Space-Time Curves and Surfaces - Gauge-institute.org

Gauge Institute Journal,
H. Vic Dannon
Space-Time Curves and
Surfaces
H. Vic Dannon
vic0@comcast.net
June, 2013
Abstract
The Differential Geometry of Space-Time
Curves and surfaces, requires the Cross-Product of 4 -
vectors.
A 3-space curve
xs ()
, parametrized by its arc-length s , is
characterized by two curvature functions κ () s , and τ () s . At
each point along xs (), the Tangent unit vector
x '( s)
Ts () ≡ 1x
'( s)
x '( s)
=
,
and the Normal unit vector
T '( s)
Ns () ≡ = 1T
,
'( s)
T '( s)
define the Binormal unit vector
1x 1y 1
z
Bs () ≡ T× N = T T T
1 2 3
N N N
1 2 3
.
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Gauge Institute Journal,
H. Vic Dannon
The Derivatives T '( s)
, N '( s)
, B'( s)
, in the Frenet Frame, T ,
N
, B
, satisfy the Frenet Differential Equation.
A 4-space curve xs (), parametrized by its arc-length s , is
characterized by three curvature functions κ () s , τ()
s , and
ω () s . At each point along xs (), the Tangent unit vector
x '( s)
Ts () ≡ 1x
'( s)
x '( s)
=
,
and the Normal unit vector
T '( s)
Ns () ≡ = 1T
,
'( s)
T '( s)
define the Binormal unit vector
Bs () ≡ T×
N,
which is believed to have six components.
Clearly, the Curve Equations cannot be written as a mixture
of 4-vectors, and 6-vectors.
Thus, the Fundamental Differential Equations of Space-time
Curves were not developed.
A 3-space surface xuv (,), with components along the axes
xyz , , ,
is parametrized by two linearly independent families
of curves, u , and v .
At each point on xuv (,), the Tangent Vector along u
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Gauge Institute Journal,
H. Vic Dannon
x
(,) u v x
≡∂ (,) u v ,
and the Tangent Vector along v
x
(,) u v x
≡∂ (,) u v ,
u
v
define the Normal to the surface,
x
u(,) u v x
v(,) u v n
× ≡ (,) u v .
The Derivatives xuu (,) u v , x
uv(,) u v x
vu(,)
u v
= , and xvv (,) u v in
the Gauss Frame, x , x
, x × x , satisfy the Gauss
u
v
Differential Equations.
The Derivatives ∂ , and ∂ in the Gauss Frame satisfy
u n
v n
the Weingarten Differential Equations.
A Space-time surface xuv (,), with components along the axes
u
v
u
v
xyzt , , ,
is parametrized by two linearly independent families
of curves, u , and v .
At each point on xuv (,), the Tangent Vector along u
x
(,) u v x
≡∂ (,) u v ,
and the Tangent Vector along v
x
(,) u v x
≡∂ (,) u v ,
define the Normal to the surface,
x
(,) u v x
× (,) u v ,
which is believed to have six components.
u
v
u
u
v
v
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Gauge Institute Journal,
H. Vic Dannon
Clearly, the Space-time Differential Equations cannot be
written as a mixture of 4-vectors, and 6-vectors.
Thus, the Fundamental Differential Equations of Space-time
Surfaces were not developed.
Recently, we showed that the cross product of 4-vectors is a
4-vector, and supplied the correct formula for it.
Applying this formula to Space-time Vectors, we obtain the
Differential Equations for Space-time Curves, for Space-time
surfaces, and for the Normals to a surface in Space time.
Keywords: Infinitesimal, Infinite-Hyper-real, Hyper-real,
Cardinal, Infinity. Non-Archimedean, Calculus, Limit,
Continuity, Derivative, Integral, Gradient, Divergence, Curl,
Space-time Vectors Fields
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30.
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Gauge Institute Journal,
H. Vic Dannon
Contents
Introduction
1. 4-Space Curl
2. Cross Product of 4-Vectors
3. Space-time Curve Frame
4. Space-time Curve Equations
5. Space-time Surface Frame
6. Space-time Surface Curvatures
7. Space-time Surface Equations, and Christoffel Curvatures
8. Space-time Normals Equations, and Curvatures
References.
