Solving Einstein Notation: Summing Subscripts and Superscripts

In summary, the conversation discusses using the Einstein summation rule for matched superscripts and subscripts and the use of the dual basis to simplify the notation. The conversation also mentions the need for a Ʃ when summing over indices and the possibility of avoiding this. Suggestions are given for using the dual basis to simplify the notation.
  • #1
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
4,116
1,712
[itex]\vec{v} = v^i\vec{e}_i = g(\vec{v},\vec{e}_i)\vec{e}_i[/itex]

The last bit is a sum over i but will need a Ʃ because the Einstein rule only applies to matched superscripts and subscripts and here bot the i are subscripts.

Even if I write out the metric in the basis it doesn't work:

[itex]g(\vec{v},\vec{e}_i)\vec{e}_i=g_{ab}v^ae^{b}_{i} \vec{e}_i[/itex]

In everything else I've ever done the indices have always been where they needed to be for Einstein summation but for some reason in this one they're not. It's no hardship to write the [itex]\Sigma^{n}_{i=1}[/itex] before it but it just feels as though there should be a way to avoid that.

Any suggestions or comments? Thanks very much.
 
Physics news on Phys.org
  • #2
You can use the dual basis, defined by

[tex]e^i (e_j) = \delta^i{}_j[/tex]
Then you have

[tex]v = v^i e_i = e^i (v) \, e_i[/tex]
 
  • #3
Thanks very much. I thought there had to be a way.

[itex]\vec{v}=v^i\vec{e}_i=\tilde{e}^i(\vec{v})\vec{e}_i[/itex]

Beautiful!
 

1. What is Einstein notation and when is it used?

Einstein notation, also known as Einstein summation convention, is a shorthand notation used to express and manipulate mathematical equations involving vectors, matrices, and tensors. It is commonly used in physics and engineering to simplify the notation of complex equations.

2. How do you sum subscripts and superscripts in Einstein notation?

In Einstein notation, repeated indices in an equation indicate a summation over those indices. To sum subscripts and superscripts, simply replace the repeated index with the summation symbol (Σ) and the range of values over which the index will be summed. For example, in the equation AijBj, the index j is repeated, so it will be summed over the appropriate range.

3. Can Einstein notation be used for any type of mathematical equation?

No, Einstein notation is specifically designed for use with tensors, which are multidimensional arrays representing quantities that are independent of the coordinate system. It cannot be used for other types of equations, such as scalar equations or equations involving only vectors.

4. What are the advantages of using Einstein notation?

Einstein notation allows for the concise representation of complex equations, making them easier to read and write. It also simplifies the process of manipulating equations, particularly when dealing with higher-dimensional tensors. Additionally, it helps to identify errors in equations and reduces the chances of making mistakes during calculations.

5. Are there any common mistakes to avoid when using Einstein notation?

One common mistake is not properly identifying the range of values over which a repeated index should be summed. Another mistake is using Einstein notation for non-tensor equations. It is important to understand the principles and limitations of Einstein notation before applying it to equations.

Similar threads

  • Special and General Relativity
Replies
8
Views
2K
  • Special and General Relativity
Replies
9
Views
3K
Replies
2
Views
1K
  • Special and General Relativity
Replies
4
Views
740
  • Linear and Abstract Algebra
Replies
1
Views
780
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Special and General Relativity
Replies
4
Views
2K
  • Special and General Relativity
Replies
14
Views
21K
  • Quantum Physics
Replies
6
Views
940
  • Other Physics Topics
Replies
3
Views
1K
Back
Top