INDICIAL NOTATION (Cartesian Tensor)
INDICIAL NOTATION (Cartesian Tensor)
INDICIAL NOTATION (Cartesian Tensor)
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Curl:<br />
∂U<br />
k<br />
( ∇ × U )<br />
i<br />
= εijk<br />
= εijk<br />
∂x<br />
j<br />
Here, ε<br />
ijk<br />
is the permutation symbol. The permutation symbol is used for<br />
evaluating the determinant. e.g.,<br />
det A = ε A A A ,<br />
ijk<br />
1i<br />
2 j<br />
3k<br />
U<br />
k, j<br />
The inner product of the permutation symbol satisfy the following identity:<br />
ε<br />
ijk<br />
ε<br />
imn<br />
= δ<br />
jm<br />
δ<br />
kn<br />
− δ<br />
jn<br />
δ<br />
km<br />
Laplacian:<br />
∇<br />
2<br />
ϕ =<br />
2<br />
∂ ϕ<br />
∂x<br />
∂x<br />
i<br />
i<br />
= ϕ<br />
,ii<br />
Isotropic <strong>Tensor</strong>s<br />
Isotropic tensors are tensors, which are form invariant under all possible rotations<br />
of the frame of reference. The most general forms of the isotropic tensors are:<br />
Rank zero: All scalars isotropic<br />
Rank one: None<br />
Rank two: αδij<br />
, α a scalar and δ<br />
ij<br />
= Kronec ker delta<br />
Rank three:<br />
αε<br />
ijk<br />
Rank four: αδ δ + β δ δ + δ δ ) + γ(<br />
δ δ − δ δ )<br />
ij<br />
kl<br />
(<br />
ik jl il jk ik jl il jk<br />
ME639<br />
2<br />
G. Ahmadi
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