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Gauge Institute Journal,
H. Vic Dannon
Introduction
0.1 Space-time Curve
A 3-space curve xs (), parametrized by its arc-length s , is
characterized by two curvature functions κ () s , and τ () s . At
each point along xs (), the Tangent unit vector
x '( s)
Ts () ≡ 1x
'( s)
x '( s)
=
,
and the Normal unit vector
T '( s)
Ns () ≡ = 1T
,
'( s)
T '( s)
define the Binormal unit vector
1 1 1
Bs () ≡ T× N = T T T
N N N
x y z
1 2 3
1 2 3
The Derivatives T '( s)
, N '( s)
, B'( s)
, in the Frenet Frame T ,
N , B , satisfy the Frenet Differential Equations.
⎡T '( s) ⎤ ⎡ 0 κ( s) 0 ⎤ ⎡T( s)
⎤
N '( s) κ( s) 0 τ( s)
= −
N( s)
.
B'( s) 0 −τ( s) 0
⎢ ⎥ ⎢⎣
⎥⎦
⎢B( s)
⎣ ⎦ ⎣ ⎥⎦
The Differential Geometry of Space-time Curves requires the
.
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Gauge Institute Journal,
H. Vic Dannon
Cross-Product of 4-vectors.
A space-time curve xs (), parametrized by its arc-length s , is
characterized by three curvature functions κ () s , τ()
s , and
ω () s . At each point along xs (), the Tangent unit vector
x '( s)
Ts () ≡ 1x
'( s)
x '( s)
=
,
and the Normal unit vector
T '( s)
Ns () ≡ = 1T
,
'( s)
T '( s)
define the Binormal unit vector
Bs () ≡ T×
N,
provided the cross product of space-time vectors is a spacetime
vector.
It is not self evident how the Spatial Cross-Product, with its
1 x , 1y , and 1z components may be generalized to a Spacetime
Cross-product with 1 x , 1y ,1z , and 1t
components.
For instance, we can add a column to obtain
1x 1y 1z 1t
Ax Ay Az At
,
Bx By Bz Bt
? ? ? ?
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Gauge Institute Journal,
H. Vic Dannon
but what will be a fourth raw of that 4 × 4 determinant?
It is less evident how the fact that there are six terms of the
form
AB
− AB
i j j i
that determine the 4-space Cross Product, lead to the belief
that the 4-Space cross product is six dimensional.
Clearly, the Curve Differential Equations cannot be written
as a mixture of 4-vectors, and 6-vectors.
Thus, the Fundamental Differential Equations of Space-time
Curves were not developed.
0.2 Space-time Surface
A 3-space surface xuv (,), with components along the axes
xyz , , ,
is parametrized by two linearly independent families
of curves, u , and v .
At each point on xuv (,), the Tangent Vector along u
x
(,) u v x
≡∂ (,) u v ,
and the Tangent Vector along v
x
(,) u v x
≡∂ (,) u v ,
define the Normal to the surface,
x
(,) u v x
(,) u v n
× ≡ (,) u v .
u
u
v
v
u
v
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Gauge Institute Journal,
H. Vic Dannon
The Derivatives xuu (,) u v , x
uv(,) u v x
vu(,)
u v
= , and xvv (,) u v in
the Gauss Frame, x , x
, x × x , satisfy the Gauss
u
v
Differential Equations.
The Derivatives ∂ , and ∂ in the Gauss Frame satisfy
u n
the Weingarten Differential Equations.
v n
The Differential Geometry of Space-time Surfaces requires
the Cross-Product of 4-vectors.
A Space-time surface xuv (,), with components along the axes
u
v
xyzt , , ,
is parametrized by two linearly independent families
of curves, u , and v .
