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Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Lecture 3: Position Vector and Coordinate Systems (cont.); Vectors: ScalarProduct; Vector Product; Dyadic Product

Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 42 / 77
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MFEFT - Lecture 3

1 Introduction

2 Vector and Tensor Algebra

3 Position Vector and Coordinate SystemsCartesian CoordinatesEinsteins Summation ConventionDifferentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)Cylinder Coordinate SystemOrthogonal Curvilinear Coordinate System

Spherical Coordinate SystemDupin Coordinates

4 Vectors: Scalar Product; Vector Product; Dyadic ProductScalar Product

5 Vector and Tensor Analysis

6 Distributions7 Complex Analysis

8 Special Functions

9 Fourier Transform

10 Laplace Transform

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Orthogonal Curvilinear Coordinate System

Uniqueness of the Transformation Formulas

The explicit request of a NEW Orthonormal Coordinate System according to

eiek

= ik (110)

transfers this requirement into

ijexj

klexl

= ik (111)

ijkl exj exl = jl

= ik (112)

and further to

ijkj = ik on the left-hand sidesum over j from 1 to 3 (113)with the definition

exjexl

= jl . (114)

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Orthonormal Relation; Transformation Dyad

Orthonormal Relation; Transformation Dyad

The Orthonormal Relation (see Eq. (113))

exjexl

= jl (115)

says, that the Transposed T

of the Dyada)

is equal to the Inverse, i.e., the dot product

T is equal to the unit dyadic I:

T = I (116)

= 1 (117)

is an Orthogonal Dyad.a)Compared to with the elements ij has

T the elements ji .


P i i V d C di S O h l C ili C di S
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Via the computation of the products ei exj we can illustrate the meaning of ij ; it follows

with the application of the orthonormal property ofexj

eiexj

= ilexl exj (118)

= illj (119)

= ij (120)

= cos(ei , exj ) . (121)


Position Vecto and Coo dinate S stems O thogonal C ilinea Coo dinate S stem
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The ij are

ij = cos(ei ,exj ) (122)

the so-called Direction Cosines of the NEW orthonormal tripoda vectors relative to the OLDCartesian Coordinate System. The ij are determine the Local Rotation of the NEWorthonormal tripod at every point in space. If this rotation is independent of position, then thecoordinate transformation in Eq. (96) is a simple rotation of a cartesian coordinate system.

aTripod is a word generally used to refer to a three-legged object, generally one used as a platform of some sort, and comesfrom the Greek tripous, meaning three feet.



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Handiness of the Coordinate SystemThe Handiness of the NEw Coordinate System compared to the OLD one is given by the

Determinant of. With the properties detI

= 1 and det

T

= det

it follows

det

( T)

= det

det

T

= 1 . (123)and

(det

)2 = 1 (124)

and

det

= 1 (125)

=

+1 the tripod e1 , e2 ,e3 is right-handed

1 the tripod e1 , e2 ,e3 is left-handed(126)


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If the OLD Coordinate System is not a Cartesian Coordinate System, but also CurvilinearOrthogonal, than we have the Transformation j i, and Eq. (109) reads for this general case:

ij =hj

hi

j

i

=h

i

hj

ij

. (127)



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Orthogonal Curvilinear Coordinate SystemIllustration of the Meaning of the Metric Coefficients hi

Illustration of the Meaning of the MetricCoefficients hi

Like the ij we can illustrate the meaning of themetric coefficients hi . h has been alreadydiscussed. We define a Line Element ds as the

magnitude of the differential change dR of theposition vector, i. e., a change ofR to R+ dR(see Fig. 3):

ds2 = dR dR . (128)

In order the compute ds in the orthogonal

curvilinear coordinates we build the total differentialofR with regard to the dependence of 1, 2, 3

dR =R

1d1 +

R

2d2 +

R

3d3 (129)

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................

O

R

R+ dR

dR, |dR| = ds

Figure 3: Definition of the Line Elements



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y g y


Illustration of the Meaning of the Metric Coefficients hi

We multiply and make use of the summation convention

ds2

=

R

i di

R

j dj (130)

= dihi ei ej = ij

hj dj (131)

= h2j d2j (132)

= h21d21 + h22d22 + h23d23 . (133)


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y g y


Illustration of the Meaning of the Metric Coefficients hi

Because of the invariance of the Scalar Line Element ds when the Coordinate System ischanging, we can use Eq. (133)

ds2 = h21d21 + h

22

d22 + h23

d23

to interpret the Metric Coefficients:In the Cartesian Coordinate System ds read

ds2 = dx2 + dy2 + dz2 [m2] (134)

this means ds2 is given by the Theorem of Pythagoras by adding the squares of the threeMetric Differential Changes dx, dy, dz in the direction of the Coordinate Lines.

