Vector analysis

1
VECTOR ANALYSIS
SOLO HERMELIN
http://www.solohermelin.com

2
Vector AnalysisSOLO
TABLE OF CONTENT
Algebras History
Vector Analysis History
Vector Algebra
Reciprocal Sets of Vectors
Vector Decomposition
The Summation Convention
The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee 

Change of Vector Base, Coordinate Transformation
Vector Space
Differential Geometry
Osculating Circle of C at P
Theory of Curves
Unit Tangent Vector of path C at a point P
Curvature of curve C at P
Osculating Plane of C at P
Binormal
Torsion
Seret-Frenet Equations
Involute
Evolute
Vector Identities Summary
Cartesian Coordinates

3
Vector AnalysisSOLO
TABLE OF CONTENT (continue – 1)
Conjugate Directions
Surfaces in the Three Dimensional Spaces
First Fundamental Form:
Arc Length on a Path on the Surface
Surface Area
Change of Coordinates
Second Fundamental Form
Principal Curvatures and Directions
Asymptotic Lines
Scalar and Vector Fields
Vector Differentiation
Ordinary Derivative of Scalars and Vectors
Partial Derivatives of Scalar and Vectors
Differentials of Vectors
The Vector Differential Operator Del (, Nabla)
Scalar Differential
Vector Differential
Differential Identities

4
Vector AnalysisSOLO
Curvilinear Coordinates in a Three Dimensional Space
Covariant and Contravariant Components of a Vector in Base .321
,, uuu
rrr

Coordinate Transformation in Curvilinear Coordinates
Covariant Derivative
Covariant Derivative of a Vector .A

Vector Integration
Ordinary Integration of Vectors
Line Integrals
Surface Integrals
Volume Integrals
Simply and Multiply Connected Regions
Green’s Theorem in the Plane
Stoke’s Theorem
Divergence Theorem
Gauss’ Theorem Variations
Stokes’ Theorem Variations
Green’s Identities
Derivation of Nabla (  ) from Gauss’ Theorem
The Operator .

5
Vector AnalysisSOLO
Fundamental Theorem of Vector Analysis for a Bounded Region V
(Helmholtz’s Theorem)
Reynolds’ Transport Theorem
Poisson’s Non-homogeneous Differential Equation
Kirchhoff’s Solution of the Scalar Helmholtz
Non-homogeneous Differential Equation
Derivation of Nabla (  ) from Gauss’ Theorem
The Operator .
Orthogonal Curvilinear Coordinates in a Three Dimensional Space
Vector Operations in Various Coordinate Systems
Applications
Laplace Fields
Harmonic Functions
Rotations

6
Synthetic Geometry
Euclid 300BC
Algebras HistorySOLO
Extensive Algebra
Grassmann 1844
Binary Algebra
Boole 1854
Complex Algebra
Wessel, Gauss 1798
Spin Algebra
Pauli, Dirac 1928
Syncopated Algebra
Diophantes 250AD
Quaternions
Hamilton 1843
Tensor Calculus
Ricci 1890
Vector Calculus
Gibbs 1881
Clifford Algebra
Clifford 1878
Differential Forms
E. Cartan 1908
First Printing
1482
http://modelingnts.la.asu.edu/html/evolution.html
Geometric Algebra
and Calculus
Hestenes 1966
Matrix Algebra
Cayley 1854
Determinants
Sylvester 1878
Analytic Geometry
Descartes 1637
Table of Content

7
Vector Analysis HistorySOLO
John Wallis
1616-1703
1673
Caspar Wessel
1745-1818
“On the Analytic Representation
of Direction; an Attempt”, 1799
bia 
Jean Robert Argand
1768-1822
1806
1i
Quaternions
1843
William Rowan Hamilton
1805-1865
3210 qkqjqiq 
Extensive Algebra
1844
Herman Günter Grassmann
1809-1877
“Elements of Vector
Analysis” 1881
Josiah Willard Gibbs
1839-1903
Oliver Heaviside
1850-1925
“Electromagnetic
Theory” 1893
3. R.S. Elliott, “Electromagnetics”,pp.564-568
http://www-groups.dcs.st-and.ac.uk/~history/index.html
Table of Content
Edwin Bidwell Wilson
1879-1964
“Vector Analysis”
1901

8
Vector AnalysisSOLO
ba


Vector Algebra
b

a

a

Addition of Vectors Parallelogram
Law of addition
Subtraction of Vectors
 baba

1
Parallelogram
Law of subtraction
b

a

b


b
 a

ba


Multiplication of Vector by a Scalar
am

a

a

am

b

a

ba


ba


b


Geometric Definition of a Vector
A Vector is defined by it’s Magnitude and Directiona
 a

a


9
Vector AnalysisSOLO








bababa

,sin
b

a

ba

,

b

a

ba

,

ba


abba


Scalar product of two vectors ba

,
Vector product of two vectors ba

,
 
2/1
2/1
,cos


















aaaaaaa
Magnitude of Vector a

Unit Vector (Vector of Unit Magnitude)aa 1ˆ  a
a
aa


1
:1:ˆ 
Zero Vector (Vector of Zero Magnitude)0

aa

0
0:0 

00 a

   ababbababa ba
  ,cos, , 







Vector Algebra (continue – 1)

10
Vector AnalysisSOLO
    n||
ˆˆˆˆ aanannana n

 
nˆ
a


n
a
n
a
ˆ
ˆ
n||





n
a
n
a
n
ˆ
ˆ





Vector decomposition in two orthogonal directions nn ,||
Vector decomposition in two given directions (geometric solution)
1
ˆn
a

2
ˆn
A B
C
Given two directions and , and the vector a

1
ˆn 2
ˆn
anBCnCA

 21
ˆˆ
1
ˆn
a

2
ˆn
A B
Draw lines parallel to those directions passing
through both ends A and B of the vector .
The vectors obtained are in the desired directions
and by rule of vector addition satisfy
a

Vector Algebra (continue – 2)
Table of Content

11
SOLO
Triple Scalar Product
Vectors & Tensors in a 3D Space
3321 ,, Eeee 

are three non-coplanar vectors, i.e.
1e

2e

3e

    0:,, 321321  eeeeee

       
      0,,
,,,,
123123213
132132132321


eeeeeeeee
eeeeeeeeeeee


Reciprocal Sets of Vectors
The sets of vectors and are called Reciprocal Sets or Systems
of Vectors if:
321 ,, eee
 321
,, eee

DeltaKroneckertheis
ji
ji
ee
j
i
j
i
j
i







1
0
Because is orthogonal to and then2e

3e
1
e

     
 321
321321
1
132
1
,,
1
,,1
eee
keeekeeekeeeeke 


and in the same way and are given by:2
e
 3
e
1
e

 
 
 
 
 
 321
213
321
132
321
321
,,,,,, eee
ee
e
eee
ee
e
eee
ee
e 







 






12
SOLO Vectors & Tensors in a 3D Space
Reciprocal Sets of Vectors (continue)
By using the previous equations we get:
   
 
     
   321
3
2
321
13323132
2
321
133221
,,,,,, eee
e
eee
eeeeeeee
eee
eeee
ee 











 
 
 
 
 
 321
213
321
132
321
321
,,,,,, eee
ee
e
eee
ee
e
eee
ee
e 







 





   
   
0
,,
1
,,
,,
321321
3
3321321



eeeeee
ee
eeeeee 


Multiplying (scalar product) this equation by we get:3
e

In the same way we can show that:
Therefore are also non-coplanar, and:321
,, eee

   1,,,, 321
321
eeeeee

 
 
 
 
 
 321
21
3321
13
2321
32
1
,,,,,, eee
ee
e
eee
ee
e
eee
ee
e 







 





1e

2e

3e

1
e

2
e

3
e

Table of Content
1e

2e

3e


13
Vector Decomposition
Given we want to find the coefficients and such that:3
EA

321 ,, AAA 321
,, AAA






3
1
3
3
2
2
1
1
3
1
3
3
2
2
1
1
j
j
j
i
i
i
eAeAeAeA
eAeAeAeAA


3,2,1, iee i
i

are two reciprocal
vector bases
Let multiply the first row of the decomposition by :j
e

Let multiply the second row of the decomposition by :ie

j
i
j
i
i
i
j
i
ij
AAeeAeA   
3
1
3
1


i
j
i
j
j
j
i
j
ji
AAeeAeA   
3
1
3
1


Therefore:
ii
jj
eAAeAA

 &
Then:
       
       





3
1
3
3
2
2
1
1
3
1
3
3
2
2
1
1
j
j
j
i
i
i
eeAeeAeeAeeA
eeAeeAeeAeeAA


Table of Content
1e

2e

3e


14
The Summation Convention
j
j
j
j
j
eAeAeAeAeA

 
3
1
3
3
2
2
1
1
The last notation is called the summation convention, j is called the dummy
index or the umbral index.
     