At each point on xuv (,), the Tangent Vector along u
x
(,) u v x
≡∂ (,) u v ,
and the Tangent Vector along v
x
(,) u v x
≡∂ (,) u v ,
define the Normal to the surface,
x
(,) u v x
× (,) u v ,
which is believed to have six components.
u
v
u
Clearly, the Surface Differential Equations cannot be
written as a mixture of 4-vectors, and 6-vectors.
u
v
v
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Gauge Institute Journal,
H. Vic Dannon
Thus, the Fundamental Differential Equations of Space-time
Surfaces were not developed.
0.3 Correct Space-time Cross Product
Recently, we showed that the cross product of 4-vectors is a
4-vector, and supplied the correct formula for it.
In [Dan4], we showed that the 4-space curl is a 4-vector, and
supplied the correct formula for it. In [Dan5], we obtained
the Cross-Product of 4-vectors, for space-time
Electromagnetic Vector Fields.
Applying this formula to Space-time Vectors, we obtain the
Differential Equations for Space-time Curves, for Space-time
surfaces, and for the Normals to a surface in Space time.
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Gauge Institute Journal,
H. Vic Dannon
1.
Curl of 4-Vectors
Let Pxyzt (,,,), Qxyzt (,,,), Rxyzt (,,,), and Sxyzt (,,,) be hyperreal
differentiable functions, defined on an infinitesimal area
element dS .
the unit vectors
dS projects onto six 2-planes generated by
1 x ,1y ,1z , and 1t .
x-y projection with area dxdy and normal 1x
× 1y
= 1 z
y-z projection with area dydz and normal 1y
× 1z
= 1 t
z-t projection with area dz dt and normal 1z
× 1t
= 1 x
t-x projection with area dt dx and normal 1t
× 1x
= 1 y
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Gauge Institute Journal,
H. Vic Dannon
t-y projection with area dtdy and normal
1t × 1
y
= (1y × 1
z) × 1y = 1
z
(1y ⋅1 y) −1 y(1z ⋅ 1
y) = 1
z
1 0
z-x projection with area dzdx and normal
1z × 1
x
= (1x × 1
y) × 1x = 1
y(1x ⋅1 x) −1 x
(1y ⋅ 1
x) = 1
y
1 0
The projected areas are
dSx = 1x ⋅ 1ndS = dzdt ,
dSy = 1y ⋅ 1ndS = dtdx + dzdx ,
dSz = 1z ⋅ 1ndS = dxdy + dtdy ,
dS = 1 ⋅ 1 dS = dydz
t t n
The projections areas are walls of a box with vertex at
( x0, y0, z0, t0)
and sides dx , dy , dz , and dt .
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Gauge Institute Journal,
H. Vic Dannon
Given positive orientation of a right hand system,
⎡Pxyzt
(,,,) ⎤ 1
∇× = 1 Pxyzt ( , , , )1 ⋅ 1 dl+ Qxyzt ( , , , )1 ⋅ dl
Qxyzt (,,,)
∫ y
1l
,
⎢⎣
⎥⎦
z x l
dxdy ∂ ( dS )
z
=
∂
x
∂
Pxyzt (,,,) Qxyzt (,,,)
y
x , y , z , t
0 0 0 0
1
z
⎡Qxyzt
(,,,) ⎤ 1
∇× = 1 Qxyzt ( , , , )1 ⋅ 1 dl+ Rxyzt ( , , , )1 ⋅ dl
Rxyzt (,,,)
∫ z
1l
,
⎢⎣
⎥⎦
t y l
dydz ∂ ( dS )
t
=
∂
y
∂
Qxyzt (,,,) Rxyzt (,,,)
z
x , y , z , t
0 0 0 0
1
t
⎡Rxyzt
(,,,) ⎤ 1
∇× = 1 Rxyzt ( , , , )1 ⋅ 1 dl+ Sxyzt ( , , , )1 ⋅ dl
Sxyzt (,,,)
∫ t
1l
,
⎢⎣
⎥⎦
x z l
dzdt ∂ ( dS )
x
=
∂
z
∂
Rxyzt (,,,) Sxyzt (,,,)
t
x , y , z , t
0 0 0 0
1 .
