INDICIAL NOTATION (Cartesian Tensor)

INDICIAL NOTATION (Cartesian Tensor)
Basic Rules
i) A free index appears only once in each term of a tensor equation. The equation
then holds for all possible values of that index.
ii) Summation is implied on an index, which appears twice.
iii) No index can appear more than twice in any term.
Definition (Cartesian Tensors)
Consider a change of frame
x = Q x ,
*
i
ij
j
*
x
j
= Qijx
i
, det Qij = ± 1, QijQ ik
= δ
jk
, QijQ kj
= δik
The quantities T(
x ) , v (x)
, and t (x)
are said to be a scalar, a vector, or a second order
tensor if they transform according to the rules:
T * *
= T, v = Q ⋅ v ,
or in component forms:
τ
*
= Q ⋅ t ⋅Q
T
*
*
v
i
= Qijv
j
, t
ij
= QikQ
jlt
kl
Similarly, a third order tensor λ transforms as:
λ
*
ijk
= Q Q Q λ
im
jn
kl
mnl
Tensor Operations
Gradient:
∂ϕ
( ∇ϕ)
i
= = ϕ
∂x
i
∂v
j
( ∇v)
ij
= = v
∂x
ε
,i
j,i
Divergence:
∇ ⋅ v = v i,i
∂τij
( ∇ ⋅ τ )
j
= = τ
∂x
i
ij,i
ME637
G. Ahmadi

Curl:
∂U
k
( ∇ × U )
i
= εijk
= εijk
∂x
j
Here, ε
ijk
is the permutation symbol. The permutation symbol is used for
evaluating the determinant. e.g.,
det A = ε A A A ,
ijk
1i
2 j
3k
U
k, j
The inner product of the permutation symbol satisfy the following identity:
ε
ijk
ε
imn
= δ
jm
δ
kn
− δ
jn
δ
km
Laplacian:
Isotropic Tensors
2
2 ∂ ϕ
∇ ϕ =
∂x
∂x
i
i
= ϕ
,ii
Isotropic tensors are tensors, which are form invariant under all possible rotations
of the frame of reference. The most general forms of the isotropic tensors are:
Rank zero: All scalars isotropic
Rank one: None
Rank two: αδ
ij
, α a scalar and δ
ij
= Kronecker delta
Rank three: αε
ijk
Rank four: αδ δ + β δ δ + δ δ ) + γ(
δ δ − δ δ )
ij
kl
(
ik jl il jk ik jl il jk
ME637
G. Ahmadi