      i
i
i
i
j
j
j
j
j
j
j
j
j
i
i
i
i
i
eAeeAeeAeeA
eAeeAeeAeeAA








3
1
3
1
Instead of summation notation we shall use the shorter notation
first adopted by Einstein

3
1j
j
j
eA
 j
j eA

Table of Content

15
Let define:

jiijjiij
geeeeg 

3321 ,, Eeee 

the metric covariant tensors of
By choosing we get:
       
j
ijiii
j
jiiiii
egegegeg
eeeeeeeeeeeee




3
3
2
2
1
1
3
3
2
2
1
1
i
eA


or: j
iji ege


For i = 1, 2, 3 we have:























































3
2
1
332313
322212
312111
3
2
1
333231
232221
131211
3
2
1
e
e
e
eeeeee
eeeeee
eeeeee
e
e
e
ggg
ggg
ggg
e
e
e












1e

2e

3e


16
We want to prove that the following determinant (g) is nonzero:


























332313
322212
312111
333231
232221
131211
detdet:
eeeeee
eeeeee
eeeeee
ggg
ggg
ggg
g



g is the Gram determinant of the vectors 321 ,, eee

Jorgen Gram
1850 - 1916
Proof:
Because the vectors are non-coplanars the following equations:321 ,, eee

03
3
2
2
1
1
 eee


is true if and only if 0321
 
Let multiply (scalar product) this equation, consecutively, by :321 ,, eee












































0
0
0
0
0
0
3
2
1
332313
322212
312111
33
3
23
2
13
1
32
3
22
2
12
1
31
3
21
2
11
1






eeeeee
eeeeee
eeeeee
eeeeee
eeeeee
eeeeee






Therefore α1= α2= α3=0 if and only if g:=det {gij}≠0 q.e.d.

17
Because g ≠ 0 we can take the inverse of gij and obtain:

where Gij = minor gij having the following property: j
i
kj
ik
ggG 




















































3
2
1
333231
232221
131211
3
2
1
333231
232221
131211
3
2
1
1
e
e
e
ggg
ggg
ggg
e
e
e
GGG
GGG
GGG
g
e
e
e









and:
g
G
g
gminor
g
ij
ijij

Therefore:
g
g
g
g
G
gg
j
i
kj
ik
kj
ik 
 j
i
kj
ik
gg 
Let multiply the equation by gij and perform the summation on ij
iji ege


jj
ij
ij
i
ij
eeggeg


Therefore: i
ijj
ege


Let multiply the equation byk
kjj
ege

 i
e
 iji
k
kjijji
geegeeee 

jiijjiij
geeeeg 

i
jkj
ik
ggG 
The Operator .

18

Let find the relation between g and    321321 :,, eeeeee


We shall write the decomposition of in the vector base32 ee

 321 ,, eee

3
3
2
2
1
1
32
eeeee

 
Let find λ1, λ2, λ3. Multiply the previous equation (scalar product) by .1e

    i
i
ggggeeeeee 113
3
12
2
11
1
321321
,,  

Multiply this equation by g1i:   ii
i
ii
ggeeeg  


1
1
1321
1
,,
Therefore:  321
1
,, eeeg ii 

Let compute now:
           321
1
0
323
3
0
322
2
321
1
3232 eeeeeeeeeeeeeeee





 
    
 
  
 
    
       321
11
321
11
321
2
233322
321
3322332
321
3232
321
32321
,,,,,,,,
,,,,
eee
gg
eee
G
eee
ggg
eee
eeeeeee
eee
eeee
eee
eeee















From those equations we obtain:
 321
11
1
,, eee
gg

Finally:     geeeeee  321
2
321
1
,,,,


We can see that if are collinear than and g are zero.321 ,, eee
 321
,, eee
 Table of Content

19
Change of Vector Base, Coordinate Transformation
Let choose another base and its reciprocal 321
,, fff

 321
,, fff

 













































3
2
1
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
e
e
e
L
e
e
e
f
f
f
ef j
j
ii














where j
i
j
i
ef


By tacking the inverse of those equations we obtain:
 
















































3
2
1
1
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
f
f
f
L
f
f
f
e
e
e
fe i
j
ij














where j
ij
i ef


Because are the coefficients of the inverse matrix with coefficients :j
i
 j
i
i
j
i
k
k
j  

20
Change of Vector Base, Coordinate Transformation (continue – 1)
Let write any vector in those two bases:A

 



















































3
2
1
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
e
e
e
L
e
e
e
f
f
f
ef
ef
ji
j
i
j
j
ii
















then:
 
















































 
3
2
1
1
3
2
1
3
3
3
2
3
1
2
3
2
2
2
1
1
3
1
2
1
1
3
2
1
E
E
E
L
E
E
E
F
F
F
ef
EF T
j
ii
j
i
j
ji






i
i
j
j
fFeEA

 iijj
fAFeAE

 &
i
j
ji
i
i
j
j
j
j
i
i
EFfEeEfF  

or:
But we remember that:
We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is
not similar, contravariant, to the relation between the two bases of vectors
to . Therefore we define F1, F2, F3 and E1, E2, E3 as the
contravariant components of the bases and .
 321
,, fff

 321 ,, eee

 321
,, fff

 321 ,, eee

where

21
 











































3
2
1
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
E
E
E
L
E
E
E
F
F
F
EF j
iji




i
i
j
j fFeEA

 iijj fAFeAE

 &
then:
Let write now the vector in the two bases andA

 321
,, fff

 321
,, eee

where
and j
ij
ef
ijjjjii EfeEeEAfAF j
iji


 

We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is
similar, covariant, to the relation between the two bases of vectors
to . Therefore wew define F1, F2, F3 and E1, E2, E3 as the covariant
components of the bases and .
 321
,, fff

 321
,, eee

 321
,, fff

 321
,, eee


22
We have:
 
  j
j
j
j
i
i
i
i
eeAeA
eeAeAA



 Ai contravariant component
Aj covariant component
Let find the relation between covariant and the contravariant components:
j
j
j
ij
i
ege
i
i
eAegAeAA j
iji

 

i
i
i
ij
j
ege
j
j
eAegAeAA
i
ijj

 

Therefore: ij
j
i
ij
i
j gAAgAA  &
Let find the relation between gij and gij defined in the bases and to
and defined in the bases and .
ie
 i
e

i
f

i
f

ij
g ij
g
m
m
kkj
j
ii efef

  &
jm
m
k
j
imj
m
k
j
ikiik geeffg  

Hence: jm
m
k
j
iik gg 
This is a covariant relation of rank two, (similar, two times, to relation
between to .i
f

j
e


23
 



















































3
2
1
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
e
e
e
L
e
e
e
f
f
f
ef
ef
ji
j
i
j
j
ii
















Since we have:k
iki
fgf

   m
jm
j
i
ege
j
j
i
ef
k
iki
egefgf m
jmjj
j
ii

 
 

and:     m
jm
j
i
km
kjm
j
i
gg
k
ik
egfgfg
mjm
m
k
j
iik




Therefore, by equalizing the terms that multiply we obtain:
 















































3
2
1
3
2
1
3
3
3
2
3
1
2
3
2
2
2
1
1
3
1
2
1
1
3
2
1
f
f
f
L
f
f
f
e
e
e
fe
Tkm
k
m














 jm
j
i g
We found the relation:

24
Therefore:
 












































 
3
2
1
1
3
2
1
3
3
3
2
3
1
2
3
2
2
2
1
1
3
1
2
1
1
3
2
1
e
e
e
L
e
e
e
f
f
f
ef
Tmj
m
j














Let take the inverse of the relation by multiplying by and summarize on m:j
m
km
k
m
fe


jkj
k
km
k
j
m
mj
m
fffe

 
From the relation: mk
m
kjj
i
i
efef

  &
we have: jmk
m
j
i
mjk
m
j
i
kiik
geeffg  

or: jmk
m
j
i
ik
gg  This is a contravariant relation of rank two.
From the relation:
m
m
kk
jj
i
i
efef

  &
we have: m
j
m
k
i
jm
jm
k
i
j
i
kk
i
eeff  

or: This is a relation once covariant and once contravariant of
rank two.
m
j
m
k
i
j
i
k  
Table of Content