x
⎡Sxyzt
(,,,) ⎤ 1
∇× = 1 Sxyzt ( , , , )1 ⋅ 1 dl+ Pxyzt ( , , , )1 ⋅ dl
Pxyzt (,,,)
∫ x
1l
,
⎢⎣
⎥⎦
y t l
dtdx ∂ ( dS )
y
=
∂
t
∂
Sxyzt (,,,) Pxyzt (,,,)
x
x , y , z , t
0 0 0 0
1
y
⎡Sxyzt
(,,,) ⎤ 1
∇× = 1 Sxyzt ( , , , )1 ⋅ 1 dl+ Qxyzt ( , , , )1 ⋅ dl
Qxyzt (,,,)
∫ y
1l
,
⎢⎣
⎥⎦
z t l
dtdy ∂ ( dS )
z
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Gauge Institute Journal,
H. Vic Dannon
=
∂
t
∂
Sxyzt (,,,) Qxyzt (,,,)
y
x , y , z , t
0 0 0 0
1
z
⎡Pxyzt
(,,,) ⎤ 1
∇× = 1 Pxyzt ( , , , )1 ⋅ 1 dl+ Rxyzt ( , , , )1 ⋅ dl
Rxyzt (,,,)
∫ z
1l
,
⎢⎣
⎥⎦
y x l
dzdx ∂ ( dS )
y
=
∂
x
∂
Pxyzt (,,,) Rxyzt (,,,)
z
x , y , z , t
0 0 0 0
1
y
The 4-space Curl is the sum of the six area curls. That is,
⎡Pxyzt
(,,,) ⎤
Qxyzt (,,,)
∇× Rxyzt (,,,)
=
Sxyzt (,,,)
⎢⎣
⎥⎦
⎡R⎤ ⎡S ⎤ ⎡P⎤ ⎡P⎤ ⎡S ⎤ ⎡Q⎤
= ∇× S +∇× +∇× +∇× +∇× +∇×
P R Q Q
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢R
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎣ ⎦ ⎣ ⎥⎦
∂z ∂
t
∂t ∂
x
∂x ∂
z
∂x ∂
y
∂t ∂
y
∂y ∂
z
1x 1y 1y 1z 1z
1
R S S P P R P Q S Q Q R t
⎛ Sz
R ⎞
−
t
Pt Sx Rx P
− + −
z
= .
Qx Py Qt S
− + −
y
⎜
⎜⎝ Ry
−Q
z ⎠⎟
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Gauge Institute Journal,
H. Vic Dannon
2.
Cross-Product of 4-Vectors
2.1 The Cross-product of 4-vectors is the sum of six crossproducts
of 2-vectors, That is,
⎡A
⎤ ⎡B
⎤
x
x
A y
B ⎡
y
A ⎤ ⎡
z
B ⎤ ⎡
z
A ⎤ ⎡
t
B ⎤ ⎡
t
A ⎤ ⎡
x
B ⎤
x
A × z
B = z
A × t
B + t
A × x
B + x
A ×
z
B
⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ z ⎥⎦
A
t
B
⎢ ⎥ ⎢ t ⎥ Az A
t
At A
x
Ax A
⎣ ⎦ ⎣ ⎦ z
1x
1y
1
B B B B B B y
z t t x x z
+
⎡A ⎤ ⎡B ⎤ ⎡A ⎤ ⎡B ⎤ ⎡A ⎤ ⎡B
⎤
+ × + × + ×
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣
⎦ ⎣ ⎦
x x t t y
y
A y
B y
A y
B y
A z
B
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ z ⎥
Ax A
y
At A
y
Ay A
z
1z
1z
1
B B B B B B t
x y t y y z
⎡ Az
A ⎤
t
Bz
B
t
At Ax Ax A
z
+
Bt Bx Bx B
z
= Ax Ay At A
y
+
Bx By Bt By
Ay
Az
⎢ By
B
⎣
z ⎥
⎦
.
