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SUMMARY OF VECTOR AND TENSOR NOTATION-Bird, Stewart and Lightfoot "Transport Phenomena" -Bird, Armstrong and Hassager "Dynamics of Polymeric Liquids" The Physical quantities encountered in the theory of transport phenomena can be categorised into: - Scalars (temperature, energy, volume, and time) - Vectors (velocity, momentum, acceleration, force) - Second-order tensors (shear stress or momentum flux tensor) While for scalars only one type of multiplication is possible, for vectors and tensors several kinds are possible which are: - single dot . - double dot : - cross x The following types of parenthesis will also be used to denote the results of various operations. ( ) = scalar (u . w), ( : ) [ ] = vector [ u x w], [ . u] { } = tensor { . } The multiplication signs can be interpreted as follows: Multiplication sign None x . Order of Result -1 -2

: -4 ________________________________________ Scalars can be interpreted as 0th order tensors, and vectors as first order tensors. Examples: s uxw : order is order is order is 0+2=2 which is a 2nd order tensor 1+1-1=1 which is a vector 2+2-4=0 which is a scalar

2 Definition of a Vector: A vector is defined as a quantity of a given magnitude and direction. |u| is the magnitude of the vector u Two vectors are equal when their magnitudes are equal and when they point in the same direction. Addition and Subtraction of Vectors: w u+w u Dot Product of two Vectors: (u . w) = |u| |w| cos() commutative (u . v) = (v . u) not associative (u . v)w u(v . w) distributive (u . [v + w]) = (u . v) + (u . w) Cross Product of two Vectors: [uxw] = |u| |w| sin() n where n is a vector (unit magnitude) normal to the plane containing u and w and pointing in the direction that a right-handed screw will move if we turn u toward w by the shortest route. u w w u Area=(u.w) u w u-w

Area of this equals the length of [uxw]

3 not commutative distributive [uxw] = -[wxu] [[u + v] x w] = [u x w] + [v x w]

not associative [u x [v x w]] [[u x v] x w]

4

VECTOR OPERATIONS FROM AN ANALYTICAL VIEWPOINTDefine rectangular co-ordinates: 1, 2, 3 x, y, z respectively Many formulae can be expressed more compactly in terms of the kronecker delta ij and the alternating unit tensor ijk, which are defined as: ij = 1 if ij =0 if and ijk=1 ijk= -1 ijk =0 if if if ijk=123, 231, 312 ijk=321, 132, 213 any two indices are alike i=j ij

We will use the following definitions, which can be easily proved:

j k ijk hjk = 2 ihand

k ijk mnk = im

jn - in jm

The determinant of a three-by-three matrix may be written as:

11 12 13 21 22 23 = i j k ijk 1i 2j 3k 31 32 33

5

DEFINITION OF A VECTOR AND ITS MAGNITUDE: THE UNIT VECTORSA vector u can be defined completely by giving the magnitudes of its projections u1, u2, and u3 on the co-ordinate axis 1, 2, and 3 respectively. Thus one may write3

u = 1u1 + 2u 2 + 3u3 = i uii =1

where 1, 2, and 3 are the unit vectors in the direction of the 1, 2 and 3 axes respectively. The following identities between the vectors can be proven readily:1u3 u2 u u1

1.1 = 2 . 2 = 3 . 3 = 12

1. 2 = 2 . 3 = 3 . 1 = 0 1 1 = 2 2 = 3 3 = 0

3

[ 1 2 ] = 3

[ 2 3 ] = 1

[ 3 1] = 2

[ 2 1] = - 3

[ 3 2 ] = -1

[ 1 3 ] = - 2

All these relations can be summarized as:

( i . j) = ij3

[ i j ] = ijk kk =1

6 Addition of vectors:

u + w = i i u i + i i w i = i i ( u i + w i )Multiplication of a Vector by a Scalar:

s u = s [i i ui] = i i (su i)Dot Product:( u . w ) = [i i ui] . [ j j w j] = i j ( i . j) u i w j =

= i j ij ui w j = i ui wi

Cross Product:[ u w ] = [( j j u j) (k k w k )]

= j k [ j k ] u j w k = i j k ijk i u j w k

1= u1 w1

2u2 w2

3u3 w3

Proofs of Identities (Example):

Prove the following identity u [v w ] = v ( u . w ) - w ( u . v )

This identity will be proven for the i-component, so the summation i will be dropped out for the

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sake of simplicity.(u x [v x w])i =

j k ijk uj [ v x w ]k = j k ijk uj [l mklmvl wm] = j k l m ijk klm uj vl wm j k l m ijk lmk uj vl wm j l m (iljm - im jl) uj vl wm j l m iljm uj vl wm - j l m im jl uj vl wm vi j m jm uj wm - wi j l jl uj vl vi j uj wj - wi j uj vj vi ( u . w) - wi ( u . v)v (u . w ) - w (u . v )

= = = = = = = =

set l=i in the first term and m=i in the second term set m=j in the first term and l=j in the second term

8

VECTOR DIFFERENTIAL OPERATIONSDefine first the del operator, which is a vector +2 +3 = i i x1 x2 xi x3