25
Cartesian Coordinates
Three dimensional cartesian coordinates are define as coordinates in a orthonormal
basis such that:     zyxorkjieee 1,1,1ˆ,ˆ,ˆ,, 321


ix ˆ1 
jy ˆ1 
O
kz ˆ1 
x1
y1z1
0111111
1111111


zyzxyx
zzyyxx
yzxxyzzxy
yxzxzyzyx
111111111
111111111


    11111,1,12
 zyxzyxg
The reciprocal set is identical to the original set






ji
ji
ggg ij
ijj
iij
0
1

Given
  










z
y
x
zyx
A
A
A
zyxzAyAxAA 111111

 











z
y
x
A
A
A
AMatrix notation
of a vector

26
Cartesian Coordinates (continue – 1)
ix ˆ1 
jy ˆ1 
O
kz ˆ1 
Given  











z
y
x
zyx
A
A
A
zyxzAyAxAA 111111

 











z
y
x
A
A
A
AMatrix notation
of a vector
 











z
y
x
zyx
B
B
B
zyxzByBxBB 111111

     
        ABABBABABABA
B
B
B
AAAzByBxBzAyAxABA
T
zzyyxx
T
z
y
x
zyxzyxzyx













 111111
 











z
y
x
B
B
B
B

27
   
     
zyx
zyxxyyxzxxzyzzy
zyxzyx
BBB
AAA
zyx
zBABAyBABAxBABA
zByBxBzAyAxABA
111
det111
111111



         ABAB
A
A
A
BB
BB
BB
BA
B
B
B
AA
AA
AA
BA
z
y
x
xy
xz
yz
z
y
x
xy
xz
yz


















































0
0
0
0
0
0
ix ˆ1 
jy ˆ1 
O
kz ˆ1 
Given  











z
y
x
zyx
A
A
A
zyxzAyAxAA 111111

 











z
y
x
A
A
A
AMatrix notation
of a vector
 











z
y
x
zyx
B
B
B
zyxzByBxBB 111111

 











z
y
x
B
B
B
B

28
Table of Content
   
     
       
        ACBACBCBACBCBACBCB
BACBACACBACACBACAC
CBABACBABACBABA
CCC
BBB
AAA
BBB
AAA
CCC
zCyCxC
BBB
AAA
zyx
CBA
zxyyxyzxxzxyzzy
zxyyxyzxxzxyzzy
zxyyxyzxxzxyzzy
zyx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
zyx






 detdet111
111
det
                 ABCABCBAC
B
B
B
AA
AA
AA
CCCBAC
TT
z
y
x
xy
xz
yz
zyx


























0
0
0
Given  











z
y
x
zyx
A
A
A
zyxzAyAxAA 111111

 











z
y
x
A
A
A
A
Matrix notation
of a vector
 











z
y
x
zyx
B
B
B
zyxzByBxBB 111111

 











z
y
x
B
B
B
B
 











z
y
x
zyx
C
C
C
zyxzCyCxCC 111111

 











z
y
x
C
C
C
C

29
Vector AnalysisSOLO
           bacbacacbacbcbacba

,,,,:,, 
      cbabcacba


      
    
     cbdadbca
dcbcdba
dcbadcba






       
   adcbbdca
dcbacdbadcba


,,,,
,,,,


       2
,, cbaaccbba


           
     
     feabdcfebadc
dcefbadcfeba
fedcbafecdbafedcba



,,,,,,,,
,,,,,,,,
,,,,,,,,



Vector Identities Summary
      0 bacacbcba

Table of Content

30
SOLO
VECTOR SPACE
Given the complex numbers .C ,,
A Vector Space V (Linear Affine Space) with elements over C if its elements
satisfy the following conditions:
Vzyx 

,,
I. Exists a operation of Addition with the following properties:
xyyx

 Commutative (Abelian) Law for Addition1
   zyxzyx

 Associative Law for Addition2
xx

 0 Exists a unique vector0

3
II. Exists a operation of Multiplication by a Scalar with the following properties:
0..

 yxtsVyVx4 Inverse
xx

15
   xx

  Associative Law for Multiplication6
  xxx

  Distributive Law for Multiplication7
  yxyx

  Commutative Law for Multiplication8
   
 
 
 
00101010 3
575 
 xxxxxxxxWe can write:
Vector Analysis

31
SOLO
Scalar Product in a Vector Space
The Scalar Product of two vectors is the operation with the
symbol with the following properties:
Vyx 

,   Cyx 

,
   xyyx

,, 
   yxyx

,,  
     zyzxzyx

,,, 
    00,&0,

 xxxxx
Distance Between Two Vectors
The Distance between two Vectors is
defined by the following properties:
Vyx 

,   yxyxd

,
    00,&0,

 xxxdxxd
   xydyxd

,, 
     yzdzxdyxd

,,, 
Vector Analysis
Table of Content

32
SOLO
Differential Geometry is the study of geometric figures using the methods of Calculus.
Here we present the curves and surfaces embedded in a three dimensional space.
Properties of curves and surfaces which depend only upon points close to a particular
point of the figure are called local properties.. The study of local properties is called
differential geometry in the small.
Those properties which involve the entire geometric figure are called global properties.
The study of global properties is called differential geometry in the large.
Hyperboloid
of RotationToroyd
Mobius
Movement
Differential Geometry in the 3D Euclidean Space
Table of Content

33
SOLO
A curve C in a three dimensional space is
defined by one parameter t,  tr

 ur

rd
P
O
a
b
C
Theory of Curves
Regular Parametric Representation of a Vector Function:
parameter t, defined in the interval I and:
  Ittrr  ,

 tr

(i) is of class C1 (continuous and 1st order differentiable) in I
Arc length differential:      td
td
rd
td
td
rd
td
rd
trdtrdsd









2/1
2/1
:
We also can define      sdtrdtrdsd 
2/1*
:

(ii)
  It
td
trd
 0

  Iinconstantnottr


Arc length as a parameter: 
t
t
td
td
rd
s
0

Regular Curves:
A real valued function t = t (θ), on an interval Iθ, is an allowable change of parameter if:
(i) t (θ) is of class C1 in Iθ (ii) d t/ d θ ≠ 0 for all θ in Iθ
A representation on Is is a
representation in terms of arc length or a
natural representation
 srr


Table of Content

34
SOLO
A curve C in a three dimensional space is defined by one parameter t,  tr

 ur

rd
P
O
a
b
C
- arc length differential     td
td
rd
td
rd
trdtrdsd
2/1
2/1
: 








td
rd
td
rd
r
sd
rd
t /::   - unit tangent vector of path C at P
(tangent to C at P)
1x
2x
3x
td
rd
r '
 - tangent vector of path C at P
(tangent to C at P)
0,0,sincos 321
 baetbetaetar

Example: Circular Helix
0,0,cossin' 321
 baebetaeta
td
rd
r


  2/122
2/1
ba
td
rd
td
rd
td
rd








   321
2/122
cossin/: ebetaetaba
td
rd
td
rd
t 

Theory of Curves (continue – 1)
We also can define      sdtrdtrdsd 
2/1*
:

t
sd
rd
sd
rd 
*
Unit Tangent Vector of path C at a point P
Table of Content

35
SOLO
The earliest investigations by means of analysis were made
by René Descartes in 1637.
 tr

 ur

rd
P
O
a
b
C
René Descartes
1596 - 1650
Pierre Fermat
1601 - 1665
Christian Huyghens
1629 - 1695
Gottfried Leibniz
1646 - 1716
The general concept of tangent was introduced in
seventeenth century, in connexion with the basic concepts of
calculus. Fermat, Descartes and Huyghens made important
contributions to the tangent problem, and a complete
solution was given by Leibniz in 1677.
The first analytical representation of a tangent was given
by Monge in 1785.
Gaspard Monge
1746 - 1818

36
SOLO
A curve C in a three dimensional space is defined by one parameter t,  tr

- arc length differential     td
td
rd
td
rd
trdtrdsd
2/1
2/1
: 








'/'/:: rr
td
rd
td
rd
r
sd
rd
t


- unit tangent vector of path C at P
(tangent to C at P)
Normal Plane to at P:t

  00
 trr

We also can define - arc length differential     sdtrdtrdsd 
2/1*
:

t
sd
rd
sd
rd 
*
O
a
C
t

P
r

b
0
r

NormalPlane   00
 trr

Return to Table of Contents

37
SOLO
O
a
C
t

P
r

b
NormalPlane   00
 trr

0
r

Curvature of curve C at P: rt
sd
td
k 

:
Since 01  tkttt
sd
td
tt



Define nnkkkkkkn



1
/1:&/: 
ρ – radius of curvature of C at P
k – curvature of C at P
A point on C where k = 0 is called a point of inflection and the radius of curvature
ρ is infinite.
'' st
td
sd
sd
rd
td
rd
r




      "'"'"'
'
'''''
22
stskstststs
td
sd
sd
td
td
sd
ts
td
td
st
td
d
r
td
d
r






    32
'"''''' skntstskstrr


'' sr 

3
1
'''' skntrr


 3
'
'''
r
rr
k 



Let compute k as a function of and :'r

''r


38
SOLO
1x
2x
3x
t

k

0,0,sincos 321

Example 2: Circular Helix
0,0,cossin' 321
 baebetaeta
td
rd
r


    2/1222/122
2/1
bardsdba
td
rd
td
rd
td
rd









   321
2/122
cossin/: ebetaetaba
td
rd
td
rd
t 

 2122
sincos/ etet
ba
a
td
sd
td
rd
t
sd
td
k 





1
x
2x
3
x
t

k

0,sincos 21
 aetaetar

Example 1: Circular Curve
0,cossin' 21
 aetaeta
td
rd
r


    2/1222/122
2/1
bardsdba
td
rd
td
rd
td
rd









 21
cossin/: etaetaa
td
rd
td
rd
t 

 21 sincos
1
/ etet
atd
sd
td
rd
t
sd
td
k 



Table of Content

39
SOLO
O
a
C
t

P
b
ntk


1

NormalPlane
Osculating
Plane
  00
 trr

0r

  00  ktrr
Osculating Plane of C at P is the plane that contains
and P:     00
 ktrr
kt