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Gauge Institute Journal,
H. Vic Dannon
2.2 1 × 1 = 1
x
y
1 × 1 =
y
z
z
1
1 × 1 = 1
z
t
t
x
1 × 1 = 1
t
x
y
Proof:
⎡1⎤
⎡0⎤
0 1 Ax
A
y
1 0
1x × 1y = 1z 1z
0 × 0 = = .
Bx
By
0 1
0 0
⎢⎣ ⎥⎦
⎢⎣ ⎥⎦
2.3 A× B = − B×
A
2.4 A× ( B× C) ≠ ( A⋅C) B −(
A⋅B)
C
Proof: 1 × ( 1 × 1 ) = 1 × 1 = 1
y x y y z
(1 ⋅1 )1 −(1 ⋅ 1 )1 = 1
y y x y x y
x
t
d
As () ⋅ Bs () = A '() s ⋅ Bs () + As () ⋅ B '() s
ds
2.5 { }
d
{ As () × Bs ()}
= A '() s × Bs () + As () × B '() s
ds
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Gauge Institute Journal,
H. Vic Dannon
3.
Space-time Curve Frame
Let xs () be a Curve parametrized by its arc-length s ,
xs ()
⎡ x1()
s ⎤
x2()
s
= x3()
s
x4()
s
⎢⎣
⎥⎦
,
with continuous linearly independent derivative functions,
dx()
s
x '( s)
= ,
ds
2
dxs ()
x ''( s)
= ,
2
ds
3
dxs ()
x '''( s)
= ,
3
ds
4
dxs ()
x ''''( s)
= .
4
ds
3.1 The Tangent unit vector At s ,
x '( s)
Ts () ≡ 1x
'( s)
x '( s)
=
.
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Gauge Institute Journal,
H. Vic Dannon
3.2 T '( s) ⊥ T( s)
T '( s)
is Normal to the Tangent.
Proof: Ts () ⋅ Ts () = 1,
T '( s) ⋅ T( s) + T( s) ⋅ T '( s) = 0 ,
T '( s) ⋅ T( s) = 0,
3.3 The Normal unit vector at s
Ns () ≡ 1 T s
'( )
T '() s = T '() s N()
s
3.4 The Curvature of xs () at s
κ () s = T
'() s
3.5 The First Frenet Equation
T '( s) = κ( s) N( s)
3.6 The Bi-Normal unit vector at s
Bs () ≡ Ts () × Ns ()
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Gauge Institute Journal,
H. Vic Dannon
3.7 The Tri-Normal unit vector at s
Ws () ≡ Ns () × Bs ()
3.8 { TNBW , , , } is an Orthonormal Vector System at s
Bs () × Ws () = Ts ()
Ws () × Ts () = Ns ()
3.10 T , and N
span the Osculating (=Tangent) Plane
Span{ TN , } = { λT+ μN: λ, μ ∈ }
3.11 N , and B
span the Normal Plane
Span{ NB , } = { λN+ μB: λ, μ ∈ }
3.12 B , and W span the Bi-Normal Plane
Span{ BW , } = { λB+ μW
: λ, μ ∈ }
3.13 W , and T span the Tri-Normal Plane
Span{ WT , } = { λW + μT
: λ, μ ∈ }
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Gauge Institute Journal,
H. Vic Dannon
4.
Space-time Curve Equations
The Vector Functions
T '( s)
, N '( s)
, B'( s)
, and W '( s)
are in
Span{ Ts ( ), Ns ( ), Bs ( ), Ws ( )}.
From 3.2,
4.1 T '( s) ⊥ T( s)
, and
T '( s)
has no component along Ts ().
Similarly,
4.2 N '( s) ⊥ N( s)
, and
N '( s)
has no component along Ns (),
B'( s) ⊥ B( s), and
B'( s)
has no component along Bs (),
W '( s) ⊥ W( s), and
W '( s)
has no component along Ws ().
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Gauge Institute Journal,
H. Vic Dannon
By 3.3, T '( s)
only component is along Ns (). That is,
4.3 T '( s)
has no component along Bs (),
T '( s) ⋅ B( s) = 0 .
T '( s)
has no component along Ws (),
T '( s) ⋅ W( s) = 0 .
