How to prove that the Kronecker delta is the unique isotropic tensor of order 2?












1














Is there a way to prove that the Kronecker delta $delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = lambda delta_{ij}$ for some constant $lambda$?



By rotational invariance I mean:
$$
T_{ij} = T^prime_{ij} = R_{ip} R_{jq} T_{pq}text{,}
$$
where the matrices $R_{ij}$ are orthogonal.



It is very straightforward to show that $delta_{ij}$ is invariant, but how can I show that it is unique?










share|cite|improve this question






















  • If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
    – gebruiker
    Mar 30 '16 at 13:28
















1














Is there a way to prove that the Kronecker delta $delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = lambda delta_{ij}$ for some constant $lambda$?



By rotational invariance I mean:
$$
T_{ij} = T^prime_{ij} = R_{ip} R_{jq} T_{pq}text{,}
$$
where the matrices $R_{ij}$ are orthogonal.



It is very straightforward to show that $delta_{ij}$ is invariant, but how can I show that it is unique?










share|cite|improve this question






















  • If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
    – gebruiker
    Mar 30 '16 at 13:28














1












1








1


1





Is there a way to prove that the Kronecker delta $delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = lambda delta_{ij}$ for some constant $lambda$?



By rotational invariance I mean:
$$
T_{ij} = T^prime_{ij} = R_{ip} R_{jq} T_{pq}text{,}
$$
where the matrices $R_{ij}$ are orthogonal.



It is very straightforward to show that $delta_{ij}$ is invariant, but how can I show that it is unique?










share|cite|improve this question













Is there a way to prove that the Kronecker delta $delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = lambda delta_{ij}$ for some constant $lambda$?



By rotational invariance I mean:
$$
T_{ij} = T^prime_{ij} = R_{ip} R_{jq} T_{pq}text{,}
$$
where the matrices $R_{ij}$ are orthogonal.



It is very straightforward to show that $delta_{ij}$ is invariant, but how can I show that it is unique?







rotations tensors invariance






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jul 29 '14 at 14:07









mSSM

14617




14617












  • If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
    – gebruiker
    Mar 30 '16 at 13:28


















  • If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
    – gebruiker
    Mar 30 '16 at 13:28
















If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
– gebruiker
Mar 30 '16 at 13:28




If any of the answeres below were useful to you, then you should upvote all answers you find useful and accept the one that was most useful to you. It is a way to show that you have found the answer to your question and it shows your appreciation. Now it seems like you still need help. If answers are not useful to you, then it helps if you say why not. This helps others to help you. For more information read this.
– gebruiker
Mar 30 '16 at 13:28










2 Answers
2






active

oldest

votes


















1














Step 1: Realize $T$ is diagonal. Let
$$R_{kl}=begin{cases}
-1 & text{if }k=l=j\
delta_{kl} & text{otherwise}\
end{cases}$$
therefore $R$ is the reflection w.r.t to the hyperplane perpendicular to the j-th vector in the standard ordered basis.
$$R=R^Tland R^2=IRightarrow R^TR=RR^T=I$$
Therefore:
$$T_{ij}=sum_{p,q}R_{ip}R_{jq}T_{pq}=R_{ii}R_{jj}T_{ij}\
ineq jRightarrow T_{ij}=-T_{ij}Rightarrow T_{ij}=0$$





Step 2: Realize $T_{jj} = T_{11}$. Let
$$P_{kl}=begin{cases}
delta_{jl} & text{if } k=1\
delta_{1l} & text{if } k=j\
delta_{kl} & text{otherwise}
end{cases}$$
therefore $P$ is the permutation matrix that interchanges the 1st and j-th rows on left multiplication.
$$(P^TP)_{kl}=sum_{m}P^T_{km}P_{ml}=sum_{m}P_{mk}P_{ml}=sum_{mneq1,j}P_{mk}P_{ml}+sum_{m=1,j}P_{mk}P_{ml}\
=sum_{mneq1,j}delta_{mk}delta_{ml}+delta_{jk}delta_{jl}+delta_{1k}delta_{1l}=sum_{m}delta_{mk}delta_{ml}=delta_{kl}$$
Therefore:
$$T_{jj}=sum_{p,q}P_{jp}P_{jq}T_{pq}=sum_{q}P_{jq}^2T_{qq}=sum_{q}delta_{1q}^2T_{qq}=sum_{q}delta_{1q}T_{qq}=T_{11}$$






share|cite|improve this answer



















  • 1




    wow, whole answer in math mode :D
    – MyUserIsThis
    Jun 6 '15 at 8:16










  • It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
    – Najib Idrissi
    Dec 17 '15 at 14:13










