1. Lecture 2 Mathematical preliminaries and tensor analysis
CEE 262A HYDRODYNAMICS
Lecture 2
Mathematical preliminaries and tensor analysis

2. Where are components of
Right-handed, Cartesian coordinate system
Unit vectors
Position vector
Where are components of

3. Vector: Kundu- “…Any quantity whose components change like the components of a position vector under the rotation of the coord. system.”
Scalar: Any quantity that does NOT change with rotation or translation of the coord. system
e.g. density (r) or temperature (T)

4. Tensor: Assigns a vector to each direction in space ( 2nd order)
e.g.
Rows
Columns
Isotropic – Components are unchanged by a rotation of frame of reference (i.e. independent of direction - e.g “Kronecker Delta dij”)
Symmetric : Aij = Aji (in general Aij = ATji)
Anti-symmetric: Aij = -Aji
Useful result: Aij = 1/2 (Aij+Aij)+1/2 (Aji-Aji)
= 1/2 (Aij+Aji)+1/2 (Aij-Aji)
= Symmetric+ Anti-symmetric

5. Einstein summation convention
A) If an index occurs twice in a term a summation over the repeated index is implied
e.g.
* Result is a scalar quantity When a summation occurs over a repeated index contraction
B) Higher-order tensors can be formed by multiplying lower order tensors:
a) If Ui and Vj are 1st-order tensors then their product Ui Vj = Wij is a 2nd-order tensor. Also known as vector outer product ( ).
b) If Aij and Bkl are 2nd-order tensors then their product Aij Bkl = Pijkl is a 4th-order tensor.

6. C) Lower-order tensors are obtained from contractions
(a) Contraction of two 2nd-order tensors
(b) Tensor multiplied by a vector
(c) Double-contraction of two 2nd-order tensors

7. Isotropic tensor of 2nd order
D) Kronecker delta
=1 if i=j
=0 otherwise
Isotropic tensor of 2nd order *
Expand: If

8. FLOI = first(il)last (jm) - outer(im)inner(jl)
E) Alternating tensor
= ijk in cyclic order e.g. 123,231,312
= ijk in anticyclic order e.g. 321,132,213
= if any two indices are equal
(c)
Epsilon -
Delta Relation jl im jm il
klm
ijk d e =
FLOI = first(il)last (jm) - outer(im)inner(jl)

9. Basic vector operations
A) Dot Product (“Inner” Product)
Vector . Vector = Scalar
* “Magnitude of one vector times component of other in direction of first vector”
implies

10. form a right-handed system”
B) Outer product ( )
e.g.
C) Cross product
“…is the vector, , whose magnitude is , and whose direction is perpendicular to the plane formed by and such that
form a right-handed system” ～ W

11. Cross-product rules
(a)
(b)
(c)
Now since
If:

12. The “Del” ( ) operator A) Gradient – “Grad”: increase tensor order
Vector
“…( ) is perpendicular to
lines and gives magnitude
and direction of maximum spatial
rate of change of ”
If we apply  to a vector, we produce a second-order tensor

13. B) Divergence – “div”: Reduce tensor order
[ Scalar ]
[ Vector ]
e.g.
Our application will be to the divergence of a flux of various quantities.

14. C) Curl
e.g.
i=1:
If j=1 or k=1

15. Important div/grad/curl identities
(b)
(c)
If is a scalar
Now But +1 -1

16. “Curl of a vector is non divergent”
(see above)
(e)
But: jl im jm il
klm
ijk d e - =

17. We will make good use of this result!
1 if
We will make good use of this result!

18. Integral theorems ò A) Gauss’ Theorem
(outward unit normal vector to surface element)
(infinitesimal surface area)
(infinitesimal volume) “ …
Relates a vo
lume integral ò
to a surface integral A ” V

19. If is a scalar, vector, or any order tensor
Specifically, if is a vector or
“Divergence Theorem”: Integral over volume of divergence of flux = integral over surface of the flux itself

20. Examples… (a)
(b)
(c) or
Divergence of flux within volume = Net flux across

21. B) Stokes' Theorem
C (bounding curve)
A (open surface)