Prove that $ E^{ijk}=frac{1}{g} E_{ijk}$
Prove that the components of Contravariant Levi-Civita tensor can be given in terms of the components of the Covariant Levi-Civita tensor as follows:
$$ E^{ijk}=frac{1}{g} E_{ijk},$$ where $g$ is the metric tensor.
Answer:
We know,
$ begin{align}E_{ijk} &=+1, if (i,j,k) is (1,2,3), (3,1,2) and (2,3,1) \ &=-1, if (i,j,k) is (1,3,2), (3,2,1) and ( 2,1,3) \ &=0, if i=j or j=k or i=k end{align}$
Now,
$ large bar{E}^{ijk}= large frac{partial bar x^{i}}{partial x^a} frac{partial bar x^{j}}{partial x^b} frac{partial bar x^{k}}{partial x^c} E^{abc}$ and $g=g_{ijk} dx^{i} otimes dx^{j} otimes dx^{k} $
But how to do next?
Help me
differential-geometry tensors
asked Nov 26 at 0:43
arifamath
995
add a comment |
Prove that the components of Contravariant Levi-Civita tensor can be given in terms of the components of the Covariant Levi-Civita tensor as follows:
$$ E^{ijk}=frac{1}{g} E_{ijk},$$ where $g$ is the metric tensor.
Answer:
We know,
$ begin{align}E_{ijk} &=+1, if (i,j,k) is (1,2,3), (3,1,2) and (2,3,1) \ &=-1, if (i,j,k) is (1,3,2), (3,2,1) and ( 2,1,3) \ &=0, if i=j or j=k or i=k end{align}$
Now,
$ large bar{E}^{ijk}= large frac{partial bar x^{i}}{partial x^a} frac{partial bar x^{j}}{partial x^b} frac{partial bar x^{k}}{partial x^c} E^{abc}$ and $g=g_{ijk} dx^{i} otimes dx^{j} otimes dx^{k} $
But how to do next?
Help me
differential-geometry tensors
asked Nov 26 at 0:43
arifamath
995
What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58
add a comment |
Prove that the components of Contravariant Levi-Civita tensor can be given in terms of the components of the Covariant Levi-Civita tensor as follows:
$$ E^{ijk}=frac{1}{g} E_{ijk},$$ where $g$ is the metric tensor.
Answer:
We know,
$ begin{align}E_{ijk} &=+1, if (i,j,k) is (1,2,3), (3,1,2) and (2,3,1) \ &=-1, if (i,j,k) is (1,3,2), (3,2,1) and ( 2,1,3) \ &=0, if i=j or j=k or i=k end{align}$
Now,
$ large bar{E}^{ijk}= large frac{partial bar x^{i}}{partial x^a} frac{partial bar x^{j}}{partial x^b} frac{partial bar x^{k}}{partial x^c} E^{abc}$ and $g=g_{ijk} dx^{i} otimes dx^{j} otimes dx^{k} $
But how to do next?
Help me
differential-geometry tensors
asked Nov 26 at 0:43
arifamath
995
Prove that the components of Contravariant Levi-Civita tensor can be given in terms of the components of the Covariant Levi-Civita tensor as follows:
$$ E^{ijk}=frac{1}{g} E_{ijk},$$ where $g$ is the metric tensor.
Answer:
We know,
$ begin{align}E_{ijk} &=+1, if (i,j,k) is (1,2,3), (3,1,2) and (2,3,1) \ &=-1, if (i,j,k) is (1,3,2), (3,2,1) and ( 2,1,3) \ &=0, if i=j or j=k or i=k end{align}$
Now,
$ large bar{E}^{ijk}= large frac{partial bar x^{i}}{partial x^a} frac{partial bar x^{j}}{partial x^b} frac{partial bar x^{k}}{partial x^c} E^{abc}$ and $g=g_{ijk} dx^{i} otimes dx^{j} otimes dx^{k} $
But how to do next?
Help me
differential-geometry tensors
differential-geometry tensors
asked Nov 26 at 0:43
arifamath
995
asked Nov 26 at 0:43
arifamath
995
edited Nov 26 at 0:57
asked Nov 26 at 0:43
arifamath
995
asked Nov 26 at 0:43
arifamath
995
asked Nov 26 at 0:43
arifamath
995
995
What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58
add a comment |
What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58
What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58
What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58
add a comment |
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What is $frac{1}{g}$ in this expression?
– Matt
Nov 26 at 8:58