  • The very first argument: why do you assume that $R_{ij}$ has that form?
    – mSSM
    Mar 31 '16 at 11:05










  • @mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
    – Jack's wasted life
    Mar 31 '16 at 12:16



















1














There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows



Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.



That is, the proof is simplicity itself and is given in detail in the reference.






share|cite|improve this answer























  • That is indeed a very nice proof. Thank you very much!
    – mSSM
    Dec 3 '18 at 15:47











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f881571%2fhow-to-prove-that-the-kronecker-delta-is-the-unique-isotropic-tensor-of-order-2%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









1














Step 1: Realize $T$ is diagonal. Let
$$R_{kl}=begin{cases}
-1 & text{if }k=l=j\
delta_{kl} & text{otherwise}\
end{cases}$$
therefore $R$ is the reflection w.r.t to the hyperplane perpendicular to the j-th vector in the standard ordered basis.
$$R=R^Tland R^2=IRightarrow R^TR=RR^T=I$$
Therefore:
$$T_{ij}=sum_{p,q}R_{ip}R_{jq}T_{pq}=R_{ii}R_{jj}T_{ij}\
ineq jRightarrow T_{ij}=-T_{ij}Rightarrow T_{ij}=0$$





Step 2: Realize $T_{jj} = T_{11}$. Let
$$P_{kl}=begin{cases}
delta_{jl} & text{if } k=1\
delta_{1l} & text{if } k=j\
delta_{kl} & text{otherwise}
end{cases}$$
therefore $P$ is the permutation matrix that interchanges the 1st and j-th rows on left multiplication.
$$(P^TP)_{kl}=sum_{m}P^T_{km}P_{ml}=sum_{m}P_{mk}P_{ml}=sum_{mneq1,j}P_{mk}P_{ml}+sum_{m=1,j}P_{mk}P_{ml}\
=sum_{mneq1,j}delta_{mk}delta_{ml}+delta_{jk}delta_{jl}+delta_{1k}delta_{1l}=sum_{m}delta_{mk}delta_{ml}=delta_{kl}$$
Therefore:
$$T_{jj}=sum_{p,q}P_{jp}P_{jq}T_{pq}=sum_{q}P_{jq}^2T_{qq}=sum_{q}delta_{1q}^2T_{qq}=sum_{q}delta_{1q}T_{qq}=T_{11}$$






share|cite|improve this answer



















  • 1




    wow, whole answer in math mode :D
    – MyUserIsThis
    Jun 6 '15 at 8:16










  • It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
    – Najib Idrissi
    Dec 17 '15 at 14:13










  • The very first argument: why do you assume that $R_{ij}$ has that form?
    – mSSM
    Mar 31 '16 at 11:05










  • @mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
    – Jack's wasted life
    Mar 31 '16 at 12:16
















1














Step 1: Realize $T$ is diagonal. Let
$$R_{kl}=begin{cases}
-1 & text{if }k=l=j\
delta_{kl} & text{otherwise}\
end{cases}$$
therefore $R$ is the reflection w.r.t to the hyperplane perpendicular to the j-th vector in the standard ordered basis.
$$R=R^Tland R^2=IRightarrow R^TR=RR^T=I$$
Therefore:
$$T_{ij}=sum_{p,q}R_{ip}R_{jq}T_{pq}=R_{ii}R_{jj}T_{ij}\
ineq jRightarrow T_{ij}=-T_{ij}Rightarrow T_{ij}=0$$





