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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Lecture 2: Position Vector and Coordinate Systems (cont.)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 22 / 71
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
MFEFT - Lecture 2
1 Introduction
2
Vector and Tensor Algebra3 Position Vector and Coordinate Systems
Cartesian CoordinatesEinsteins Summation ConventionDifferentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)Cylinder Coordinate SystemOrthogonal Curvilinear Coordinate SystemSpherical Coordinate SystemDupin Coordinates
4 Vectors: Scalar Product; Vector Product; Dyadic Product
5 Vector and Tensor Analysis
6 Distributions
7 Complex Analysis
8 Special Functions
9 Fourier Transform
10 Laplace Transform
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 23 / 71
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R(40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
( )
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
P i i V d C di S Diff i i f h P i i V K k S b l (K k D l )
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
Position Vecto and Coo dinate S stems Diffe entiation of the Position Vecto K onecke S mbol (K onecke Delta)
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
Cartesian Coordinate SystemPosition Unit Vector
Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):
R =R
R(37)
=xex + yey + zezx2 + y2 + z2 . (38)
If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form
R
xiexi
= R (39)
= R (40)
with the so-called Nabla Operator
=
xex +
yey +
zez =
3i=1
xiexi
=
xiexi
. (41)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 24 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
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y ; y ( )
Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2x21 + x22 + x23 12
Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
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y y ( )
Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2x21 + x22 + x23 12
Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2x21 + x22 + x23 12
Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2x21 + x22 + x23 12
Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2x21 + x22 + x23 12
Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2 x21 + x22 + x23 12 Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
R
xiexi
=3
i=1
R
xiexi
(42)
=R
x1ex1
+R
x2ex2
+R
x3ex3
(43)
= x1
x21 + x
22 + x
23 ex1
+ x2
x21 + x
22 + x
23 ex2
+ x3
x21 + x
22 + x
23 ex3
.
(44)
We find by applying the chain rule
x1x21 + x22 + x23 = x1 x21 + x22 + x23
1
2
=
1
2 x21 + x22 + x23 12 Derivative ofouter function
2 x1Derivative ofinner function
(45)
=x1
x21 + x22 + x231
2
=x1
x21 + x22 + x23. (46)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 25 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
Analog follows for the other two deriviatives
x2x21 + x
22 + x
23 =
x2
x21 + x22 + x23(47)
x3
x21 + x
22 + x
23 =
x3x21 + x
22 + x
23
(48)
or in general
xi
x21 + x
22 + x
23 = x
ix21 + x
22 + x
23
i = 1, 2, 3 . (49)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 26 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
Analog follows for the other two deriviatives
x2x21 + x
22 + x
23 =
x2
x21 + x22 + x23(47)
x3
x21 + x
22 + x
23 =
x3x21 + x
22 + x
23
(48)
or in general
xi
x21 + x22 + x23 = xix21 + x
22 + x
23
i = 1, 2, 3 . (49)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 26 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
=x1
x21 + x22 + x
23
ex1+
x2x21 + x
22 + x
23
ex2+
x3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1ex1
+ x2ex2
+ x3ex3
x21 + x22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
= x1
x21 + x22 + x
23
ex1+ x
2x21 + x
22 + x
23
ex2+ x
3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1ex1
+ x2ex2
+ x3ex3
x21 + x22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
= x1
x21 + x22 + x
23
ex1+ x
2x21 + x
22 + x
23
ex2+ x
3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1ex1
+ x2ex2
+ x3ex3
x21 + x22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
= x1
x21 + x22 + x
23
ex1+ x
2x21 + x
22 + x
23
ex2+ x
3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1e
x1+ x2e
x2+ x3e
x3x21 + x
22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
= x1
x21 + x22 + x
23
ex1+ x
2x21 + x
22 + x
23
ex2+ x
3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1e
x1+ x2e
x2+ x3e
x3x21 + x
22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemExample: Proof ofR = R
Example (Proof ofR = R)
We summarize:
R
xiexi
=
x1x21 + x
22 + x
23 ex1
+
x2x21 + x
22 + x
23 ex2
+
x3x21 + x
22 + x
23 ex3
(50)
= x1
x21 + x22 + x
23
ex1+ x
2x21 + x
22 + x
23
ex2+ x
3x21 + x
22 + x
23
ex3(51)
=1
x21 + x22 + x
23
x1ex1 + x2ex2 + x3ex3
(52)
=x1e
x1+ x2e
x2+ x3e
x3x21 + x
22 + x
23
(53)
=R
R(54)
= R . (55)
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 27 / 71
Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)
C i C di S
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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Cartesian Coordinate SystemProjection of the Position Vector in the Direction of the Axes of the Cartesian Coordinate System
Projection of the Position Vector in the Direction ofthe Axes of the Cartesian Coordinate System
If we project the position vector in the direction ofthe Axes of the Cartesian Coordinates System wefind the components ofR:
x = R ex (56)
y = R ey (57)
z = R ez , (58)
this means also
R = (R ex)ex + (R ey)ey + (R ez)ez (59)
Cartesian Coordinates of the spatialpoint P and the related positionvector
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O
x
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x
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P (x,y,z)
R
ez
ex
ey
Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 28 / 71
Position Vector and Coordinate Systems Cylinder Coordinate System
(Ci l )C li d C di S
http://-/?-http://-/?-http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?- -
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(Circular)Cylinder Coordinate SystemCoordinates; Unit Vectors, Magnitude of the Position Vector
Coordinates; Unit Vectors, Magnitude ofthe Position Vector
Cylinder Coordinates: r,,z
in the limits 0 r < , 0 < 2,
< z