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Today we introduce the application of index notations in vector and tensor algebra.

Index notation:

e.g. Kronecker delta: . In matrix form:

=

= ji

jiij

,0

,1

100

010

001

Dot product of vectors: ( ) ( ) ( ) iiijjijijijjii babaeebaebeaba ==== vvvvvv

Cross product of vectors: ,bacvvv

=

The magnitude of cross product is: sinabc =v

The direction of cross product is:

av

bv

cv

For base vectors 1ev

, 2ev

, :3ev

011 =eevv

, 321 eeevvv

= , 231 eeevvv

=

312 eeevvv

= , 022 =eevv

, 132 eeevvv

=

213 eeevvv

= , 123 eeevvv

= , 033 =eevv

Or we can write the nine equations in a concise form as:

kijkji eeevvv =

where (called permutation symbol)

=

=

=

otherwise,0

321,132,213ijk,1

312,231,123ijk,1

ijk

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2 3

Vector operations are useful in representing areas and volumes in a solid body.

Area of a parallelogram:

av

bv

nv

(nv to the plane of a

vand b

v)

baabA vv== sin

banAAvvvr

==

yxAvvv

ddd = ydv

xdv

Volume of a parallelepiped:

av

bv

cv

( ) ( )bacAcncAVvvvvvvv

===

xdv

ydvzd

v

( ) ( ) ( 132321213 dddddddddd xxxxxxxxxV )vvvvvvvvv

===

In index notation, we can express volume of a parallelepiped

as

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( ) ( ) ( ) kjiijkkjiijkkjikjikkjjii cbabaceeebacebeaecbacV =====vvvvvvvvv

,

We could also write this in matrix notation as

( ) ( ) ( ) kjiijk cbacbcbacbcbacbcbaccc

bbb

aaa

V =+== 122131331223321

321

321

321

For an arbitrary matrix

=

333231

232221

131211

AAA

AAA

AAA

A , its determinant is ( ) kjiijk AAAA 321det = .

More generally,

( ) pqrrkqjpiijk AAAA det=

relation:

=

krkqkp

jrjqjp

iriqip

pqrijk

Proof:

For any matrix,

( )

333231

232221

131211

det

AAA

AAA

AAA

A =

For any change in rows, the calculation rule of a determinant obeys

( )

321

321

321

det

kkk

jjj

iii

ijk

AAA

AAA

AAA

A =

Similarly, for any change in column,

( )

krkqkp

jrjqjp

iriqip

pqrijk

AAA

AAA

AAA

A =det

Set ijijA = ( =A ), ( ) 1det =I ,

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=

krkqkp

jrjqjp

iriqip

pqrijk

In particular,

kqjrkrjq

krkqki

jrjqji

iriqii

iqrijk

=

=

The above are called relations.

By using the relationship, we can calculate

( )ijkjik

rjkirjrik

rqkrqijrqkrijqkqijqkji

ee

e

eeeeeee

vv

v

vvvvvvv

=

=

===

( )

( ) ( )acbbcaeacbebca

eecbaeeecbacba

iijjjjii

ijkjikkjikjikji

vvvvvv

vv

vvvvvvvv

=

=

==

Gradient of a scalar field ( )zyxT ,,

3

3

2

2

1

1x

ex

ex

e

+

+

=

vvv

ii

i

i Tex

Te

x

Te

x

Te

x

TeT ,

3

3

2

2

1

1

vvvvv=

=

+

+

=

Divergence of a vector field ( )zyxu ,,v

:

ii

i

iij

i

j

jj

i

i

ux

u

x

u

x

u

x

u

x

ueu

xeu

,

3

3

2

2

1

1 =

+

+

=

=

=

=

vvv

e.g.

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iijij

jij

uf

t

uf

&&

vv

=+

=

=+

,

,

2

2

Curl of a vector field ( )zyxu ,,v :

kijkijjj

i

i eueux

euvvvv

,=

=

Tensors:

(Second rank) tensors are linear transformations that map one vector into another vector. Higher

order tensors can map one tensor into another tensor. One can also say that vectors are first rank

tensors and scalar are zeroth rank tensors.

For example,

nv

tv

If is the normal vector of a tiny area on the surface,nv

tv

is surface traction, we will show that

there exists a tensor which maps tonv

tv

,

tnvv

=

where will turn out to be the stress tensor (discussed later).

An example of higher order tensor is the elastic modulus tensor. If is stress tensor and is

strain tensor, then we need a 4th

order tensor to relate them to each other:

C=

where C is called the modulus tensor (to be discussed later).

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This relationship, called the generalized Hookes law, is the 3D generalization of the 1D Hookes

law,

E= .

Base tensors:

1ev

2ev

3ev

=

0

0

1

1ev

, ,

=

0

1

0

2ev

=

1

0

0

3ev

=

000

000

001

11 eevv

, ,

=

000

000

010

21 eevv

=

000

000

100

31 eevv

where is the tensor product (or dyad product) symbol.

=

0

0

ji eevv 1 columnjth

rowith

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