Problem with index notation

  • Thread starter andrien
  • Start date
  • #1
1,024
32

Main Question or Discussion Point

(r×∇).(r×∇)=r.∇×(r×∇)
now in index notation it is written as,
=xijxij-xijxji
but when I tried to prove it ,it just came out twice.can anyone tell how it is correct(given is the correct form).i really mean that i was getting four terms which gave twice of above after reshuffling so prove it.
 

Answers and Replies

  • #2
vanhees71
Science Advisor
Insights Author
Gold Member
2019 Award
14,695
6,232
All your formulae are correct.

The left-hand side of your first equation reads in index notation
[tex](\vec{r} \times \vec{\nabla}) \cdot (\vec{r} \times \vec{\nabla}) \phi= \epsilon_{jkl} r_k \partial_l (\epsilon_{jmn} r_m \partial_n \phi).[/tex]
Now using
[tex]\epsilon_{jkl} \epsilon_{jmn}=\delta_{km} \delta_{ln}-\delta_{kn} \delta_{lm},[/tex]
you indeed get
[tex](\vec{r} \times \vec{\nabla}) \cdot (\vec{r} \times \vec{\nabla}) \phi = r_{k} \partial_l(r_k \partial_l \phi)-r_k \partial_l(r_l \partial_k \phi).[/tex]

The right-hand side of your first equation is
[tex]\vec{r} \cdot [\vec{\nabla} \times (\vec{r} \times \vec{\nabla}) \phi] = r_j \epsilon_{jkl} \partial_k (\epsilon_{lmn} r_m \partial_n \phi).[/tex]
Again we have
[tex]\epsilon_{jkl} \epsilon_{lmn}=\epsilon_{ljk} \epsilon_{lmn}=\delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}.[/tex]
Thus we have
[tex]\vec{r} \cdot [\vec{\nabla} \times (\vec{r} \times \vec{\nabla}) \phi] = r_j \partial_k (r_j \partial_k \phi)-r_j \partial_k (r_k \partial_j \phi).[/tex]
This shows that indeed both expressions of your first equations are equal, because the only difference is the naming of the dummy-summation indices :-).
 
  • #3
1,024
32
[tex]\epsilon_{jkl} \epsilon_{jmn}=\delta_{km} \delta_{ln}-\delta_{kn} \delta_{lm},[/tex]
I was aware of it which I have seen in butkov an year ago.but this is the first use of it.so thanks,van I think i am just becoming lazy.
 

Related Threads on Problem with index notation

Replies
4
Views
428
  • Last Post
Replies
0
Views
2K
Replies
5
Views
11K
Replies
20
Views
25K
Replies
16
Views
3K
Replies
4
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
3
Views
661
Replies
1
Views
6K
  • Last Post
Replies
3
Views
2K
Top