= 1

The Gradient of a Scalar Field:

s = 1 not commutative not associative distributive

s s s s + 2 +3 = i i x1 x2 xi x3

s sL (r)s (rs) (r+s) = r + s

The Divergence of a Vector Field:

( . u ) = i i xi

. j j u j = i j [ i . j] u j xi

[

]

= i j ij

ui u j = i xi xi

not commutative not associative distributive

( . u ) ( u . ) (. s ) u (s . u ) . ( u + w ) = ( . u ) + ( . w )

The Curl of a Vector Field:

9

[ u ] =

j j x [ k k u k ] x j

= j k [ j k ] uk xj

1= x1 u1

2 x2 u2

3 x3 u3

u u = 1 3 - 2 x 2 x3

+ 2 u1 - u 3 x3 x1

+ 3 u 2 - u1 x1 x 2

[ x u ] = curl (u) = rot (u) It is distributive but not commutative or associative.The Laplacian Operator:

The Laplacian of a scalar is:

( . s ) = i

2 2 s 2 s 2 s + 2 s= + xi2 xi2 x2 x3 2

The Laplacian of a vector is:

2 u = ( . u ) - [ [ u ] ]The Substantial Derivative of a Scalar Field:

If u is assumed to be the local fluid velocity then:D = + (u . ) Dt t

The substantial derivative for a scalar is:

10

s s Ds = + i u i xi Dt t

The substantial derivative for a vector is:Du u + (u . ) u = i i = Dt t ui + (u . ) ui t

This expression is only to be used for rectangular co-ordinates. For all co-ordinates:(u.)u = 1 (u.u ) [u [ u ]] 2

11

SECOND - ORDER TENSORSA vector u is specified by giving its three components, namely u1, u2, and u3. Similarly, a secondorder tensor is specified by giving its nine components.

11 = 21 31

12 22 32

13 23 33

The elements 11, 22, and 33 are called diagonal while all the others are the non-diagonal elements of the tensor. If 12=21, 31=13, and 32=23 then the tensor is symmetric. The transpose of is defined as:

11 * = 12 13If is symmetric then =*.

21 22 23

31 32 33

Dyadic Product of Two Vectors:

This is defined as follows: u1 w1 uw = u 2 w1 u3 w1 u1 w 2 u2 w 2 u3 w 2 u1 w 3 u 2 w3 u3 w 3

12 Unit Tensor:

1

0 1 0

0 0 1

= 00

The components of the unit tensor are ij (kronecker delta for i,j=1,3)

Unit Dyads:

These are just the dyadic products of unit vectors, mn in which m,n=1,2,3. 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0

1 1 = 00 0

1 2 = 00 0

1 3 = 00

3 2 = 00

Thus, a tensor can be represented as:

= i j i j ijand the dyadic product of two vectors as:u w = i j i j u i w j

13

Also note the following identities: ( i j : k l ) = il jk scalar

[ i j . k ] = i jk

vector

[ i . j k ] = ij k

vector

i j . k l = jk i l

tensor

Addition of Tensors:

+ = i j i j ij + i j i j ij= i j i j ( ij + ij )

Multiplication of a Tensor by a Scalar:s = s i j i j ij = i j i j ( s ij )

Double Dot Product of Two Tensors:

: = (i j i j ij) : (k l k l kl)

14

= i j k l ( i j : k l) ij kl

= i j k l il

jk ij kl

set l=i and k=j to simplify to:= i j ij ji

which is a scalar 2 + 2 - 4 = 0

Dot Product of Two Tensors:

. = (i j i j ij) . (k l k l kl)= i j k l ( i j . k l) ij kl = i j k l ( jk i l) ij kl

= i l i l ( j ij jl )

Vector Product (or Dot Product) of a Tensor with a Vector:

[ . u ] = [ ( i j i j ij ) . ( k k u k ) ]

= i j k i jk ij u k

= i i ( j ij u j )

15 Differential Operations: [ . ] = i i j k j k jk x i

[

]

= i j k [ i . j k ] = i j k ij k jk xi

jk xi

= k k i ik xi

Some other identities which can readily be proven are: uk xi

w . u = i k k w i

: u = i j ij

ui xj

16

INTEGRAL THEOREMS FOR VECTORS AND TENSORSGauss - Ostrogradskii Divergence Theorem :

If V is a closed region in space surrounded by a surface S then ( . u ) dV = ( n . u ) dS = ( u . n ) dS = u n dSV S S S

where n is the outwardly directed normal vector.

s dV = n s dSV S

where s is a scalar quantity. [ . ] dV = [ n . ] dSVS

where is a tensor.The Stokes Curl Theorem:

If S is a surface bounded by a close curve C, then: ( [ x u ] . n ) dS = (u.t) dCSC

where t is the tangential vector in the direction of the integration and n is the unit vector normal to S in the direction that a right-handed screw would move if its head were twisted in the direction of integration along C.

17 The Leibnitz Formula for Differentiating a Triple Integral:

s d s dV = dV + s ( u s . n ) dS dt V V t S where us is the velocity of