[tex]
\epsilon_{ijk}a_ib_j\mathbf{e_k}=a_2b_3\mathbf{e_1}-a_3b_2\mathbf{e_1}+a_3b_1\mathbf{e_2}-a_1b_3\mathbf{e_2}+a_1b_2\mathbf{e_3}-a_2b_1\mathbf{e_3}=(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-b_2a_1)
[/tex]

Yes, if you right things out properly then it is obvious. What was written down above was ambiguous.
 
  • #13
Many thanks all..
so is anyone able to verify the comments i made in post 6? Also any thoughts on post 7?

Thank you
 
  • #14
The first four look good now. As for 5 compare it to your definition of the cross product. For 6 do something similar to 4) and think what it means for the Levi-Civita symbol if two of the indices are the same.
 
  • #15
aha ok get them now..

5) = jth component of a x b

and 6) = 0

finally ( i promise) any thoughts on my question in post 7? ty!
 
  • #16
dont worry if you arent sure about that.. i have another q in the meantime which is:

im meant to find div and curl of (a x r) where a is a fixed vector and r = (x,y,z)

after simplifying the expression for div i get:

e_ijk d/dx_i x_j a_k

but isn't this (del x r)times a which is a vector times a vector with no dot/cross = illegal?
 
  • #17
6) is correct. 5) is not entirely right remember that the Levi-Civita symbol is antisymmetric.

Yes that is illegal, but it is not what they are asking.

We have [itex]\nabla \times (a \times b)[/itex] first compute axb, which yields a vector and then compute the curl of that vector which again yields a vector. For the divergence we have [itex]\nabla \cdot (axb)[/itex]. First compute (a \times b) again which yields a vector then take the divergence which yields a scalar.
 
  • #18
Also in another question, I am asked to find the div and curl of this expression, where r = (x,y,z) and a and b are fixed vectors..

(a.r)b

so i took the div, which i found i could rearrange to be (a.r)(del.b) but can i simplify further? what does it mean when it says a and b are fixed vectors? does it mean that all components are constants, in which case, del.b = 0? thanks :)

Yes it means that all components are constant, but r is not a constant vector so you can't just put it in front of the del operator.
 
  • #19
hmm so is it -(axb)_j i.e. -1 times the jth component?
 
  • #20
Yes and -(axb) is?
 
  • #21
ok so for div (a.r) b my working is:

d/dx_i e_i . (a_j x_j b_k e_k)

why can't i just rewrite this as a_i x_i d/dx_j bj?

how do i deal with it then?

thanks
 
  • #22
Cyosis said:
Yes and -(axb) is?

aha (bxa)!
 
  • #23
Yes that is correct.

why can't i just rewrite this as a_i x_i d/dx_j bj?

For the same reason you can't write [itex]\frac{d}{dx}f(x)[/itex] as [itex]f(x)\frac{d}{dx}[/itex]. You are differentiating when taking the divergence.
 
  • #24
Cyosis said:
Yes that is correct.



For the same reason you can't write [itex]\frac{d}{dx}f(x)[/itex] as [itex]f(x)\frac{d}{dx}[/itex]. You are differentiating when taking the divergence.

ah ok i see..well I am not sure how to deal with it in that case..so far when doing divergence I've just dotted the e_i in the divergence term with the later e_j and simplified to d_ij etc..

what can i do now? thanks
 
  • #25
The handling of the unit vectors are fine, they do not depend on x y or z. Could you compute the divergence of r for me with r=(x,y,z) the 'normal' way (with that I mean don't use index notation)?
 
  • #26
yes that would just be 3, right? dx/dx + dy/dy + dz/dz
 
  • #27
Yes that is correct. Can you now compute [itex]\vec{r} \cdot \nabla[/itex] for me. Again the normal way?
 
  • #28
i thought dot product was commutative..so I am not sure..that would suggest 3

but intuitively xd/dx + yd/dy + zd/dz
 
  • #29
but no actually..result should be a scalar..so 3
 
  • #30
Post 28 is correct if you change the commas with plusses, post 29 is wrong. The dot product is only commutative if the elements of the vectors are commutative. The elements of the divergence vector are differential operators. To illustrate this I would like you to calculate [itex](\nabla \cdot \vec{r})x^2 [/itex] and [itex](\vec{r} \cdot \nabla)x^2[/itex]. Calculate it the normal way by taking the definition of the divergence and what you know about derivatives.
 
