Twnsor Product

1. Tensor (None) (x) (.) (:)
Operations
Inner Product Cross Product Outer Product Dyadic Product
0 -1 -2 -4
Scalar=0 , Vector=1, Tensor=2
Inner Product
 ∂   ∂u ∂v ∂w 
   ∂x 
 ∂x   ∂x ∂x 
 ∂  ∂u ∂v ∂w
∇u =   (u v w )=
 ∂y


∂y ∂y ∂y
   
 ∂   ∂u ∂v ∂w 
   ∂z 
 ∂z   ∂z ∂z 
∇P =vector (none) scalar=1+0+0=1 (vector)
 ∂   ∂P ∂ (0) ∂ (0) 
   ∂x ∂y ∂x 
 ∂x  P 0 0  
 ∂   ∂ (0) ∂P ∂ (0) 
∇P =  0 P 0 = 
∂y  ∂y ∂y ∂y 
 0
 0  
P 
 ∂   ∂ (0) ∂ (0) ∂P 
   ∂z
 ∂z   ∂z ∂z  
Inner Product will always increase the order of tensor

2. Outer Product
Divergence of velocity = ∇.u =vector (outer) vector=1-2+1=0 (scalar)
u . ∇u
inner = vector.tensor=1-2+2=1 (vector)
outer
 ∂u ∂v ∂w 
 ∂x ∂x ∂x 
 
∂u ∂v ∂w
u . ∇u = (u v w )
 ∂y


∂y ∂y
inner
 
outer
 ∂u ∂v ∂w 
 ∂z
 ∂z ∂z 

 ∂u ∂u ∂u  ɵ  ∂v ∂v ∂v   ∂w ∂w ∂w 
= u +v +w  i + u +v +w  j + u +v +w k
 ∂x ∂y ∂z   ∂x ∂y ∂z   ∂x ∂x ∂x 
Outer Product will always reduce the order of tensor

3. Dyadic Product
σ : τ =2-4+2=0 (scalar)
σ xx σ xy σ xz  τ xx τ xy τ xz 
   
σ : τ = σ yx σ yy σ yz  : τ yx τ yy τ yz 
σ zx σ zy σ zz  τ zx τ zy τ zz 
   