Mixing
Permutation of an element
$begingroup$
An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. The number of independent components in this array is
I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.
Sorry for my bad english, this is my first question.
permutations
edited Jan 17 at 5:06
Daniel Mathias
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
$endgroup$
add a comment |
$begingroup$
An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. The number of independent components in this array is
I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.
Sorry for my bad english, this is my first question.
permutations
edited Jan 17 at 5:06
Daniel Mathias
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
$endgroup$
add a comment |
$begingroup$
An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. The number of independent components in this array is
I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.
Sorry for my bad english, this is my first question.
permutations
edited Jan 17 at 5:06
Daniel Mathias
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
$endgroup$
An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. The number of independent components in this array is
I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.
Sorry for my bad english, this is my first question.
permutations
permutations
edited Jan 17 at 5:06
Daniel Mathias
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
edited Jan 17 at 5:06
Daniel Mathias
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
edited Jan 17 at 5:06
Daniel Mathias
1,30518
edited Jan 17 at 5:06
Daniel Mathias
1,30518
edited Jan 17 at 5:06
Daniel Mathias
1,30518
1,30518
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
asked Jan 17 at 4:13
Harshit TiwariHarshit Tiwari
32
32
add a comment |
add a comment |
1 Answer
1
active
oldest
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$begingroup$
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
$$
{i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
$$
in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
$$
{i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
$$
in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
$endgroup$
add a comment |
$begingroup$
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
$$
{i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
$$
in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
$endgroup$
add a comment |
$begingroup$
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
$$
{i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
$$
in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
$endgroup$
You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.
If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
$$
{i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
$$
in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).
If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
answered Jan 17 at 14:19
Vasily MitchVasily Mitch
2,3141311
2,3141311
add a comment |
add a comment |
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