T '( s)
has no component along Ts (),
T '( s) ⋅ T( s) = 0 .
and by 3.5, the Frenet Equation for T '( s)
is
T '( s) = κ( s) N( s)
,
Similarly, we obtain the Differential Equations for N '( s)
,
B'( s)
, and W '( s)
.
4.4 The component of N '( s)
along Ts () is
N '( s) ⋅ T( s) = −κ( s)
.
Proof: N ⋅ T = 0 ,
N '( s) ⋅ T( s) + N( s) ⋅ T '( s) = 0 .
κ() sNs ()
κ()
s
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Gauge Institute Journal,
H. Vic Dannon
4.5 The component of N '( s)
along Bs () is
N '( s) ⋅ B( s) = −N( s) ⋅B'( s)
.
Proof: Ns () ⋅ Bs () = 0,
N '( s) ⋅ B( s) + N( s) ⋅ B'( s) = 0 .
4.6 The Bi-Curvature (=Torsion) of xs () at s
τ () s = N '() s ⋅B()
s
4.7 The component of N '( s)
along Ws () is
N '( s) ⋅ W( s) = 0 .
Proof: By the Schmidt Orthogonalization Process
T = x ' ,
x
''
N = ∈ Span{ x ', x ''},
x ''
x ''' −( x ''' ⋅T) T −( x ''' ⋅N)
N
B = ∈ Span{ x ', x '', x '''},
x ''' −( x ''' ⋅T) T −( x ''' ⋅N)
N
N ' ∈ Span{ x ', x '', x '''} = Span{ T, N, B },
The projection of N
' on W
is zero.
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Gauge Institute Journal,
H. Vic Dannon
4.8 The Differential Equation for N '( s)
N '( s) =− κ( sT ) ( s) + τ( s) B( s)
,
4.9 The component of B'( s)
along Ts () is
B'( s) ⋅ T( s) = 0 .
Proof: B ⋅ T = 0 ,
B'
⋅ T + B ⋅ T ' = 0.
T ' ⋅ B=
0
4.10 The component of B'( s)
along Ns () is
B'( s) ⋅ N( s) = −τ( s)
.
Proof: Bs () ⋅ Ns () = 0,
B'
⋅ N + B ⋅ N ' = 0
B'
⋅ N = −B ⋅N
'.
τ()
s
4.11 The component of B'( s)
along Ws () is
B'() s ⋅ W() s = −W '() s ⋅B()
s .
Proof: Bs () ⋅ Ws () = 0,
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Gauge Institute Journal,
H. Vic Dannon
B'
⋅ W + B
⋅ W ' =
W ' ⋅B
0.
4.12 The Tri-Curvature of xs () at s
ω () s = B'() s ⋅W()
s
4.13 The Differential Equation for B'( s)
B'( s) =− τ( s) N( s) + ω( sW ) ( s)
,
4.14 The component of W '( s)
along Ts () is
W '( s) ⋅ T( s) = 0
Proof: W ⋅ T = 0,
W ' ⋅ T + W ⋅ T ' = 0.
T ' ⋅ W=
0
4.15 The component of W '( s)
along Ns () is
W '( s) ⋅ N( s) = 0 .
Proof: Ws () ⋅ Ns () = 0,
W ' ⋅ N + W ⋅ N ' = 0.
N ' ⋅ W=
0, by (4.7)
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Gauge Institute Journal,
H. Vic Dannon
4.16 The component of W '( s)
along Bs () is
W '( s) ⋅ B( s) = −ω( s)
.
Proof: Ws () ⋅ Bs () = 0,
W ' ⋅ B + W ⋅ B'
= 0.
BW ' ⋅ =ω( s)
4.17 The Differential Equation for W '( s)
W '( s) =−ω( s) B( s)
,
4.18 The Space-time Curve Differential Equations
⎡T '( s) ⎤ ⎡ 0 κ( s) 0 0 ⎤⎡T( s)
⎤
N '( s) −κ() s 0 τ() s 0
N( s)
=
B'( s) 0 τ( s) 0 ω( s)
−
B( s)
W '( s) 0 0 −ω( s) 0
⎢ ⎥ ⎢⎣
⎥⎦⎢W( s)
⎣ ⎦ ⎣
⎥
⎦
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Gauge Institute Journal,
H. Vic Dannon
5.