Step 2: Realize $T_{jj} = T_{11}$. Let
$$P_{kl}=begin{cases}
delta_{jl} & text{if } k=1\
delta_{1l} & text{if } k=j\
delta_{kl} & text{otherwise}
end{cases}$$
therefore $P$ is the permutation matrix that interchanges the 1st and j-th rows on left multiplication.
$$(P^TP)_{kl}=sum_{m}P^T_{km}P_{ml}=sum_{m}P_{mk}P_{ml}=sum_{mneq1,j}P_{mk}P_{ml}+sum_{m=1,j}P_{mk}P_{ml}\
=sum_{mneq1,j}delta_{mk}delta_{ml}+delta_{jk}delta_{jl}+delta_{1k}delta_{1l}=sum_{m}delta_{mk}delta_{ml}=delta_{kl}$$
Therefore:
$$T_{jj}=sum_{p,q}P_{jp}P_{jq}T_{pq}=sum_{q}P_{jq}^2T_{qq}=sum_{q}delta_{1q}^2T_{qq}=sum_{q}delta_{1q}T_{qq}=T_{11}$$






share|cite|improve this answer



















  • 1




    wow, whole answer in math mode :D
    – MyUserIsThis
    Jun 6 '15 at 8:16










  • It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
    – Najib Idrissi
    Dec 17 '15 at 14:13










  • The very first argument: why do you assume that $R_{ij}$ has that form?
    – mSSM
    Mar 31 '16 at 11:05










  • @mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
    – Jack's wasted life
    Mar 31 '16 at 12:16














1












1








1






Step 1: Realize $T$ is diagonal. Let
$$R_{kl}=begin{cases}
-1 & text{if }k=l=j\
delta_{kl} & text{otherwise}\
end{cases}$$
therefore $R$ is the reflection w.r.t to the hyperplane perpendicular to the j-th vector in the standard ordered basis.
$$R=R^Tland R^2=IRightarrow R^TR=RR^T=I$$
Therefore:
$$T_{ij}=sum_{p,q}R_{ip}R_{jq}T_{pq}=R_{ii}R_{jj}T_{ij}\
ineq jRightarrow T_{ij}=-T_{ij}Rightarrow T_{ij}=0$$





Step 2: Realize $T_{jj} = T_{11}$. Let
$$P_{kl}=begin{cases}
delta_{jl} & text{if } k=1\
delta_{1l} & text{if } k=j\
delta_{kl} & text{otherwise}
end{cases}$$
therefore $P$ is the permutation matrix that interchanges the 1st and j-th rows on left multiplication.
$$(P^TP)_{kl}=sum_{m}P^T_{km}P_{ml}=sum_{m}P_{mk}P_{ml}=sum_{mneq1,j}P_{mk}P_{ml}+sum_{m=1,j}P_{mk}P_{ml}\
=sum_{mneq1,j}delta_{mk}delta_{ml}+delta_{jk}delta_{jl}+delta_{1k}delta_{1l}=sum_{m}delta_{mk}delta_{ml}=delta_{kl}$$
Therefore:
$$T_{jj}=sum_{p,q}P_{jp}P_{jq}T_{pq}=sum_{q}P_{jq}^2T_{qq}=sum_{q}delta_{1q}^2T_{qq}=sum_{q}delta_{1q}T_{qq}=T_{11}$$






share|cite|improve this answer














Step 1: Realize $T$ is diagonal. Let
$$R_{kl}=begin{cases}
-1 & text{if }k=l=j\
delta_{kl} & text{otherwise}\
end{cases}$$
therefore $R$ is the reflection w.r.t to the hyperplane perpendicular to the j-th vector in the standard ordered basis.
$$R=R^Tland R^2=IRightarrow R^TR=RR^T=I$$
Therefore:
$$T_{ij}=sum_{p,q}R_{ip}R_{jq}T_{pq}=R_{ii}R_{jj}T_{ij}\
ineq jRightarrow T_{ij}=-T_{ij}Rightarrow T_{ij}=0$$