,
The name “osculating plane” was
introduced by D’Amondans
Charles de Tinseau (1748-1822) in
1780.
O
a
C
t

P
b
ntk


1

NormalPlane
Osculating
Plane
  00
 trr

0r

  00  ktrr

The osculating plane can be also defined as the limiting position of a plane passing
through three neighboring points on the curve as the points approach the given point.
If the curvature k is zero along a curve C then:
tarrconstartt

 0
0
The curve C is a straight line. Conversely if C is a straight line:
0//0
 tkaa
td
rd
td
rd
ttarr 
C a regular curve of class ≥2 (Cclass) is a straight line if and only if k = 0 on C
Table of Content

40
SOLO
Osculating Circleof C at P is the plane that contains
and Pkt

,
The osculating circle of a curve C at a given point P is the circle that has the same
tangent as C at point P as well as the same curvature.
Just as the tangent line is the line best approximating a curve at a point P,
the osculating circle is the best circle that approximates the curve at P.
http://mathworld.wolfram.com/OsculatingCircle.html
O
a
C
t

P
b
ntk


1

Normal Plane
Osculating
Plane
  00  trr

0r

  00  ktrr

Osculating
Circle
Osculating Circles on the Deltoid
The word "osculate" means "to kiss."

41
SOLO
Osculating Circleof C at P is the plane that contains
and P
kt

,
Theory of Curves (continue – 6a)
O
a
C
t

P
b
ntk


1

Normal Plane
Osculating
Plane
  00  trr

0r

  00  ktrr

Osculating
Circle
3
xy 
xy /1
xy cos xy sin http://curvebank.calstatela.edu/osculating/osculating.htm
xy tan
Table of Content

42
SOLO
O
a
C
t

P
b
ntk


1

NormalPlane
Osculating
Plane
  00
 trr

0r

  00  ktrr
b

Rectifying
Plane
  00  krr

Binormal ntb

:
Tangent Line:
Principal Normal Line:
Binormal Line:
Normal Plane:
Rectifying Plane:
Osculating Plane:
tmrr

 0
nmrr

 0
bmrr

 0
  00
 trr

  00  nrr

  00
 brr

The name binormal was introduced by
Saint-Venant
Jean Claude Saint-Venant
1797 - 1886
Fundamental Planes:Fundamental Lines:
Table of Content

43
SOLO
Torsion
Suppose that is a regular curve of class ≥ 3 (Cclass) along which is of
class C1. then let differentiate to obtain:
 srr

  sn

     snstsb


                     snstsnstsnsnksnstsnstsb 

Since                 001  snsnsnsnsnsnsnsn 
Therefore is normal to , meaning that is in the rectifying plane,
or that is a linear combination of and .
n
n
t

b

         sbsstssn

 
                    snssbsstsstsnstsb

 
O
a
C
t

P
b
n

0r

b

The continuous function τ (s) is called the second curvature
or torsion of C at P.
     snsbs



44
SOLO
Torsion (continue – 1)
Suppose that the torsion vanishes identically (τ ≡0) along a curve , then srr


        0
0 bsbsnssb

 
O
a
C
t

P
b
n

0r

0
b

Since and are orthogonal st

 sb

        constbsrbtbsr
sd
d
bsr
sd
d
 0000
0

Therefore is a planar curve confined to the plane srr

   constbsr  0

C a regular curve of class ≥3 (Cclass) is a planar curve if and only if τ = 0 on C
1x
2x
3x
t

k

0,0,sincos 321

Example 2: Circular Helix
   321
2/122
cossin ebetaetabat 

 21
sincos etetn 

     
   321
2/122
21321
2/122
cossin
sincoscossin
eaetbetbba
etetebetaetabantb




     21
1222/122
sincos etbetbbaba
td
bd
sd
td
td
bd
sd
bd
b 



 122 
 babnb



45
SOLO
Torsion (continue – 2)
Let compute τ as a function of and :'',' rr

'''r

ttr
sd
td
td
rd
sd
rd
r




 '      tbknkttrtrtr
sd
d
trtr
sd
d
r


 2
"''''
 
    tkbkbknkbktnkbktbktbkbk
trttrtrtrttrttrtrtrtr
sd
d
r





2
332
'''"3''''"2"'"'



     
        
       
             2
0
3
1
2
0
26
6
0
3
0
4
0
22
5232
32
,,,,,,'''",'
'''",'',",'''',','",','3
'''"'"''''"'3'
'''"3'"'',,
ktntkbntknntkktkbknknktrrrt
rrrtrrrttrrrttrrrtt
rrtrrttrrttrrtttr
trttrtrtrtrtrrrr



















'
1
/
1
rtdsdsd
td
t  
3
'
'''
r
rr
k 


We also found:
     6
2
2
6
'
'''
'
'''",'
,,
r
rr
k
r
rrr
rrr 



 



 
2
'''
'''",'
rr
rrr





Table of Content

46
SOLO
Seret-Frenet Equations
We found and      snssb

     snskst


Let differentiate      stsbsn


                             stsksbssnsbskstsnsstsbstsbsn


 
We obtain
         sbsnskstst

00 
           sbsbsnstsksn

 0
         sbsnsstsb

00  
or
 
 
 
 
   
 
 
 
 



































sb
sn
st
s
ssk
sk
sb
sn
st






00
0
00


Jean Frédéric Frenet
1816 - 1900
Those are the Serret – Frenet Equations of a curve.
Joseph Alfred Serret
1819 - 1885

47
SOLO
Let compute:
Seret-Frenet Equations (continue – 1)
Let show that if two curves C and C* have the same curvature k (s) = k* (s) and
torsion τ (s) = τ*(s) for all s then C and C* are the same except for they position in
space. Assume that at some s0 the triads and
coincide.
     999
,, sbsnst

     999
*,*,* sbsnst

   ********
*
nttnknkttnktttttt
sd
d kk 


         ************
*
*
bnnbnttnkbtknnbtknnnnnn
sd
d kk 



 
   ********
*
nbbnnbbnbbbbbb
sd
d 




Adding the equations, we obtain:   0***
 bbnntt
sd
d 
Integrating we obtain:     30
******
 sbbnnttconstbbnntt

Since: and1,,1 ***
 bbnntt

  3***
 bbnntt

we obtain: 1***
 bbnntt

Finally since:         constsrsr
sd
rd
stst
sd
rd
 *
*
* 




48
SOLO
Existence Theorem for Curves
Let k (s) and τ (s) be continuous functions of a real variable s for s0 ≤ s ≤ sf.
Then there exists a curve , s0 ≤ s ≤ sf, of class C2 for which k is the curvature,
τ is the torsion and s is a natural parameter.
 srr


332211332211332211
,, ebebebbenenennetetett


       tnktttttt
sd
d 
 2          nbntknnnnnn
sd
d 
 22
   bnbbbbbb
sd
d 





 




  2
with:
Proof: Consider the system of nine scalar differential equations:
                      3,2,1,,,  isnssbsbsstsksnsnskst iiiiiii


and initial conditions:       302010
,, esbesnest


           btttktnkntntnt
sd
d 
           nnbbbtkbnbnbn
sd
d 






  
       ntbnkbtbtbt
sd
d 





  
and initial conditions:
            1,0,1,0,0,1 000000
 ssssss bbbnnnbtnttt


49
SOLO
Existence Theorem for Curves (continue – 1)
         
               
         bnbb
sd
d
ntbnkbt
sd
d
nnbbbtkbn
sd
d
btttktnknt
sd
d
nbntknn
sd
d
tnktt
sd
d









2
222
Proof (continue – 1):
and initial conditions:             1,0,1,0,0,1 000000
 ssssss bbbnnnbtnttt

We obtain:
The solution of this type of differential equations with given initial conditions has
a unique solution and since
is a solution, it is unique.
            1,0,1,0,0,1  bbbnnnbtnttt

The solution is an orthonormal triad.bnt

,,
We now define the curve:    
s
s
dtsrr
0
: 

We have: and , therefore k (s) is the curvature.1 tr

        1&  snsnskst

Finally since:       nbtttknnkntntbntb

 
Therefore τ (s) is the torsion of  srr

 q.e.d.

