Lecture 3: Position Vector and Coordinate Systems (cont ...

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

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

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 42 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

MFEFT - Lecture 3

1 Introduction

2 Vector and Tensor Algebra

3 Position Vector and Coordinate Systems
Cartesian Coordinates
Einstein’s Summation Convention
Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cylinder Coordinate System
Orthogonal Curvilinear Coordinate System
Spherical Coordinate System
Dupin Coordinates

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

5 Vector and Tensor Analysis

6 Distributions

7 Complex Analysis

8 Special Functions

9 Fourier Transform

10 Laplace Transform

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 43 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Uniqueness of the Transformation Formulas
The explicit request of a NEW Orthonormal Coordinate System according to

eξi · eξk = δik (110)
(111)
transfers this requirement into (112)

γij exj · γklexl = δik (113)
γij γkl exj · exl = δik (114)

= δjl

and further to

γij γkj = δik on the left-hand side
sum over j from 1 to 3

with the definition

exj · exl = δjl .

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 44 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthonormal Relation; Transformation Dyad

Orthonormal Relation; Transformation Dyad
The Orthonormal Relation (see Eq. (113))

exj · exl = δjl (115)

says, that the Transposed ΓT of the Dyada) Γ is equal to the Inverse, i.e., the dot product
Γ · ΓT is equal to the unit dyadic I:

Γ · ΓT = I (116)
= Γ · Γ−1 (117)

Γ is an Orthogonal Dyad.

a)Compared to Γ with the elements γij has ΓT the elements γji.

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 45 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Uniqueness of the Transformation Formulas

Uniqueness of the Transformation Formulas

Via the computation of the products eξi · exj we can illustrate the meaning of γij ; it follows —
with the application of the orthonormal property of exj —

eξi · exj = γilexl · exj (118)
= γilδlj
= γij (119)
= cos ∠(eξi , exj ) . (120)
(121)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 46 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Uniqueness of the Transformation Formulas
The γij are

γij = cos ∠(eξi , exj ) (122)

the so-called Direction Cosines of the NEW orthonormal tripoda vectors relative to the OLD
Cartesian Coordinate System. The γij are determine the Local Rotation of the NEW
orthonormal tripod at every point in space. If this rotation is independent of position, then the

coordinate 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 comes
from the Greek tripous, meaning ”three feet”.

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 47 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Handiness of the Coordinate System

The Handiness of the NEw Coordinate System compared to the OLD one is given by the
Determinant of Γ. With the properties det I = 1 and det ΓT = det Γ it follows

det (Γ · ΓT) = det Γ det ΓT
= 1.
(123)

and

(det Γ )2 = 1 (124)

and

det Γ = ±1 (125)

= +1 the tripod eξ1 , eξ2 , eξ3 is right-handed (126)
−1 the tripod eξ1 , eξ2 , eξ3 is left-handed

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 48 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Uniqueness of the Transformation Formulas

If the ”‘OLD”’ Coordinate System is not a Cartesian Coordinate System, but also Curvilinear
Orthogonal, than we have the Transformation ξj → ξi, and Eq. (109) reads for this general case:

γij = hξj ∂ξj
hξi ∂ξi

= hξi ∂ξi . (127)
hξj ∂ξj

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 49 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Illustration of the Meaning of the Metric Coefficients hξi

Illustration of the Meaning of the Metric ..............R................................................................R.............+.............d........R...d........R...........,.....|....d.......R. | = ds
Coefficients hξi
O
Like the γij we can illustrate the meaning of the Figure 3: Definition of the Line Elements
metric coefficients hξi . hϕ has been already
discussed. We define a Line Element ds as the
magnitude of the differential change dR of the
position vector, i. e., a change of R to R + dR
(see Fig. 3):

ds2 = dR · dR . (128)

In order the compute ds in the orthogonal
curvilinear coordinates we build the total differential
of R with regard to the dependence of ξ1, ξ2, ξ3

∂R ∂R ∂R
dR = ∂ξ1 dξ1 + ∂ξ2 dξ2 + ∂ξ3 dξ3 (129)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 50 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Illustration of the Meaning of the Metric Coefficients hξi

Illustration of the Meaning of the Metric Coefficients hξi
We multiply and make use of the summation convention

ds2 = ∂R dξi · ∂R dξj (130)
∂ξi ∂ξj (131)

= dξihξi eξi · eξj hξj dξj (132)
(133)
= δij
= h2ξj dξj2
= hξ21 dξ12 + h2ξ2 dξ22 + h2ξ3 dξ32 .

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 51 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Orthogonal Curvilinear Coordinate System

Illustration of the Meaning of the Metric Coefficients hξi

Illustration of the Meaning of the Metric Coefficients hξi

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

ds2 = h2ξ1 dξ12 + h2ξ2 dξ22 + hξ23 dξ32

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 three
Metric Differential Changes dx, dy, dz in the direction of the Coordinate Lines.

