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Tensors
Transformation Rule

x3
x3
x2

A Cartesian vector can be
defined by its transformation
rule.
Another transformation
matrix T transforms similarly.
x2
x j l ij xi
x1
x1
xi  lij x j
Tpq l ipl jqTij
Tij l ip l jqTpq
Order and Rank


For a Cartesian coordinate
system a tensor is defined
by its transformation rule.
The order or rank of a tensor
determines the number of
separate transformations.
• Rank 0: scalar
• Rank 1: vector
• Rank 2 and up: Tensor

The Kronecker delta is the
unit rank-2 tensor.
ss
Scalars are independent
of coordinate system.
x j l ij xi
Tpq l ipl jqTij
Tp1pn l i1 p1 l in pnTi1in
 pq l ip l jq  ij
Direct Product

A rank 2 tensor can be
represented as a matrix.

Two vectors can be
combined into a matrix.
• Vector direct product
• Old name dyad
• Indices transform as
separate vectors
C11 C12 C13 
C  C21 C22 C23 
C31 C32 C33 
   T
C  AB  A  B
 a1b1
C  a2b1
 a3b1
a1b2
a2b2
a3b2
a1b3 
a2b3 
a3b3 
Tensor Algebra

1T  T
Tensors form a linear vector
space.
• Tensors T, U
• Scalars f, g
f ( gT)  ( fg )T
( f  g )T  fT  gT
f ( T  U )  fT  fU
T U  UT
T  U  Tij  U ij

Tensor algebra includes
addition and scalar
multiplication.
• Operations by component
• Usual rules of algebra
Contraction


The summation rule applies
to tensors of different ranks.
ci  Aijb j
• Dot product
• Sum of ranks reduce by 2
Tik   ijk v j
A tensor can be contracted
by summing over a pair of
indices.
• Reduces rank by 2
• Rank 2 tensor contracts to
the trace
s  ai bi
Tij  Tii
3
tr T   ijTij   Tii
i 1
Symmetric Tensor


The transpose of a rank-2
tensor reverses the indices.
~
T T ij  Tij  T ji
• Transposed products and
products transposed
(TU ) T  U T TT
A symmetric tensor is its own
transpose.
• Antisymmetric is negative
transpose

All tensors are the sums of
symmetric and
antisymmetric parts.
Sij  S ji
Aij   A ji

 

~
~
T  12 T  T  12 T  T
TSA
Eigenvalues
viTij  v j

viTij  vi ij
vi Tij   ij   0
T11  
T12
T13
det Tij   ij   T21
T22  
T23  0
T31
T32
T33  
A tensor expression
equivalent to scalar
multiplication is an
eigenvalue equation.
• Equivalent to determinant
problem

The scalars are eigenvalues.
• Corresponding eigenvectors
• Left and right eigenvectors
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