dr. wang xingbo fall ， 2005

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Einstein Conventions Einstein Conventions


Coordinate Systems Coordinate Systems

31 2 3

1 3 31

k kk k

k

x x x x x

i i i i ikkAA i

k changes from 1 to n in an n-dimensional space


Coordinate SystemsCoordinate Systems

,i ji jA A A e B e

( ) ( ) j ji j ijA B A B g A B

n ni j i i

i ji=1 j=1

A B = e e e e

Einstein Conventions Einstein Conventions


Curvilinear coordinate systems Curvilinear coordinate systems

change the value of , we will get a series of “parallel surfaces” in the space and for each point in the space, there exists one and only one surface passing through it.

1 2 3( , , )x x x



have a unique intersection point M,

Take three different surfaces 1 1 1 2 3

2 1 1 2 3

3 1 1 2 3

( , , )

( , , )

( , , )

u u x x x

u u x x x

u u x x x

u1

u2

M

u3

X3

X1 O X2



curvilinear coordinate of M.

the two ordered triples of numbers 1 2 3( , , )x x x 1 2 3( , , )u u u

are all coordinates of point M because both of them determine M uniquely in the space

We call the ordered triples of numbers 1 2 3( , , )u u u

Coordinate Transformation

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Curvilinear Coordinate SystemsCurvilinear Coordinate Systems

is Cartesian coordinate System then

1 2 3( ) ( , , )i i j iu x x x x 1 2 3, ,x x x

1 2 3( , , )u u u

is a curvilinear coordinate System

i-th coordinate surface passing through P0



ui-line: ui-changge while the other two remain constant

1 2 30( , , )i ix x x u

Local basis



1 2 3

1 2 3 ( 1,2,3)i i i i

x x xi

u u u u

r

e e e

Notice



Therefore, the local basis exist.

1 2 3

1 2 3 1 2 3

( , , )( )

( , , )

x x x

u u u u u u

r r r

Arc-length



1 2 3 1 2 3( , , ) ( , , )i ii iu u u x f u u u r e e

2( ) i jj

i j i ji j ij

ds d d du duu u

du du g du du

r rr r =

e e

i

| | ( 1, 2,3)i ih i

u

r

Normalized Basis



1, 1,2,3k

k kk k

kh u h

er

i

Gradient of a Scalar Field F

1 2 31 1 2 2 3 3

1 1 1F F FF

h u h u h u

i i i

Divergence of vector Field



1 2 32 3 1 3 1 21 2 3

1 2 3

1[ ( ) ( ) ( )]div F h h F h h F h h

h h h u u u

F

Curl of vector Field

1 1 2 2 3 3

1 2 31 2 3

1 2 31 2 3

1

h h h

curlh h h u u u

F h F h F h

i i i

F

Two Important things To be considered



two

1. Jacobian matrix is the fundamental of the transformation. If the matrix is regular, it means the transformation is invertible; otherwise the transformation is not invertible.

2. The metric in the new frame is important because it determine many metric properties such as distance, area, etc.

Suppose we have a complicated geometry object M a small part


A Further Investigation on Local Coordinate SystemA Further Investigation on Local Coordinate System

M



{ }ixO be described by a differential function 1 2( , ,..., )nx x x lay a smooth parameterized curve on

c=c()

P

h=h()



directional derivative of along the curve

c=c()

P

h=h()

1

1...

n i

n i

d dx dx dx

d x d x d x d



many differential functions like , etc. to describe

c=c()

P

h=h()

,...,

i

i

d dx

d d x



consider two curves passing through the same point P

c=c()

P

h=h()

i

i

d dx

d d x

i

i

d dx

d d x



the linear combination

c=c()

P

h=h()

( )i i

i

d d dx dxa b a bd d d d x

there must exist a curve

( )i i

i

d dx dxa b

ds d d x



Linear space: Tangent Space TP

c=c()

P

h=h()

i id dx dxa b

ds d d ( )

d d d da a ad d d d

( )d d d

a b a bd d d

( ) ( )d d

ab a bd d

1d d

d d



Basis vectors of coordinate system

c=c()

P

h=h()

kkii i k k i

d dx

dx dx x x x

1 2(1,0,...,0), (0,1,...,0),..., (0,0,...,1)

nx x x



A surface with its tangent space is an impersonally in existence while we can subjectively illustrate them by many functions . There exists a function such that are unit tangent vectors no matter what the actual form of is .

is a basis of the “ideal” state

{ }ix

{ }ix



The tangent space has infinite frames at a point P but the frame with an “ideal” basis can simplify problem. That is why people usually use the ideal basis in theoretical research. However, we cannot have an ideal basis in engineering practice. This is way is of importance.

In later parts of this book, we will, by convention, use the ideal basis for theoretical study.