In an arbitrary Orthogonal Curvilinear Coordinate System the hξi determine the Metric of
the Coordinate Lines (”‘in meter”’).

The dξi can represent as dϕ in cylinder coordinates and dϑ in spherical coordinates a
Change in Angle Direction.

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 52 / 77

Position Vector and Coordinate Systems Orthogonal Curvilinear Coordinate System

Illustration of the Meaning of the Metric Coefficients hξi

Differential Volume Element

Differential Volume Element

This interpretation of the hξi makes clear, that a Differential Volume Element dV in Cartesian
Coordinate System

dV = dx dy dz (135)

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

dV = hξ1 dξ1 hξ2 dξ2 hξ3 dξ3 (136)
= hξ1 hξ2 hξ3 dξ1 dξ2 dξ3 (137)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 53 / 77

Position Vector and Coordinate Systems Spherical Coordinate System

Spherical Coordinate System

Coordinates; Unit Vectors, Magnitude of the Position Vector

z

Spherical Coordinates; Unit Vectors, eR(q,j)
Magnitude of the Position Vector ej(j)

Cartesisan Coordinates: R, ϑ, ϕ in the P(R,q,j)
limits 0 ≤ R < ∞, 0 ≤ ϑ ≤ π, q
0 ≤ ϕ < 2π
R eq(q,j)
ϑ: polar angle; ϕ: azimuth angle R

Orthonormal Unit Vectors: eR, eϑ, eϕ O
with |eR| = |eϑ| = |eϕ| = 1a and y
eR ⊥ eϑ ⊥ eϕb
The straight line from the coordinate j
origin O to the (observation) point P is
illustrating the position vector R of P x

the magnitu√de of the position vector is Figure 4: Spherical Coordinates of the spatial
|R| = R = R2 point P and the related position vector

a|·| stands for the magnitude of the argument
b⊥ stands for perpendicular

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 54 / 77

Position Vector and Coordinate Systems Spherical Coordinate System

Spherical Coordinate System

Coordinates; Unit Vectors, Magnitude of the Position Vector

Coordinate Transformation Formulas (138)
The Transformation Formulas are following from Fig. 4: (139)
(140)
x = R sin ϑ cos ϕ
y = R sin ϑ sin ϕ
z = R cos ϑ

Cartesian Position Vector as a Funtion of the Spherical Coordinates

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

R= x ex + y ey + z ez (141)
(142)
R sin ϑ cos ϕ R sin ϑ sin ϕ R cos ϑ

= R sin ϑ cos ϕ ex + R sin ϑ sin ϕ ey + R cos ϑ ez

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 55 / 77

Position Vector and Coordinate Systems Spherical Coordinate System

Spherical Coordinate System

Metric Coeffecients; Orthonormal Unit Vectors

Metric Coefficients (143)
The three Metric Coefficients of the Spherical Coordinate System are (144)
(145)
hR = 1
hϑ = R
hϕ = R sin ϑ .

Orthonormal Unit Vectors

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

eR (ϑ, ϕ) = sin ϑ cos ϕ ex + sin ϑ sin ϕ ey + cos ϑ ez (146)
eϑ (ϑ, ϕ) = cos ϑ cos ϕ ex + cos ϑ sin ϕ ey − sin ϑ ez (147)
(148)
eϕ (ϕ) = − sin ϕ ex + cos ϕ ey .

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 56 / 77

Position Vector and Coordinate Systems Spherical Coordinate System

Spherical Coordinate System

Position 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 Coordinate
Transformation from the Cartesian Coordinate to the Spherical Coordinate System. The result is:

R = R eR (ϑ, ϕ) . (149)

The position vector in the spherical coordinate system has only ONE vector component

R eR (ϑ, ϕ) with the scalar vector component R. The dependencies of the angles ϕ and ϑ are
hidden in the unit vector eR (ϑ, ϕ).

Position Vector in the Spherical Coordinate System

For Unit Position Vector it follows then

Rˆ = R = R eR (ϑ, ϕ) = eR (ϑ, ϕ) . (150)
R R

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 57 / 77

Position Vector and Coordinate Systems Dupin Coordinates

Dupin Coorindates

Dupin Coorindates Figure 5: Dupin Coordinates of the surface =

The (Circular)Cylindrical and Spherical ξ1ξ2 plane with the unit normal vector
Coodinate System are Special Cases of n = eξ1 ×eξ2
the so-called general Dupin Coordinate
System [Tai, 1992], which are very ξ1..............e.....ξ...1.......n..O.............................e.....ξ..2.....................................ξ2
important in the Vector and Tensor
Analysis of Surfaces.