Space-Time Surface Frame
Let
xuv (,)
be a Space-time Surface Patch parametrized by
two linearly independent families of curves, u , and v ,
xuv (,)
⎡ x1(,)
u v ⎤
x2(,)
u v
= x3(,)
u v
x4(,)
u v
⎢⎣
⎥⎦
,
with continuously differentiable partial derivatives of any
necessary order.
5.1 Tangent along the u -Curves
At each point on xuv (,), the Tangent Vector along u is
x
(,) u v x
≡∂ (,) u v ,
and the Unit Tangent Vector along u is
∂uxuv
(,)
1
x
=
u ∂ xuv (,)
u
u
u
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Gauge Institute Journal,
H. Vic Dannon
5.2 Tangent along the v -Curves
At each point on xuv (,), the Tangent Vector along v is
x
(,) u v x
≡∂ (,) u v ,
and the Unit Tangent Vector along v is
∂vxuv
(,)
1
x
=
v ∂ xuv (,)
v
v
v
5.3 The Metric Tensor at ( uv) ,
g
ij
⎡
x ⋅x x ⋅x
(,) u v ≡
⎢x ⋅x x ⋅x
⎣
u u u v
v u v v
⎤
⎥⎦
5.4 The Inverse Metric Tensor at ( uv) ,
⎡
x
1
v
⋅xv −xu ⋅x
⎤
−
v ij
( gij
) ( u, v) = ≡ g ( u, v)
−xv ⋅xu xu ⋅x
⎣⎢
u ⎦⎥
5.5 Area of parallelogram generated by x
u
, and
Area( x , x ) = x × x
u v u v
x x g g g g
2 2
u
×
v
= det(
ij) =
11 22
−( 12)
x
u
Proof:
x
u
× x
v
2
expanded into its components by 2.1, equals
27

Gauge Institute Journal,
H. Vic Dannon
g g
11 22 12
2
− ( g )
expanded into components.
5.6 The Normal Unit Vector at ( uv) ,
xu xv xu xv
Nuv (,) ×
1
×
≡ =
xu×
x
=
v
xu
× xv
g g − ( g )
11 22 12
2
5.7 The Bi-Normal Unit Vector at ( uv) ,
Buv (,) ≡ 1 × Nuv (,)
x u
5.8 Buv (,) is in the Tangent Plane
Span{ x u, x v} = { λx u
+ μx
v
: λ, μ ∈ }.
Proof: Buv (,) is normal to Nuv (,).
5.10 The Tri-Normal unit vector at ( uv) ,
Wuv (,) ≡ Nuv (,) × Buv (,)
5.11 {1 , NBW , , } is an Orthonormal Vector System
x
u
Nuv ( , ) × Buv ( , ) = 1 ( uv , )
x u
28

Gauge Institute Journal,
H. Vic Dannon
Bus (,) × 1 (,) uv = Nuv (,)
x u
5.12 1x u
, and 1
x
v
span the Tangent Plane
Span{1 , 1 } = { λ1 + μ1 : λ, μ ∈ }
x x x x
u v u v
29

Gauge Institute Journal,
H. Vic Dannon
6.
Space-time Surface Curvatures
Since N ⋅ N = 1,
∂
⋅ N
=
u N
0
Thus,
6.1 N
u(,) u v u
N N
≡∂ ⊥ (,) u v , and
Nu (,)
u v has no component along Nuv (,),
Nv (,) u v ⊥ N(,), u v and
Nv (,)
u v has no component along Nuv (,),
Similarly,
6.2 Wu (,) u v ⊥ W(,)
u v , and
Wu (,)
u v has no component along Wuv (,),
Wv (,) u v ⊥ W(,), u v and
Wv (,)
u v has no component along Wuv (,),
Since x ⋅ N = 0,
u
x ⋅ N + x ⋅ N =
uu u u
0 .
