Translation from vector calc. notation to index notation

In summary: I suspect that you should write it like:$$\begin{align}R^j{}_i x^i &= n^j g_{hf} n^h x^f + \cos(\phi)\sqrt{\lvert g \rvert} \bigl(\epsilon^{jln}n_l x^n \bigr) n^h g_{hf} + \sin(\phi)\sqrt{\lvert g \rvert} \bigl(\epsilon^j{}_{pq} n^p x^q \bigr) \\&= n^j g_{hf} n^h x^f + \cos(\phi)\sqrt{\lvert g \rvert} \bigl(\epsilon^{
  • #1
JonnyMaddox
74
1
Hi, I want to translate this equation
[itex]R_{\hat{n}}(\alpha)\vec{x}=\hat{n}(\hat{n}\cdot\vec{x})+\cos\left(\alpha\right)(\hat{n}\times\vec{x})\times\hat{n}+\sin\left(\alpha\right)(\hat{n}\times\vec{x})[/itex]
to index notation (forget about covariant and contravariant indices).

My attempt:
[itex]R_{ji}x_{i}=n_{j}n_{k}n_{k}-cos(\phi)\epsilon_{lmj} \epsilon_{noj} n_{l}x_{m}n_{j}+sin(\phi)\epsilon_{pqj} n_{p}x_{q}[/itex]

The minus sign comes from [itex]\hat{n} \times (\vec{x} \times \hat{n})= -(\hat{n} \times \vec{x}) \times \hat{n}[/itex]

So is this right??
 
Physics news on Phys.org
  • #2
If you are not going to have any upper indices, it would be appropriate to restore the implied summations since the Einstein summation convention does not apply for indices which are all lower.

Looking at your expression though, the indices don't seem to match up very well between the left and right sides...

Also, I think your attempt at a solution, even after you have fixed the errors, would only be valid in Cartesian coordinates. In other coordinate systems, the dot product and cross product does not have such a nice form...
 
Last edited:
  • #3
Matterwave said:
If you are not going to have any upper indices, it would be appropriate to restore the implied summations since the Einstein summation convention does not apply for indices which are all lower.

Looking at your expression though, the indices don't seem to match up very well between the left and right sides...

Hello !
Ok you are right, but in the book I use, tensor calculus is introduces first in cartesian coordinates and they make no distinction between upper and lower indices but introduce the Einstein summation convention. Could you be a bit more explicit what went wrong? I think that this is wrong [itex]
cos(\phi)\epsilon_{lmj} \epsilon_{noj} n_{l}x_{m}n_{j}[/itex], but I'm not sure what to do about the n and o indices. Maybe I should put them to the left side somehow. And I read in some pdf that they put an additional basis into the epsilon tensor expression like [itex]\vec{b} \times \vec{c}=\epsilon_{jkl}b_{j}c_{k}\hat{e}_{l}[/itex] But shouldn't the expression be free of any reference to a basis??
 
  • #4
JonnyMaddox said:
Hello !
Ok you are right, but in the book I use, tensor calculus is introduces first in cartesian coordinates and they make no distinction between upper and lower indices but introduce the Einstein summation convention. Could you be a bit more explicit what went wrong? I think that this is wrong [itex]
cos(\phi)\epsilon_{lmj} \epsilon_{noj} n_{l}x_{m}n_{j}[/itex], but I'm not sure what to do about the n and o indices. Maybe I should put them to the left side somehow. And I read in some pdf that they put an additional basis into the epsilon tensor expression like [itex]\vec{b} \times \vec{c}=\epsilon_{jkl}b_{j}c_{k}\hat{e}_{l}[/itex] But shouldn't the expression be free of any reference to a basis??

Stick with Cartesian coordinates for now, restore the summations, and let's see what expression you end up with.

To get you started, the first term in the OP should read something like $$(\hat{n}(\hat{n}\cdot\vec{x}))_j=n_j(\sum_k n_k x_k)$$
 
  • #5
Matterwave said:
Stick with Cartesian coordinates for now, restore the summations, and let's see what expression you end up with.

To get you started, the first term in the OP should read something like $$(\hat{n}(\hat{n}\cdot\vec{x}))_j=n_j(\sum_k n_k x_k)$$

Ok that's clear I think. Now the second term should be, since it is just a vector [itex]x_{k}[/itex]:
[itex]x_{k}= cos(\phi)\epsilon_{kln} \epsilon_{mjn} n_{l}x_{m}n_{j}[/itex]
Now this is a contraction of the n indices and summation on the l,m and j indices. Now how should I incorporate that in my above equation? Something like this [itex][R_{ji}x_{i}]_{k}[/itex] ?? I have no idea why there is contraction of the n indices but ok...
 
  • #6
Ah I see, it's just [itex]R_{ji}x_{i}=n_{j}n_{k}n_{k}-cos(\phi)\epsilon_{jln} \epsilon_{mon} n_{l}x_{m}n_{o}+sin(\phi)\epsilon_{pqj} n_{p}x_{q}[/itex] . If this is right I'll try to convert it into upper and lower indices with metric tensors.
 
  • #7
JonnyMaddox said:
Ah I see, it's just [itex]R_{ji}x_{i}=n_{j}n_{k}n_{k}-cos(\phi)\epsilon_{jln} \epsilon_{mon} n_{l}x_{m}n_{o}+sin(\phi)\epsilon_{pqj} n_{p}x_{q}[/itex] . If this is right I'll try to convert it into upper and lower indices with metric tensors.

In notation with lower and upper indices and in all dimensions and geometry it is I think:
[itex]R^{j}_{i}x^{i}=n^{j}g_{hf}n^{h}n^{f}-cos(\phi)\sqrt{|det (g)|}\epsilon^{jln} \epsilon_{mon} n_{l}x^{m}n^{o}+sin(\phi)\sqrt{|det (g)|}\epsilon^{j}_{pq} n^{p}x^{q}[/itex]
 
Last edited:
  • #8
Hi may someone comment if this is right? :)
 
  • #9
Your first term should have an ##x## in it rather than 3 ##n##'s. I'm not sure about raising and lowering indices on the Levi-civita symbol since that will bring in factors of the metric. I can't recall the exactly correct way to take a cross product in such a notation.
 

What is vector calc. notation?

Vector calc. notation is a mathematical notation used to represent vectors and operations on vectors. It typically involves the use of symbols such as arrows, dots, and cross products.

What is index notation?

Index notation is a more concise and efficient way of representing vectors and operations on vectors. It involves the use of subscripts to represent the components of a vector and summation notation to represent vector operations.

How do you translate from vector calc. notation to index notation?

To translate from vector calc. notation to index notation, you can use the following steps:

  • Replace arrows with commas.
  • Replace dot products with summation notation.
  • Replace cross products with a determinant.
  • Replace unit vectors with subscripts.

What are the benefits of using index notation?

Index notation is more concise and efficient compared to vector calc. notation. It also allows for easier manipulation and computation of vector operations. Additionally, it is the standard notation used in many fields such as physics and engineering.

Are there any limitations to using index notation?

One limitation of index notation is that it may not be as intuitive for beginners who are more familiar with vector calc. notation. It also requires a good understanding of index notation rules and conventions for accurate translation and computation.

Similar threads

Replies
3
Views
1K
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
164
  • Precalculus Mathematics Homework Help
Replies
20
Views
637
  • Differential Geometry
Replies
12
Views
3K
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Precalculus Mathematics Homework Help
Replies
18
Views
419
  • Calculus and Beyond Homework Help
Replies
9
Views
674
  • Advanced Physics Homework Help
Replies
1
Views
730
Replies
3
Views
1K
Back
Top