The Transition and Boundary
Conditions for electromagnetic fields
from Maxwell’s equations are typically
derived using Dupin Coordinates.

Dupin Coordinates are orthogonal
curvilinear coordinates ξ1, ξ2, ξ3 with
the unit vectors eξ1 , eξ2 , n, i. e. eξ3 is
the unit normal vector n of the surface
given by eξ1 and eξ2 . The related
metric coefficient hξ3 is hξ3 = 1. The
coordinate system is right handed, if
n = eξ1 ×eξ2 .

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 58 / 77

Vectors: Scalar Product; Vector Product; Dyadic Product

MFEFT - Lecture 3

1 Introduction

2 Vector and Tensor Algebra

3 Position Vector and Coordinate Systems
Cartesian Coordinates
Einstein’s Summation Convention
Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cylinder Coordinate System
Orthogonal Curvilinear Coordinate System
Spherical Coordinate System
Dupin Coordinates

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

5 Vector and Tensor Analysis

6 Distributions

7 Complex Analysis

8 Special Functions

9 Fourier Transform

10 Laplace Transform

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 59 / 77

Vectors: Scalar Product; Vector Product; Dyadic Product

Vectors: Scalar Product; Vector Product; Dyadic Product

The Scalar, Vector, and Dyadic Product ← = Scalar! (151)
Scalar Product (Dot Product) ← = Vector! (152)
Example: ← = Dyad! (153)

A·B = C

Vector Product (Cross Product)
Example:

A×B = C

Dyadic Product
Example:

AB = D

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 60 / 77

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

Scalar Product

Scalar Product •.......................................................................................eˆ.A.......................................φ..............................................................A......................·..........e.ˆ.....................................................................................................................

Fig. 6 shows a vector A, which is projected to a unit Figure 6: Illustration des Skalarprodukts
vector eˆ, the result is given by the Scalar Product

A · eˆ = A cos φ , (154)

where φ determines the enclosed angle between A
and eˆ.

Replacing eˆ by a vector B with the magnitude B
yields the general form of Eq. (154), the
Commutative scalar product A · B (say: A dot B):

A·B = B·A (155)
= A B cos φ . (156)

Obviously is A · B = 0, if A and B are
perpendicular, A ⊥ B, to each other; this means
one can define two orthogonal vectors by a
vanishing scalar product between both.

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 61 / 77

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

Scalar Product

Scalar Product: Orthonormal Tripod (157)
The orthonormal tripod of a the cartesian coordinate system is charachterized by

exi · exj = δij fu¨r i, j = 1, 2, 3 .

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 (158)
Ay = A · ey
Az = A · ez .

We compute for the scalar product in components form of A 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)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 62 / 77

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

Scalar Product

Enclosed Angle φ between two General Vectors A and B
The Enclosed Angle between the vector A and B is if A = 0 and B = 0:

cos φ = A·B
=
AB

AxBx + Ay By + Az Bz . (161)

A2x + A2y + Az2 Bx2 + B2y + Bz2

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

A= A · A = Ax2 + Ay2 + A2z ; (162)

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

Aˆ = A = A = Ax ex + Ay ey + Az ez . (163)
A·A A A A A
WS 2007/2008 63 / 77
Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT)

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

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) in
numbered form we find

Axi = A · exi fu¨r i = 1, 2, 3 (164)
and for B we obtain instead Eq. (159)

3

A·B = Axi Bxi (165)

i=1

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)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 64 / 77

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

Scalar Product

Scalar Product: Independence of the Coordinate System

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

Aξi = A · eξi , (168)
and obtain by applying the summation convention

A = Aξi eξi (169)
= Aξi γij exj

= Axj exj

with

Axj = γij Aξi , (170)

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

a)At the point in space R(ξ1, ξ2, ξ3) we project the general position dependent vector A(ξ1, ξ2, ξ3) onto the position
dependent unit vectors eξi (ξ1, ξ2, ξ3).

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 65 / 77

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

Scalar Product

Scalar Product: Independence of the Coordinate System

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

Aξj = A · eξj

=Axi exi (171)
(172)
= Axi exi · eξj
= Axi exi · eξj

= γji
= γji Axi .

Number triples, which are transformed from the cartesian to an orthonormal curvilinear

coordinate system with Eq. (170) or Eq. (172) are in the mathematical sense (scalar) components

of vectors. Because of the inverse of Γ is equal to the transpose, Eq. (172), Aξj = γji Axi can
be derived from Eq. (170), Axj = γij Aξi , via inversion and vice versa. The vector A as a
directed physical value is independent of the coordinate system (it is koordinatenfrei), simply the

mathematical representation is coordinate dependent.

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

A · B = Axi Bxi = Aξi Bξi . (173)

Dr.-Ing. Ren´e Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 66 / 77