50
SOLO
From the previous development we can state the following theorems:
A curve is defined uniquely by the curvature and torsion
as functions of a natural parameter.
The equations k = k (s), τ = τ (s), which give the curvature
and torsion of a curve as functions of s are called the natural
or intrinsec equations of a curve, for they completely define
the curve. O
0s
C
t

P
n

0r

b

k
1
 f
s
Fundamental Existence and Uniqueness Theorem of Space
Curves
Let k (s) and τ (s) be arbitrary continuous functions on
s0≤s≤sf. Then there exists, for position in space, one and only
one space curve C for which k (s) is the curvature, τ (s) is the
torsion and s is a natural parameter along C. O
0s
C
t

P
n

b

f
s
*
C
0
r

*
0r

Table of Content

51
SOLO
Let consider a space curve C. We construct the tangent
lines to every point on C and define an involute Ci as any
curve which is normal to every tangent of C.
Involute
From the Figure we can see that the equation of the
Involute is given by:
turr

1
Differentiating this equation we obtain:
11
1
1
1
sd
sd
t
sd
ud
nkut
sd
sd
t
sd
ud
sd
td
u
sd
rd
t
sd
rd

















Scalar multiply this equation by and use the fact that and from the
definition of involute :
t

0nt

01
tt

  
1101
10
sd
sd
tt
sd
ud
ntkutttt 







01 
sd
ud
scu 
       stscsrsr

1
C
i
C
O
r
 1r

t

1
t

s
c 
Involute
Curve

52
SOLO
Involute (continue – 1)
C
i
C
O
r
 1r

t

1
t

s
c 
Involute
Curve
       stscsrsr

1

    n
sd
sd
ksc
sd
sd
t
sd
td
sc
sd
rd
sd
rd
t
t




111
1
1 











and are collinear unit vectors, therefore:1t

n

 
 kscsd
sd
sd
sd
ksc


1
1
11
The curvature of the involute, k1, is obtained from:
 
   ksc
btk
kscsd
nd
sd
sd
sd
td
nk
sd
td nt
kscsd
sd










 
11
1
1
1
1
11
1
1
Hence:
  22
22
2
1
ksc
k
k




For a planar curve (τ=0) we have:  
t
sc
nk




1
011 

53
SOLO
Involute (continue – 3)
C
i
C
O
r
 1r

t

1
t

s
c 
Involute
Curve
http://mathworld.wolfram.com/Involute.html
Table of Content

54
SOLO
The curve Ce whose tangents are perpendicular to a
given curve C is called the evolute of the curve.
Evolute
11
twrbvnurr


Differentiating this equation we obtain:
 
11
1
1
1
sd
sd
b
sd
vd
n
sd
ud
nvbtkut
sd
sd
b
sd
vd
n
sd
ud
sd
bd
v
sd
nd
u
sd
rd
t
sd
rd





















Scalar multiply this equation by and use the fact that and from the
definition of evolute :
t

0 btnt

01
tt

 
111
1
0
sd
sd
ttkutttt 







01  ku 
k
u
1
C
e
C
O
r

1r

t
1
t

Evolute
Curve
The tangent to Ce, , must lie in the plane of
and since it is perpendicular to . Therefore:
n

b

t
1t

1
1
sd
sd
n
sd
ud
vb
sd
vd
ut
























55
SOLO
Evolute (continue – 1)
   ccuv   tantan

k
u
1
C
e
C
O
r

1r

t
1
t

Evolute
Curve
We obtained:
1
1
sd
sd
n
sd
ud
vb
sd
vd
ut























    111
// wbvnuwrrt

But:
Therefore:
v
v
sd
ud
u
u
sd
vd
 


or:















 
u
v
sd
d
vu
sd
ud
v
sd
vd
u
1
22
tan
c
u
v
ds
s
s






 

1
tan
0

and:  bcnrr

  tan1
We have one parameter family that describes the evolutes to the curve C.

56
SOLO
Evolute (continue – 2)
http://math.la.asu.edu/~rich/MAT272/evolute/ellipselute.html
Evolute of Ellipse
Evolute of Logarithmic Spiral
also a Logarithmic Spiral
Evolute of Parabola
Table of Content
C
e
C
O
r

1r

t
1
t

Evolute
Curve

57
SOLO
The vector defines a surface in E3
 vur ,

vu
vu
rr
rr
N 




 vur ,

 vdvudur  ,

rd 2
rd
r
udru

vdrv

d
Nd
P
O
 vudur ,

       
     22
2
22
22
2
2
,2
2
1
,
2
1
,,,
vdudOvdrvdudrudrvdrudr
vdudOrdrdvurvdvudurvur
vvvuuuvu




The vectors and define the
tangent plane to the surface at point P.
P
u
u
r
r





P
v
v
r
r





Define: Unit Normal Vector to the surface at P
vu
vu
rr
rr
N 



:
First Fundamental Form:
      2222
22: vdGvdudFudEvdrrvdudrrudrrrdrdI vvvuuu


    

0
2
0,0,00:

















GF
FEforConditionSylvester
FEGGE
vd
ud
GF
FE
vdudrdrdI
Surfaces in the Three Dimensional Spaces
Table of Contents

58
SOLO
Arc Length on a Path on the Surface:
      









b
a
b
a
vuvu
b
a
tdvdrudrvdrudrtd
td
rd
td
rd
td
td
rd
L
2/1
2/1










































































b
a
b
a
td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
vd
G
td
vd
td
ud
F
td
ud
EL
2/1
2/1
22
2
Surface Area:
 vur ,

rd
udru

vdrv

d
P
O
 
  
vdudFGEvdud
GE
F
GE
vdud
rr
rr
rrvdudrrrr
vdudrrrrvdudrrvdrudrd
vu
vu
vuvuvu
vuvuvuvu
2
2/1
2
2/1
2
2/12
1
1,cos1
,sin






















 






  vdudFGEd 2
 vur ,

rd
udru

vdrv
P
O
a
b
Table of Contents

59
SOLO
Change of Coordinates
 vur ,

rd
udru

vdrv

d
P
O
vdrv

udru

vdrudrvdrudrd vuvu


vdudFGEvdud
vu
vu
JFGEvdudFGEd 222
,
,







   vurvurr ,,

Change of coordinates from u,v to θ,φ
 
 vuvv
vuuu
,
,

The coordinates are related by


















v
u
vv
uu
vd
ud
vu
vu
      I
vd
ud
GF
FE
vdud
vd
ud
vv
uu
GF
FE
vu
vu
vdud
vd
ud
GF
FE
vdudI
vu
vu
vv
uu





















































td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
rd
td
rd
Ld
2/12/1
2/1
























































































































vu
vu
JFGE
vv
uu
FGE
vv
uu
GF
FE
vu
vu
GF
FE
FGE
vu
vu
vu
vu
vv
uu
,
,
detdetdetdetdet 22
**
**
2
Arc Length on a Path on the Surface and Surface Area are Invariant of the Coordinates:
First Fundamental Form is Invariant to Coordinate Transformation
Table of Contents

60
SOLO
vu
vu
rr
rr
N 




 vur ,


rd 2
rd
r
udru

vdrv

d
Nd
P
O
 vudur ,

Second Fundamental Form: NdrdII :
   
     
22
2
2
2
2
:
vdNvdudMudL
vdNrvdudNrNrudNr
vdNudNvdrudrNdrdII
N
vv
M
uvvu
L
uu
vuvu





  




vdNudNNdNNdNN vu

 01
 
 
 









NrNrNrNrNr
vd
d
NrNrNrNrNr
ud
d
Nr
vuvuvuvuu
uuuuuuuuu
u



0
0
0
 
 
 









NrNrNrNrNr
vd
d
NrNrNrNrNr
ud
d
Nr
vvvvvvvvv
vuuvuvvuv
v



0
0
0

61
SOLO
vu
vu
rr
rr
N 




 vur ,


rd 2
rd
r
udru

vdrv

d
Nd
P
O
 vudur ,

      2
2
2
: vdNrvdudNrNrudNrNdrdII
N
vv
M
uvvu
L
uu


  




NrNr uuuu

 NrNr vuuv








NrNr
NrL
uuuu
uu











uvvu
vuuv
vuvu
NrNrM
NrNr
NrNr









NrNr
NrN
vvvv
vv


NrNr vuvu

 NrNr vvvv


22
2: vdNvdudMudLNdrdII 
NrL uu


NrM vu


NrN vv



62
SOLO
vu
vu
rr
rr
N 




 vur ,

O
udru

vdrv

rd
       
   
   33
3
3223
22
33
3
32
,33
6
1
2
2
1
,
6
1
2
1
,,,
vdudOvdrvdudrvdudrudr
vdrvdudrudrvdrudr
vdudOrdrdrdvurvdvudurvur
vvvvuvvuuuuu
vvvuuuvu






 
   
    IINvdudOvdNvdudMudL
NvdudOvdNrvdudNrudNr
NvdudONrdNrdNrdNr
vvvuuu
2
1
,2
2
1
,2
2
1
,
6
1
2
1
22
2
22
22
2
22
33
3
32
0