Step 2: Realize $T_{jj} = T_{11}$. Let
$$P_{kl}=begin{cases}
delta_{jl} & text{if } k=1\
delta_{1l} & text{if } k=j\
delta_{kl} & text{otherwise}
end{cases}$$
therefore $P$ is the permutation matrix that interchanges the 1st and j-th rows on left multiplication.
$$(P^TP)_{kl}=sum_{m}P^T_{km}P_{ml}=sum_{m}P_{mk}P_{ml}=sum_{mneq1,j}P_{mk}P_{ml}+sum_{m=1,j}P_{mk}P_{ml}\
=sum_{mneq1,j}delta_{mk}delta_{ml}+delta_{jk}delta_{jl}+delta_{1k}delta_{1l}=sum_{m}delta_{mk}delta_{ml}=delta_{kl}$$
Therefore:
$$T_{jj}=sum_{p,q}P_{jp}P_{jq}T_{pq}=sum_{q}P_{jq}^2T_{qq}=sum_{q}delta_{1q}^2T_{qq}=sum_{q}delta_{1q}T_{qq}=T_{11}$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 31 '16 at 12:24

























answered Jun 6 '15 at 8:06









Jack's wasted life

7,37111430




7,37111430








  • 1




    wow, whole answer in math mode :D
    – MyUserIsThis
    Jun 6 '15 at 8:16










  • It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
    – Najib Idrissi
    Dec 17 '15 at 14:13










  • The very first argument: why do you assume that $R_{ij}$ has that form?
    – mSSM
    Mar 31 '16 at 11:05










  • @mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
    – Jack's wasted life
    Mar 31 '16 at 12:16














  • 1




    wow, whole answer in math mode :D
    – MyUserIsThis
    Jun 6 '15 at 8:16










  • It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
    – Najib Idrissi
    Dec 17 '15 at 14:13










  • The very first argument: why do you assume that $R_{ij}$ has that form?
    – mSSM
    Mar 31 '16 at 11:05










  • @mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
    – Jack's wasted life
    Mar 31 '16 at 12:16








1




1




wow, whole answer in math mode :D
– MyUserIsThis
Jun 6 '15 at 8:16




wow, whole answer in math mode :D
– MyUserIsThis
Jun 6 '15 at 8:16












It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
– Najib Idrissi
Dec 17 '15 at 14:13




It is actually a horrible idea to compose your whole answer in math mode. It make it difficult to copy the text for some people (when you right click a formula a different menu pops up), screen readers will not work correctly (for visually-impaired people), on mobile devices the formulas with a big width are a pain, etc. Please consider writing only formulas in math mode.
– Najib Idrissi
Dec 17 '15 at 14:13












The very first argument: why do you assume that $R_{ij}$ has that form?
– mSSM
Mar 31 '16 at 11:05




The very first argument: why do you assume that $R_{ij}$ has that form?
– mSSM
Mar 31 '16 at 11:05












@mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
– Jack's wasted life
Mar 31 '16 at 12:16




@mSSM I don't assume every orthogonal R has that form. That R is a tensor which meets the criterion of orthogonality and which is designed in that specific way for the purpose of bringing out the diagonal nature of T.
– Jack's wasted life
Mar 31 '16 at 12:16











1














There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows



Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.



That is, the proof is simplicity itself and is given in detail in the reference.






share|cite|improve this answer























  • That is indeed a very nice proof. Thank you very much!
    – mSSM
    Dec 3 '18 at 15:47
















1














There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows



Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.



That is, the proof is simplicity itself and is given in detail in the reference.






share|cite|improve this answer























  • That is indeed a very nice proof. Thank you very much!
    – mSSM
    Dec 3 '18 at 15:47














1












1








1






There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows



Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.



That is, the proof is simplicity itself and is given in detail in the reference.






share|cite|improve this answer














There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows



Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.



That is, the proof is simplicity itself and is given in detail in the reference.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 3 '18 at 17:56

























answered Dec 3 '18 at 15:23









DrWill

185




185












  • That is indeed a very nice proof. Thank you very much!
    – mSSM
    Dec 3 '18 at 15:47


















  • That is indeed a very nice proof. Thank you very much!
    – mSSM
    Dec 3 '18 at 15:47
















That is indeed a very nice proof. Thank you very much!
– mSSM
Dec 3 '18 at 15:47




That is indeed a very nice proof. Thank you very much!
– mSSM
Dec 3 '18 at 15:47


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f881571%2fhow-to-prove-that-the-kronecker-delta-is-the-unique-isotropic-tensor-of-order-2%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Berounka

Sphinx de Gizeh

Different font size/position of beamer's navigation symbols template's content depending on regular/plain...