In an arbitrary Orthogonal Curvilinear Coordinate System the hi determine the Metric ofthe Coordinate Lines (in meter).

The di can represent as d in cylinder coordinates and d in spherical coordinates aChange in Angle Direction.



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Illustration of the Meaning of the Metric Coefficients hiDifferential Volume Element

Differential Volume Element

This interpretation of the hi makes clear, that a Differential Volume Element dV in CartesianCoordinate System

dV = dx dy dz (135)

can be generalized to a Differential Volume Element of an arbitrary Curvilinear CoordinateSystem

dV = h1 d1 h2 d2 h3 d3 (136)

= h1 h2 h3 d1 d2 d3 (137)


Position Vector and Coordinate Systems Spherical Coordinate System
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Spherical Coordinate SystemCoordinates; Unit Vectors, Magnitude of the Position Vector

Spherical Coordinates; Unit Vectors,Magnitude of the Position Vector

Cartesisan Coordinates: R,, in thelimits 0 R < , 0 ,0 < 2

: polar angle; : azimuth angleOrthonormal Unit Vectors: eR,e, ewith |eR| = |e| = |e| = 1a andeR e ebThe straight line from the coordinateorigin O to the (observation) point P is

illustrating the position vector R of Pthe magnitude of the position vector is|R| = R =

R2

a|| stands for the magnitude of the argumentb stands for perpendicular

R e

q

(q,j)

e (j)j

y

R

z

j

x

O

P R,( )jq,

eR(q,j)

q

Figure 4: Spherical Coordinates of the spatialpoint P and the related position vector


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Spherical Coordinate SystemCoordinates; Unit Vectors, Magnitude of the Position Vector

Coordinate Transformation Formulas

The Transformation Formulas are following from Fig. 4:

x = R sin cos (138)

y = R sin sin (139)

z = R cos (140)

Cartesian Position Vector as a Funtion of the Spherical Coordinates

The representation of the Position Vector in the Cartesian Coordinate System as a function ofthe Spherical Coordinates is:

R = xR sin cos

ex + yR sin sin

ey + zR cos

ez (141)

= R sin cos ex + R sin sin ey + R cos ez (142)



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Spherical Coordinate SystemMetric Coeffecients; Orthonormal Unit Vectors

Metric Coefficients

The three Metric Coefficients of the Spherical Coordinate System are

hR = 1 (143)

h = R (144)

h = R sin . (145)

Orthonormal Unit Vectors

The Orthonormal Unit Vectors of the Spherical Coordinate System in form of the VectorDecomposition in the Cartesian Coordinate System as a function of the Spherical Coordinates read

eR (, ) = sin cos ex + sin sin ey + cos ez (146)

e (, ) = cos cos ex + cos sin ey sin ez (147)e () = sin ex + cos ey . (148)


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Spherical Coordinate SystemPosition Vector in the Spherical Coordinate System

Position Vector in the Spherical Coordinate System

The Position Vector in the Spherical Coordinate System can be found via CoordinateTransformation from the Cartesian Coordinate to the Spherical Coordinate System. The result is:

R = R eR (, ) . (149)

The position vector in the spherical coordinate system has only ONE vector componentR eR (, ) with the scalar vector component R. The dependencies of the angles and arehidden in the unit vector eR (, ).

Position Vector in the Spherical Coordinate System

For Unit Position Vector it follows then

R =R

R=

R eR (, )

R= eR (, ) . (150)


Position Vector and Coordinate Systems Dupin Coordinates
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Dupin Coorindates

Dupin Coorindates

The (Circular)Cylindrical and SphericalCoodinate System are Special Cases ofthe so-called general Dupin CoordinateSystem [Tai, 1992], which are veryimportant in the Vector and TensorAnalysis of Surfaces.

The Transition and BoundaryConditions for electromagnetic fieldsfrom Maxwells equations are typicallyderived using Dupin Coordinates.

Dupin Coordinates are orthogonalcurvilinear coordinates 1, 2, 3 with

the unit vectors e1 , e2 , n, i. e. e3 isthe unit normal vector n of the surfacegiven by e1 and e2 . The relatedmetric coefficient h3 is h3 = 1. Thecoordinate system is right handed, ifn = e1e2 .