63
SOLO
N

N

N

(i) Elliptic Case (ii) Hyperbolic Case (iii) Parabolic Case
02
MNL 02
MNL
0
&0
222
2


MNL
MNL

64
SOLO
Differential Geometry in the 3D Euclidean Space (continue – 6a)
 vur ,

vdrv

P
O
N

1nr

2nr

udru

2
M
1
M
02
 MNL
Dupin’s Indicatrix
N

1n
r

2n
r

P
2
M
1
M
02
 MNL
N

1nr
2nr

P
1M
2M
0
0
222
2


MNL
MNL
http://www.mathcurve.com/surfaces/inicatrixdedupin/indicatrixdedupin.html
Pierre Charles François
Dupin
1784 - 1873
We want to investigate the curvature propertiesat a point P.
    IINvdudOvdNvdudMudLNr
2
1
,2
2
1 22
2
22


The expression
12
2
221
2
1
 xNxxMxL
was introduced by Charles Dupin in 1813 in “Développments
de géométrie”, to describe the local properties of a surface.
http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html

65
SOLO
N

N

(iv) Planar Case
0 MNL
   
 3223
33
3
3223
6
1
,33
6
1
vdDvdudCvdudBudA
vdudOvdrNvdudrNvdudrNudrNNr vvvvuvvuuuuu



DxCxBxA  23
has 3 real roots
Monkey Saddle
DxCxBxA  23
has one real root

66
SOLO

 
 vuvv
vuuu
,
,



















v
u
vv
uu
vd
ud
vu
vu
         2222
22 uuuuuvvuuvuuuuuu vNvuMuLNvrvururNrL 

    vuvuvuvuvuvvvuvuvuuvvuuuvu vvNvuuvMuuLNvvrvuruvruurNrM 

         2222
2 vvvvvvvvvuvvvvuvuuvv vNvuMuLNvruvrvururNrN 

Unit Normal Vector to the surface at P
vu
vu
vu
vu
rr
rr
rr
rr
N 








:
uvuuvuu
vrur
u
v
r
u
u
rr







 vvvuvuv
vrur
v
v
r
v
u
rr








      II
vd
ud
NM
ML
vdud
vd
ud
vv
uu
NM
ML
vu
vu
vdud
vd
ud
NM
ML
vdudII
vu
vu
vv
uu





















































Second Fundamental Form is Invariant (unless the sign) to Coordinate Transformation
Table of Contents

67
SOLO
N

Osculating
Plane of C
at P
Principal Normal
Line of C at P
Surface
t

P
k

n1
 vur ,

Normal Curvature
- Length differential  2/1
rdrdrdsd 
    tvturr ,

Given a path on a surface of class
Ck ( k ≥ 2) we define:
td
rd
td
rd
sd
rd
t /: 
 - unit vector of path C at P
(tangent to C at P)
td
rd
td
td
sd
td
k /: 

- curvature vector of path C at P



 

curvatureofradius
nn
nnk
sd
td
k

111
1
1
1

 NNkkn

: - normal curvature vector to C at P
 
   /coscos1
:


kNnk
Nkkn


- normal curvature to C at P

68
SOLO
N

Osculating
Plane of C
at P
Principal Normal
Line of C at P
Surface
t

P
k

n1
 vur ,

Normal Curvature (continue – 1)
N
Because C is on the surface, is on the tangent
plan normal to .
t

  td
Nd
tN
td
td
td
Nd
tN
td
td
Nt
td
d
Nt






 00
and
        vdrudrvdrudrvdNudNvdrudr
td
rd
td
rd
td
Nd
td
rd
td
rd
td
Nd
td
rd
td
rd
td
Nd
t
td
rd
N
td
td
N
sd
td
Nkk
vuvuvuvu
n


















/
/
///
2
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
td
vd
G
td
vd
td
ud
F
td
ud
E
td
vd
N
td
vd
td
ud
M
td
ud
L
kn













































































2
2
2
2
2
2
2
2
22
22
22
22

69
SOLO
Normal Curvature (continue – 2)
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
td
vd
G
td
vd
td
ud
F
td
ud
E
td
vd
N
td
vd
td
ud
M
td
ud
L
kn













































































2
2
2
2
2
2
2
2
22
22
22
22
- kn is independent on dt therefore on C.
- kn is a function of the surface parameters L, M, N, E, F, G
and of the direction .
vd
ud
- Because I = E du2 + 2 F du dv + G dv2 > 0 → sign kn=sign II
- kn is independent on coordinates since I and II are independent.
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
Table of Contents

70
SOLO
Principal Curvatures and Directions
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
kn




























2
2
2
2
2
2
22
22
- kn is a function of the surface parameters L, M, N, E, F, G and of the direction .vd
ud
Let find the maximum and minimum of kn as functions of the directions d u/ d v.
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
If this occurs for d u0/ d v0 we must have:
       
0&0
00
00
0000
00
00 ,
2
,,
2
,










vdud
vdvd
vdud
n
vdud
udud
vdud
n
I
IIIIII
v
k
I
IIIIII
u
k
   
  
   
  
0&0
00
00
00
00
00
00
00
00
00
00
,
,,
0
,
,,
0


















vdud
vdnvd
vdud
vdvd
vdud
n
vdud
udnud
vdud
udud
vdud
n
IkII
I
II
III
v
k
IkIII
I
II
II
u
k
Multiply by I and use
 00 ,
0
vdud
n
I
II
k 

71
SOLO
Principal Curvatures and Directions (continue – 1)
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
 
  
 
  
0&0
00
00
00
00
00
00
,
,
0
,
,
0






vdud
vdnvd
vdud
n
vdud
udnud
vdud
n
IkII
v
k
IkII
u
k
22
2: vdNvdudMudLNdrdII 
22
2: vdGvdudFudErdrdI 
00
220
vdFudEI ud
 00
220
vdGudFI vd

00
220
vdMudLII ud
 00
220
vdNudMII vd

 
  
0
00
00
00
,
,
0



vdud
udnud
vdud
n
IkII
u
k
 
  
0
00
00
00
,
,
0



vdud
vdnvd
vdud
n
IkII
v
k
    00000 0
 vdFudEkvdMudL n
    00000 0
 vdGudFkvdNudM n

72
SOLO
We found:
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
   
   





0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
or:






















0
0
0
0
00
00
vd
ud
GkNFkM
FkMEkL
nn
nn
This equation has non-trivial solution if:
0det
00
00











GkNFkM
FkMEkL
nn
nn
or expending:       02 222
00
 MNLkMFLGNEkFGE nn

73
SOLO
Study of the quadratic equation:
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
The discriminant of this equation is:
      02 222
00
    222
42 MNLFGEMFLGNE 
   
2
22
222
2
2
2222
2
2
22222424 










 





 

E
LF
LG
E
LF
MFLGNEENLLFMELF
E
FGE
LFMELFME
E
FGE
    NLFNLGE
E
MLF
LMGF
E
LF
E
LGF
LFME
E
F
LGNELFME
E
FGE 2
3
2
24222
2
2
2
44884424 










 

E
LGF
LG
E
LGF
LMGFLG
NLGE
E
LF
E
MLF
E
LGF
NLF
E
LF
22
22
22
22
2
24322
2
2
24
84884
488444


    
    024
42
2
2
2
0
2
222





















LFME
E
F
LGNELFME
E
FGE
MNLFGEMFLGNE


74
SOLO
Study of the quadratic equation (continue – 1):
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
The discriminant of this equation is:
      02 222
00
    
    024
42
2
2
2
0
2
222





















LFME
E
F
LGNELFME
E
FGE
MNLFGEMFLGNE

The discriminant is greater or equal to zero, therefore we always obtain two real solutions
that give extremum for kn: 21
, nn
kk
Those two solutions are called Principal Curvatures and the corresponding two directions
are called Principal Directions    2211 ,,, vdudvdud
0&0  LGNELFME
G
N
F
M
E
L

The discriminant can be zero if:     02&0  LFME
E
F
LGNELFME
In this case:
G
N
F
M
E
L
vdGvdudFudE
vdNvdudMudL
kn



 22
22
2
2
This point in which kn is constant
in all directions is called an
Umbilical Point.

