How Can I Simplify and Solve the Einstein Summation Convention Problem?

In summary, the solution to the given problem is $$a_3 (b \cdot c) - b_3 (c \cdot a)$$ which can be obtained by using the Kronecker deltas to simplify the expression. It is helpful to use the properties of Kronecker deltas and to sum over repeated indices to simplify the problem.
  • #1
Athenian
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Homework Statement
Solve ##a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}##
Relevant Equations
See Below ##\longrightarrow##
Attempted Solution:
$$a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}$$
$$a_i\, b_j\, c_k\, (\delta_{i3} \, \delta_{jk} - \, \delta_{ik}\, \delta_{j3})$$

Beyond this, though, I am quite lost. I know I am very close to the answer, but seeing this many terms can become fairly confusing for me. Is there a way or method to better (and simply) digest the above problem and solve it?

Any help would be greatly appreciated. Thank you!
 
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  • #2
Athenian said:
Homework Statement:: Solve ##a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}##
Relevant Equations:: See Below ##\longrightarrow##

Attempted Solution:
$$a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}$$
$$a_i\, b_j\, c_k\, (\delta_{i3} \, \delta_{jk} - \, \delta_{ik}\, \delta_{j3})$$

Beyond this, though, I am quite lost. I know I am very close to the answer, but seeing this many terms can become fairly confusing for me. Is there a way or method to better (and simply) digest the above problem and solve it?

Any help would be greatly appreciated. Thank you!
That's the correct first step. Now, each Kronecker delta can be used to get rid of one of the indices that appears in it. For example, what happens if you sum over the index "i"?
 
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  • #3
@nrqed, thank you for the helpful hint and guidance! I was finally able to figure it out as seen below:

Continuing where I left off:
$$a_i \, b_j \, c_k \, \delta_{i3} \, \delta_{jk} - a_i \, b_j \, c_k \, \delta_{ik} \, \delta_{j3}$$
$$\Rightarrow a_3 \, b_k \, c_k - a_i \, b_c \, c_i$$
$$\Rightarrow a_3 (b \cdot c) - b_3 (c \cdot a)$$

Thank you for all your help!
 
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  • #4
Athenian said:
@nrqed, thank you for the helpful hint and guidance! I was finally able to figure it out as seen below:

Continuing where I left off:
$$a_i \, b_j \, c_k \, \delta_{i3} \, \delta_{jk} - a_i \, b_j \, c_k \, \delta_{ik} \, \delta_{j3}$$
$$\Rightarrow a_3 \, b_k \, c_k - a_i \, b_c \, c_i$$
$$\Rightarrow a_3 (b \cdot c) - b_3 (c \cdot a)$$

Thank you for all your help!
Good job!
 
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1. What is the Einstein Summation Convention?

The Einstein Summation Convention is a mathematical notation used to simplify the writing and manipulation of equations involving multiple indices. It was developed by physicist Albert Einstein and is commonly used in fields such as physics and engineering.

2. How does the Einstein Summation Convention work?

The convention states that when an index appears twice in a single term of an equation, it is implicitly summed over all possible values. This eliminates the need for explicit summation symbols and simplifies the notation of equations.

3. What are the benefits of using the Einstein Summation Convention?

Using the convention can make equations shorter and easier to read, as well as reducing the chance of errors in calculations. It also allows for a more compact representation of equations, making them easier to manipulate and work with.

4. Are there any limitations to the Einstein Summation Convention?

While the convention can be very useful, it is not suitable for all types of equations. It is most commonly used for equations involving vectors, matrices, and tensors. It also requires a good understanding of index notation and can be confusing for those not familiar with it.

5. Can the Einstein Summation Convention be extended to more than two indices?

Yes, the convention can be extended to any number of indices. This is known as the Einstein Summation Convention for Tensor Notation. It follows the same principles as the original convention, but with additional indices to represent higher-dimensional objects.

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