Transformation Between Basis and CoordiTransformation Between Basis and Coordinatesnates

{ } { }i i O e O e1 2

1 11 1 11 2

2 22 2 2

1 2

n

n

nn nn n n

A A A

A A A

A A A

e e

e e

e e

,det 0ji i jA A

E = AE

e e

1 1, ( ) ( ) ,

0j j j j k j

i i i i k i i

i jB B A B A

i j

je e



A vector X in j ij ix x X e e

( )i j k j ii j k j ix x B x B e e e i i j

jx B x

( )ijC B 1( )TC A



ii i j i

j jj

ii j

j

xx B x B

x

xx x

x



e1

e2 e5

P r=r(ui)

e3 e4



1 2

1 2

( , ,..., ) ( ):

( , ,..., ) ( )

n i

n i

u u u u

u u u u

r r r

r r r

A Surfacce

1 2( , ,..., ) ( ),det 0

( )

ii i n i j

j

i i j

uu u u u u u u

u

u u u

A Frame on a surface


Properties of Scalar ProductProperties of Scalar Product

1 2, ,..., nu u u

1 2 n

r r re e e

1 2, ,..., nu u u

1 2 n

r r re e e


Frame transformationFrame transformation

j j

i j

u u

u u u u

i j i i

r re e

1

,

j

j jijii i

i ji

uu u u uuu u u uu

u

j kji

i kj

e e

e e

Two frames


Dual Frame, Covariant vectors and Contravariant Dual Frame, Covariant vectors and Contravariant Vectors Vectors

{ },{ }i ie e

for any vector i ji ja a A e e

,j ij ia a A e A e

The two frames are Dual one another



, ki j ij kg a a e e

ijjg ie e ijg j

ie ei

j ij

i ijj

a g a

a g a

1( ) ( ) ,ij ik i

ij kj jg g g g



Other important properties of dual frames

j jk jk jk ji ki ig g g g i k i ke e e e e e

i j i jk jk i jk i ijkg g g g k ke e e e e e


Covariancy and Contravariancy Covariancy and Contravariancy

Two frames

Transformation T :

{ } { }i ie e

,det( ) 0j ji i j iT T e e

, ( ) ( )i i j j j Tj i ia S a S T Coordinate :

{ }iO,e { }iO,e


Covariancy and Contravariancy Covariancy and Contravariancy

Two frames { }iO,e { }iO,e

e3 j

i i jTe e 1e

e1 O e2 i i jjaTa 2e O

3e

Dual frame is changing with the transformation

so it is with the coordinate


Covariancy and ContravariancyCovariancy and Contravariancy

{ }ie

{O, } { , }i iOe e{ } { }i ia a



Dual Relationships

Transform Relationships

{ }iO,e { , }iO e

{ , }iO e { , }iO e



Transformation relationship between { , }iO e { , }iO e

i i jjSe eAssume

1 1 1 1 11 2 3

2 2 2 2 21 2 3

3 3 3 3 31 2 3

e S S S e

e S S S e

e S S S e



Since j ji i e e j

i i jTe e

l i l i l j i l j i lk k i k i j k j i k iT T S T S T S e e e e

1 2 3 1 2 31 1 1 1 1 11 2 3 1 2 3

2 2 2 2 2 21 2 2 1 2 3

3 3 3 3 3 3

1 0 0

0 1 0

0 0 1

T T T S S S

T T T S S S

T T T S S S

TTS E



Coordinate Transform i ji jA a a e e

i j j ii j j iA a a a S e e e

j ji j i i ja a S S a

ji i ja T a



e3 i i jjTe e

1e

e1 O e2 ji i jaTa 2e O 3e



Overall relationships among frame and coordinate transformations e3 e

3

{ }ia

j ji i

ji ija ga

e e { }ia

e2 e1 O

e1 O e2

j

i i j

i i jj

T

a Ta

e e

i i jj

ji i j

T

a Ta

e e

1e 1e

{ }ia O 3e { }ia

2e

j ji i

ji ija ga

e e 2e O 3e

Tra

nsfor

mat

ions

Dual Relationships

Components of vector A in are called contra-variantcomponents

Components in the dual frame are called a covariantcomponent.


Covariancy and ContravariancyCovariancy and Contravariancy{ }ie

{ }ie

A transformation



1 2( , ,..., ), 1, 2,...,j j nx x x x x j n The Jacobian matrix of transformation is a matrix of contravariance

1 1 1

1 2

2 2 21 2

1 21 2

1 2

( , ,..., )

( , ,..., )

n

nn

n

n n n

n

x x x

x x x

x x xx x x

x x xx x x

x x x

x x x

Example of contravariant quantity



1 2( , ,..., ), 1, 2,...,i i n

ii j

j

x x x x x j n

xdx dx

x

is contravariant to the frame transformation j idx todx

Contravariant quantity



under the coordinate transformation

ii j

j

uQ Q

u

( )( ),det 0

( )

ii i j

j

uu u u

u

Covariant quantity



under the coordinate transformation ( )

( ),det 0( )

ii i j

j

uu u u

u

j

i ji

uQ Q

u

Class is Over! Class is Over!

See you!See you!


dr. wang xingbo fall ， 2005

Documents

mechanical engineeringa

hmathematical mechanicalmethod

curve c

ordered triples of numbers

coordinates of point

unique intersection

important things

metric properties