75
SOLO
Gaussian and Mean Curvatures
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
Rewrite the equation:
      02 222
00
as:
 
 
 
 
0
2
2
2
2
2
00







FGE
MNL
k
FGE
MFLGNE
k nn
We define:
 
 2
2
: 21
FGE
MFLGNE
kkH nn



 
 2
2
21
:
FGE
MNL
kkK nn



Mean Curvature
Gaussian Curvature
Karl Friederich Gauss
1777-1855

76
SOLO
Gaussian and Mean Curvatures (continue – 1)
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
 
 2
2
21
:
FGE
MNL
kkK nn


 Gaussian Curvature

 
 vuvv
vuuu
,
,



















v
u
vv
uu
vd
ud
vu
vu
    II
vd
ud
NM
ML
vdud
vd
ud
vv
uu
NM
ML
vu
vu
vdudII
vu
vu
vv
uu







































We found:     I
vd
ud
GF
FE
vdud
vd
ud
vv
uu
GF
FE
vu
vu
vdudI
vu
vu
vv
uu

































































vu
vu
vv
uu
vv
uu
GF
FE
vu
vu
GF
FE


























vu
vu
vv
uu
vv
uu
NM
ML
vu
vu
NM
ML
 
2
2
2
2
detdetdetdet












































vu
vu
vu
vu
vv
uu
FGE
vv
uu
GF
FE
GF
FE
FGE
 
2
2
2
2
detdetdetdet












































vu
vu
vu
vu
vv
uu
MNL
vv
uu
NM
ML
NM
ML
MNL
Therefore: invariant to coordinate changes
 
 
 
 2
2
2
2
21
:
FGE
MNL
FGE
MNL
kkK nn







77
SOLO
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2CStart with:    
   





0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
rewritten as :






















0
01
00000
0000
n
kvdGudFvdNudM
vdFudEvdMudL
that has a nontrivial solution (1,-kn0) only if:
0det
0000
0000








vdGudFvdNudM
vdFudEvdMudL
or:       0
2
000
2
0
 vdNFMGvdudNEGLudMEFL
or:
      0
0
0
2
0
0
















 NFMG
vd
ud
NEGL
vd
ud
MEFL

78
SOLO
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
We obtained:
This equation will define the two Principal Directions 2211 21
& vdrudrrvdrudrr vunvun


      
 
 
 
 
021
21
2
2
1
1
2
2
1
1
2112212121





























vdvdG
MEFL
NEGL
F
MEFL
NFMG
E
vdvdG
vd
ud
vd
ud
F
vd
ud
vd
ud
E
vdvdrrvdudvdudrrududrrrr vVvuuunn

      0
0
0
2
0
0
















 NFMG
vd
ud
NEGL
vd
ud
MEFL
From the equation above we have:
 
 
 
 MEFL
NFMG
vd
ud
vd
ud
MEFL
NEGL
vd
ud
vd
ud






2
2
1
1
2
2
1
1
Let compute the scalar product of the Principal Direction Vectors:
The Principal Direction Vectors
are perpendicular.

79
SOLO
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
Since the two Principal Directions are orthogonal
21 21
& vdrrudrr vnun


they must satisfy the equation:
Let perform a coordinate transformation to the Principal
Direction:  vu,
      0
2
000
2
0
 vdNFMGvdudNEGLudMEFL
   21 ,0&0, vdud
or:
  0
2
1
 udMEFL
  0
2
2
 vdNFMG 0 NFMG
01
ud
0 MEFL
02
vd
0E
0G
0
0


NrM
rrF
vu
vu


at P
Definition:
A Line of Curvature is a curve whose tangent at any point has a direction
coinciding with a principal direction at that point. The lines of curvature
are obtained by solving the previous differential equation

80
SOLO
 vur ,

rd
udru

vdrv
P
O
N
1C
k
2C
k

1C
2C
Suppose (du0,dv0) is a Principal Direction, then they must satisfy the equations:
Rodriguez Formula
NrNrL uuuu


NrNrNrM vuuvvu


NrNrN vvvv


   
   





0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
   
   





0
0
0000
0000
0
0
vdrrudrrkvdNrudNr
vdrrudrrkvdNrudNr
vvvunvvuv
vuuunvuuu


uu rrE


vu rrF


vv rrG


    
    





0
0
0000
0000
0
0
vvunvu
uvunvu
rvdrudrkvdNudN
rvdrudrkvdNudN


 
 





0
0
0
0
vn
un
rrdkNd
rrdkNd


But are in the tangent plane at P since and are, and the vectors
and are independent, therefore:
rdkNd n

0
 Nd

rd

vr
 ur

00

 rdkNd n
The direction (du0,dv0) is a Principal Direction on a point on a surface if and only if
from some scalar k, and satisfy:00
vdNudNNd vu

 00 vdrudrrd vu


rdkNd

 Rodriguez Formula
We found:
Table of Contents

81
SOLO
Conjugate Directions
 vur ,

rd
udru

vdrv
P
O
N

Q
NdN


l
Let P (u,v) and Q (u+du,v+dv) neighboring points on a
surface. The tangent planes to the surface at p and Q
intersect along a straight line L. Now let Q approach P
along a given direction (du/ dv=const= PQ), then the line l
will approach a limit LC. The directions PQ and LC are
called Conjugate Directions.
Let be the normal at P and the normal at Q.N

NdN


Let the direction of LC be given by: vrurr vu 


Since LC is in both tangential planes at P and at Q we have:
  0&0  NdNrNr

     0 vdNudNvrurNdr vuvu


        0 vdvNrvduNrudvNruduNr vvvuuvuu


We found vvuvvuuu NrNNrNrMNrL

 &&
The previous relation becomes:   0 vdvNvduudvMuduL 
Given (du,dv) there is only one conjugate direction (δu,δv) given by the previous
equation.
Table of Contents

82
SOLO
Asymptotic Lines
The directions which are self-conjugate are called asymptotic directions.
becomes:
  0 vdvNvduudvMuduL 
We see that the asymptotic directions are those for which the second fundamental
form vanishes. Moreover, the normal curvature kn vanishes for this direction.
Those curves whose tangents are asymptotic directions are called asymptotic lines.
v
u
vd
ud


If a direction (du,dv) is self-conjugate than and the equation of
conjugate lines
02 22
 vdNvdudMudL
The conjugat and asymptotic lines were introduced by Charles
Dupin in 1813 in “Dévelopments de Géométrie”.
Pierre Charles François
Dupin
1784 - 1873
http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html
Table of Contents

83
Let express the cartesian coordinates (x, y, z) of any point, in a three dimensional space as
a function of three curvilinear coordinates (u1, u2, u3), where:

dr
constu 3
i

j

k

1
1
ud
u
r



2
2
ud
u
r



3
3
ud
u
r



constu 1
constu 2
curveu1
curveu2
curveu3
     
     




zyxuuzyxuuzyxuu
uuuzuuuyuuuxx
,,,,,,,,
,,,,,,,,
332211
321321321
Those functions are single valued with continuous
derivatives and the correspondence between (x,y,z)
and (u1,u2,u3) is unique (isomorphism).
kzjyixr


or
3
3
2
2
1
1
ud
u
r
ud
u
r
ud
u
r
rd











   321 ,,,, uuuzyx     321
,,,, uuuAzyxAA


Assume now that the scalars Φ and vectors are functions of local
coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3)
A

In general ( we can not assume that Φ and are functions
of position).
A

   rAAr

 ,
Table of Contents

84
 tAA

Let a vector function of a single parameter tA

Ordinary Derivative of Scalars and Vectors
The Ordinary Derivative of the Vector is defined as
     
t
tAttA
td
tAd
t


 

0
lim
 tA


t
t
A

   tAttAA


If the limit exists we say that is continuous and differentiable in t. tA

Differentiation Formulas
If are differentiable vector functions of a scalar t and φ is a differentiable
scalar of t, then
CBA

,,
  td
Bd
td
Ad
BA
td
d



  td
Bd
AB
td
Ad
BA
td
d





  td
Ad
A
td
d
A
td
d




 
  td
Cd
BAC
td
Bd
ACB
td
Ad
CBA
td
d







  td
Bd
AB
td
Ad
BA
td
d




      
















td
Cd
BAC
td
Bd
ACB
td
Ad
CBA
td
d






Table of Contents

85
Partial Derivatives of Scalar and Vectors
   321 ,,,, uuuzyx     321
,,,, uuuAzyxAA


Assume now that the scalars Φ and vectors are functions of local
coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3)
A

The partial derivatives are defined as follows
     
1
3213211
0
1
321
,,,,
lim
,,
1
u
uuuuuuu
u
uuu
u







     
2
3213221
0
2
321
,,,,
lim
,,
2
u
uuuuuuu
u
uuu
u







     
3
3213321
0
3
321
,,,,
lim
,,
3
u
uuuuuuu
u
uuu
u







     
1
3213211
0
1
321 ,,,,
lim
,,
1
u
uuuAuuuuA
u
uuuA
u







     
2
3213221
0
2
321 ,,,,
lim
,,
2
u
uuuAuuuuA
u
uuuA
u







     
3
3213321
0
3
321 ,,,,
lim
,,
3
u
uuuAuuuuA
u
uuuA
u







Higher derivatives are also defined


























































































2
3
2
1
2
31
3
1212
2
2121
2
33
2
3
2
22
2
2
2
11
2
1
2
&&
&&
u
A
uuu
A
u
A
uuu
A
u
A
uuu
A
u
A
uu
A
u
A
uu
A
u
A
uu
A