  • #31
sure.. so 3x^2

and 2x^2 + 2yx dx/dy + 2zx dx/dz?

how can i solve my question now using Einstein convention? thanks :)
 
  • #32
The first one is correct, the second one is not.

[tex]
(\vec{r} \cdot \nabla)x^2=(x \frac{d}{dx}+y\frac{d}{dy}+z\frac{d}{dz})x^2=2x^2
[/tex]

The dx^2/dy and dx^2z terms are 0. The reason we are doing this is so that you understand what the operator does, without that understanding you cannot make the exercise properly. The dot product of a vector and a differential operator does not commute.

Back to the original problem

d/dx_i e_i . (a_j x_j b_k e_k)

This is correct

why can't i just rewrite this as a_i x_i d/dx_j bj?

This is not, can you see why now?
 
  • #33
The first one is correct, the second one is not.

[tex]
(\vec{r} \cdot \nabla)x^2=(x \frac{d}{dx}+y\frac{d}{dy}+z\frac{d}{dz})x^2=2x^2
[/tex]

The dx^2/dy and dx^2z terms are 0. The reason we are doing this is so that you understand what the operator does, without that understanding you cannot make the exercise properly. The dot product of a vector and a differential operator does not commute.

Back to the original problem

d/dx_i e_i . (a_j x_j b_k e_k)

This is correct

why can't i just rewrite this as a_i x_i d/dx_j bj?

This is not, can you see why now?
 
  • #34
ok thanks, yes..so is it d/dx_i(a_j x_j) b_k e_i.e_k

=d/dx_i (a_j x_j) b_i

how do i proceed? thanks
 
  • #35
You know that a is a constant vector therefore a_j is just some constant which you can take in front of the differential operator. Now look at the expression d/dx_i x_j, what can you say about that? If you don't see it let i and j run through the values 1 2 3 and see what this operation yields for various combinations.
 
<h2>What is the Einstein Summation Convention?</h2><p>The Einstein Summation Convention, also known as the Einstein notation or the Einstein summation notation, is a mathematical notation used to simplify expressions in mathematics and physics. It was developed by Albert Einstein in his theory of general relativity.</p><h2>How does the Einstein Summation Convention work?</h2><p>The convention works by implicitly summing over repeated indices in an expression. This means that if an index appears twice in a term, it is assumed to be summed over all possible values. This allows for more concise and readable expressions in mathematical and physical equations.</p><h2>Why is the Einstein Summation Convention used?</h2><p>The convention is used because it simplifies mathematical expressions and makes them easier to read and understand. It also allows for a more compact representation of equations, which is especially useful in the field of physics where equations can become quite complex.</p><h2>What are the benefits of using the Einstein Summation Convention?</h2><p>The main benefit of using the convention is that it reduces the amount of notation needed in mathematical expressions, making them more concise and easier to read. It also allows for a more efficient and elegant representation of equations, which can be useful in solving complex problems.</p><h2>Are there any limitations to the Einstein Summation Convention?</h2><p>While the convention is a useful tool in simplifying mathematical expressions, it does have its limitations. It cannot be used in all cases, and some equations may still require more traditional notation. Additionally, it may be confusing for those who are not familiar with the convention, so it is important to use it carefully and clearly define the summation indices. </p>

What is the Einstein Summation Convention?

The Einstein Summation Convention, also known as the Einstein notation or the Einstein summation notation, is a mathematical notation used to simplify expressions in mathematics and physics. It was developed by Albert Einstein in his theory of general relativity.

How does the Einstein Summation Convention work?

The convention works by implicitly summing over repeated indices in an expression. This means that if an index appears twice in a term, it is assumed to be summed over all possible values. This allows for more concise and readable expressions in mathematical and physical equations.

Why is the Einstein Summation Convention used?

The convention is used because it simplifies mathematical expressions and makes them easier to read and understand. It also allows for a more compact representation of equations, which is especially useful in the field of physics where equations can become quite complex.

What are the benefits of using the Einstein Summation Convention?

The main benefit of using the convention is that it reduces the amount of notation needed in mathematical expressions, making them more concise and easier to read. It also allows for a more efficient and elegant representation of equations, which can be useful in solving complex problems.

Are there any limitations to the Einstein Summation Convention?

While the convention is a useful tool in simplifying mathematical expressions, it does have its limitations. It cannot be used in all cases, and some equations may still require more traditional notation. Additionally, it may be confusing for those who are not familiar with the convention, so it is important to use it carefully and clearly define the summation indices.

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