Figure 5: Dupin Coordinates of the surface =

12 plane with the unit normal vectorn = e1

e2

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1

2

n

e1

e2


Vectors: Scalar Product; Vector Product; Dyadic Product
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MFEFT - Lecture 3

1 Introduction

2 Vector and Tensor Algebra

3 Position Vector and Coordinate SystemsCartesian CoordinatesEinsteins Summation ConventionDifferentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)Cylinder Coordinate SystemOrthogonal Curvilinear Coordinate System

Spherical Coordinate SystemDupin Coordinates

4 Vectors: Scalar Product; Vector Product; Dyadic ProductScalar Product

5 Vector and Tensor Analysis

6

Distributions7 Complex Analysis

8 Special Functions

9 Fourier Transform

10 Laplace Transform


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The Scalar, Vector, and Dyadic Product

Scalar Product (Dot Product)Example:

A B = C = Scalar! (151)

Vector Product (Cross Product)Example:

AB = C = Vector! (152)

Dyadic ProductExample:

AB = D = Dyad! (153)


Vectors: Scalar Product; Vector Product; Dyadic Product Scalar Product
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Scalar Product

Scalar ProductFig. 6 shows a vector A, which is projected to a unitvector e, the result is given by the Scalar Product

A e = A cos , (154)

where determines the enclosed angle between A

and e.

Replacing e by a vector B with the magnitude Byields the general form of Eq. (154), theCommutative scalar product A B (say: A dot B):

A B = B A (155)

= A B cos . (156)

Obviously is A B = 0, ifA and B areperpendicular, A B, to each other; this meansone can define two orthogonal vectors by avanishing scalar product between both.

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A

e

A e

Figure 6: Illustration des Skalarprodukts


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Scalar Product

Scalar Product: Orthonormal Tripod

The orthonormal tripod of a the cartesian coordinate system is charachterized byexi

exj= ij fur i, j = 1, 2, 3 . (157)

Scalar Product: Scalar Vector Components

Further, we can use the scalar sroduct to determine the scalar vector components of a vector A,

i. e. in the Cartesian Coordinate System we find

Ax = A ex

Ay = A ey (158)

Az = A ez .

We compute for the scalar product in components form ofA and B

A B = (Ax ex + Ay ey + Az ez) (Bx ex + By ey + Bz ez) (159)

and find by formal multiplication and the use of Eq. (157)

A B = AxBx + AyBy + AzBz . (160)



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Scalar Product

Enclosed Angle between two General Vectors A and B

The Enclosed Angle between the vector A and B is if A= 0 and B

= 0:

cos =A B

A B

=AxBx + AyBy + AzBz

A2x + A2y + A

2z

B2x + B

2y + B

2z

. (161)

Magnitude of a General Vector A

The Magnitude A of the Vector A is defined by the scalar product A A:

A =

A A =

A2x + A

2y + A

2z ; (162)

Unit Vector of a General Vector A

Then, the Unit Vector of the Vector A can be computed by

A =A

A A=A

A=

Ax

Aex +

Ay

Aey +

Az

Aez . (163)


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Scalar Product

Scalar Product: More Short-Hand Notations

We are going to cite two other short-hand notation of the scalar product. With Eq. ( 158) innumbered form we find

Axi = A exi fur i = 1, 2, 3 (164)

and for B we obtain instead Eq. (159)

A B =

3i=1

AxiBxi (165)

or applying the summation convention

A B = Axi

Bxi

.(166)

Obviously, this proves that the scalar product is commutative, i.e.,

A B = B A . (167)



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Scalar Product

Scalar Product: Independence of the Coordinate System

In generalization of Eq. (164) we define for the components of a vector in orthogonal curvilinearcoordinates bya)

Ai = A ei , (168)

and obtain by applying the summation convention

A = Ai ei= Ai ij exj

= Axj exj (169)

with

Axj = ij Ai , (170)

by applying the transformation formulas in Eq. (108).

a)At the point in space R(1, 2, 3) we project the general position dependent vector A(1, 2, 3) onto the positiondependent unit vectors ei

(1, 2, 3).



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Scalar Product

Scalar Product: Independence of the Coordinate System

On the other hand, by inserting the cartesian components form A = Axiexi in

Aj = A=Axiexi

ej

= Axiexi ej

= Axi exi ej = ji

(171)

= ji Axi . (172)

Number triples, which are transformed from the cartesian to an orthonormal curvilinearcoordinate system with Eq. (170) or Eq. (172) are in the mathematical sense (scalar) componentsof vectors. Because of the inverse of is equal to the transpose, Eq. (172), Aj = ji Axi can

be derived from Eq. (170), Axj = ij Ai , via inversion and vice versa. The vector A as adirected physical value is independent of the coordinate system (it is koordinatenfrei), simply themathematical representation is coordinate dependent.

The values of the Scalar Product of two Vectors is Independent of the Coordinate System:

A B = Axi Bxi = Ai Bi . (173)

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