Table of Contents

86
Differentials of Vectors
3
3
2
2
1
1
ud
u
A
ud
u
A
ud
u
A
zd
z
A
yd
y
A
xd
x
A
Ad




















If     321321 111111,,,, 321
uAuAuAzAyAxAuuuAzyxAA uuuzyx 

     321321 111111111 321321
udAudAudAuAduAduAdzAdyAdxAdAd uuuuuuzyx 

  BdABAdBAd


  BdABAdBAd


  CdBACBdACBAdCBAd


        CdBACBdACBAdCBAd


then
If are differentiable vector functions of a scalar t.CBA

,,
Table of Contents

87
The Vector Differential Operator Del (, Nabla)
We define the Vector Differential Operator Del (, Nabla) in Cartesian Coordinates as:
z
z
y
y
x
x
111:









This operator has double properties: (a) of a vector, (b) of a differential
Gradient: Nabla operates on a Scalar or Vector Field
z
z
y
y
x
x
z
z
y
y
x
x
111111:























 
zz
z
A
yz
z
A
xz
z
A
zy
y
A
yy
y
A
xy
y
A
zx
x
A
yx
x
A
xx
x
A
zAyAxAz
z
y
y
x
x
A
zyx
zyx
zyx
zyx
111111
111111
111111
111111:











































a scalar
a dyadic

88
The Vector Differential Operator Del (, Nabla) (continue)
We define the Vector Differential Operator in Cartesian Coordinates as:
z
z
y
y
x
x
111:









This operator has double properties: (a) of a vector, (b) of a differential
Divergence: Nabla performs a Scalar Product on a Vector Field
  z
A
y
A
x
A
zAyAxAz
z
y
y
x
x
A zyx
zyx






















 111111:

Curl (Rotor): Nabla performs a Vector Product on a Vector Field
 
z
y
A
x
A
y
x
A
z
A
x
z
A
y
A
AAA
zyx
zyx
zAyAxAz
z
y
y
x
x
A
xyzxyz
zyx
zyx
111
111
111111:

























































Table of Contents

89
Scalar Differential
Let find the differentials of:    321 ,,,, uuuzyx 
3
3
2
2
1
1
ud
u
ud
u
ud
u
zd
z
yd
y
xd
x
d


















  rdzzdyydxxdz
z
y
y
x
x
d















 111111
Since      zyxuuzyxuuzyxuu ,,,,,,,, 332211 
We obtain rduudrduudrduud

 332211 ,,
rdu
u
u
u
u
u
ud
u
ud
u
ud
u
d



























 3
3
2
2
1
1
3
3
2
2
1
1
Comparing with we obtainrdd




























3
3
2
2
1
13
3
2
2
1
1 u
u
u
u
u
uu
u
u
u
u
u
Using the Gradient definition: z
z
y
y
x
x
111:









or
3
3
2
2
1
1
:
u
u
u
u
u
u








 in general curvilinear coordinates
Table of Contents

90
Vector Differential
Let find the differentials of:    321
,,,, uuuAzyxAA


3
3
2
2
1
1
ud
u
A
ud
u
A
ud
u
A
zd
z
A
yd
y
A
xd
x
A
Ad




















    ArdzAyAxA
z
zd
y
yd
x
xd
zdz
z
A
y
z
A
x
z
A
ydz
y
A
y
y
A
x
y
A
xdz
x
A
y
x
A
x
x
A
Ad
zyx
zyxzyx
zyx





























































111
111111
111
  ArdA
u
u
u
u
u
urdrdu
u
A
u
u
A
u
u
A
Ad





































3
3
2
2
1
13
3
2
2
1
1
rduudrduudrduud

 332211 ,,
and
3
3
2
2
1
1
:
u
u
u
u
u
u









  ArdAd

Therefore
In Cartesian Coordinates:
In General Curvilinear Coordinates using
Table of Contents

91
Vector AnalysisSOLO
    Linearity of operator
         Differentiability of operator
  BABA

 Linearity of operator
  BABA

 Linearity of operator
  AAA

  Differentiability of operator
  AAA

  Differentiability of operator

92
Vector AnalysisSOLO
     
   
   BAAB
BAAB
BABABA
BA
BA






 
     
  AAA
cbacabcba  
2


0
0 


aa

   
0
0
baabaa
A







93
Vector AnalysisSOLO
         ABBAABBABA


     BABABA B


     ABBAAB A


       
 
   ABBABABABAAB
BA
BA

  




         BAABABBABA


     BABABA BA


     ABABBA AAA


     BABABA BBB



94
Vector AnalysisSOLO
   
   
Differential Identities Summary
  BABA


  BABA


  AAA

 
  AAA

 
         ABBAABBABA


         BAABABBABA


     BAABBA


    AAA
 2

0

 
0 A

   AAAAA






 2/
2
Table of Contents

95
Curvilinear Coordinates in a Three Dimensional Space
Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space
as a function of three curvilinear coordinates (u1, u2, u3), where:

dr
constu 3
i

j

k

1
1
ud
u
r



2
2
ud
u
r



3
3
ud
u
r



constu 1
constu 2
curveu1
curveu2
curveu3
     
     




zyxuuzyxuuzyxuu
uuuzuuuyuuuxx
,,,,,,,,
,,,,,,,,
332211
321321321
Those functions are single valued with continuous
derivatives and the correspondence between (x,y,z)
and (u1,u2,u3) is unique (isomorphism).
kzjyixr


3
3
2
2
1
1
3
333
2
222
1
111
3
3
12
2
1
1
3
3
12
2
1
1
3
3
12
2
1
1
ud
u
r
ud
u
r
ud
u
r
udk
u
z
j
u
y
i
u
x
udk
u
z
j
u
y
i
u
x
udk
u
z
j
u
y
i
u
x
kud
u
z
d
u
z
ud
u
z
jud
u
y
d
u
y
ud
u
y
iud
u
x
d
u
x
ud
u
x
kzdjydixdrd












































































































or
3
3
2
2
1
1
ud
u
r
ud
u
r
ud
u
r
rd












96
Curvilinear Coordinates in a Three Dimensional Space (continue – 1)
3
3
2
2
1
1
ud
u
r
ud
u
r
ud
u
r
rd











Let define: 3,2,1:
1



 i
u
r
r iu


If and are linear independent (i.e. if and only if
αi = 0 i=1,2,3) then they form a base of the space E3.
21
, uu
rr

3u
r

0
3
1

i
ui i
r
We have also:   3,2,1,,1
 irdzyxuud i 
We can write:         3
1
2
1
1
11
1 321
,, udurudurudurrdzyxuud uuu 

Because du1, du2, du3 are independent increments the precedent equation requires:
001 111
321
 ururur uuu

Similarly by multiplying by and we obtain:rd
 2
u 3
u








ji
ji
uru
u
r j
i
j
u
j
i
0
1
1



Therefore and are reciprocal systems of vectors.321
,, uuu
rrr
 321
,, uuu 

dr
constu 3
i

j

k

1
1
ud
u
r



2
2
ud
u
r



3
3
ud
u
r



constu 1
constu 2
curveu1
curveu2
curveu3

97
We proved that reciprocal systems of vectors are related by:
and are reciprocal systems of vectors.321
,, uuu
rrr
 321
,, uuu 
     321
21
321
13
321
32
,,
,
,,
,
,,
321
uuu
uu
uuu
uu
uuu
uu
rrr
rr
u
rrr
rr
u
rrr
rr
u 











     321
21
321
13
321
32
,,
,
,,
,
,, 321
uuu
uu
r
uuu
uu
r
uuu
uu
r uuu










and    1,,,, 321
321
 uuurrr uuu

or
1
,,
,,
,,
,, 321
321
333
222
111
333
222
111



















































zyx
uuu
J
uuu
zyx
J
z
u
y
u
x
u
z
u
y
u
x
u
z
u
y
u
x
u
u
z
u
y
u
x
u
z
u
y
u
x
u
z
u
y
u
x
where is the Jacobian of x,y,z with respect to u1, u2, u3.








321 ,,
,,
uuu
zyx
J Carl Gustav Jacob Jacobi
1804 - 1851

dr
constu 3
i

j

k

1
1
ud
u
r



2
2
ud
u
r



3
3
ud
u
r



constu 1
constu 2
curveu1
curveu2
curveu3

98
  grrr
u
z
u
y
u
x
u
z
u
y
u
x
u
z
u
y
u
x
uuu
zyx
J uuu



























321
,,det:
,,
,,
333
222
111
321

If is nonsingular the transformation from x,y,z to u1, u2, u3 is unique.







321 ,,
,,
uuu
zyx
J

dr
constu 3
i

j

k

1
1
ud
u
r



2
2
ud
u
r



3
3
ud
u
r



constu 1
constu 2
curveu1
curveu2
curveu3
g
rr
u
g
rr
u
g
rr
u
uuuuuu 211332 321
,,







     211332
321
,, uugruugruugr uuu


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