30

Gauge Institute Journal,
H. Vic Dannon
Therefore,
6.3
x
( u, v) ⋅ N( u, v) = −x ( u, v) ⋅N ( u, v)
uu u u
x (,) uv ⋅ Nuv (,) = −x (,) uv ⋅N(,)
uv
uv u v
x (,) u v ⋅ N(,) u v = −x (,) u v ⋅N (,) u v
vu v u
x (,) u v ⋅ N(,) u v = −x (,) u v ⋅N (,) u v
vv v v
Since x ⋅ W = 0 ,
u
x ⋅ W + x ⋅ W =
uu u u
0 .
Therefore,
6.4
x
(,) u v ⋅ W(,) u v = −x (,) u v ⋅W (,) u v
uu u u
x (,) uv ⋅ Wuv (,) = −x (,) uv ⋅W(,)
uv
uv u v
x (,) u v ⋅ W(,) u v = −x (,) u v ⋅W (,) u v
uv v u
x (,) u v ⋅ W(,) u v = −x (,) u v ⋅W (,) u v
vv v v
Since N ⋅ W = 0,
N
u
W N
⋅ + ⋅ W
u
= 0 .
N
W
⋅ + N
⋅ W
= 0
v
v
Therefore,
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Gauge Institute Journal,
H. Vic Dannon
6.5 N ( uv , ) ⋅ Wuv ( , ) = −Nuv ( , ) ⋅W( uv , )
u
N
(,) u v W
(,) u v N
⋅ = − (,) u v ⋅ W
(,) u v
v
u
v
32

Gauge Institute Journal,
H. Vic Dannon
7.
Space-time Surface Equations,
and Christoffel Curvatures
Since the Space-time Vector Functions
x
uu
,
xuv
= xvu
, x vv
,
are in Span{ x , x , N
, W
}, we have
7.1
7.2
7.3
7.4
u
v
x (,) u v x x ( x N) N ( x W)
W
1 2
uu
=Γ
11 u
+Γ
11 v
+
uu
⋅ +
uu
⋅
−x ⋅N −x ⋅W
u u u u
x (,) u v x x ( x N) N ( x W)
W
1 2
uv
=Γ
12 u
+Γ
12 v
++
uv
⋅ +
uv
⋅
−x ⋅N −x ⋅W
u v u v
x (,) u v x x ( x N) N ( x W)
W
1 2
vu
=Γ
21 u
+Γ
21 v
++
vu
⋅ +
vu
⋅
−x ⋅N −x ⋅W
v u v u
x (,) u v x x ( x N) N ( x W)
W
1 2
vv
=Γ
22 u
+Γ
22 v
++
vv
⋅ +
vv
⋅
−x ⋅N −x ⋅W
v v v v
where
k
Γ ij
are Christoffel Curvatures.
Since ∂ x = ∂ x
, the equations for x (,) u v , and
uv
vu
are the same, and
uv
xvu (,) u v
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Gauge Institute Journal,
H. Vic Dannon
k
ij
k
ji
Γ = Γ .
The Christoffel Curvatures can be written in terms of the
Metric Tensor. Multiplying each equation by x , and by x ,
The First Equation gives
x x x x x x
,
1
2
1 2
uu
⋅
u
= Γ11 u
⋅
u
+ Γ11
v
⋅
u
∂u( xu⋅xu)
g11 g12
g11
,
1 2
xuu ⋅ xv = Γ11 xu ⋅ xv + Γ11
xv ⋅xv
( )
1
∂u xu⋅xv − ∂ ( )
g
2 v
xu⋅xu
12
g22
g12 g11
u
v
7.5
7.6
g
1
2 11, u 12
g
1
1 12, u
− g
2 11, v
g22
11 2
g11g22 − ( g12)
Γ =
g
1
11 2 11, u
g
1
12
g
2
12, u
− g
2 11, v
11 2
g11g22 − ( g12)
Γ =
g
g
The Second (or Third) Equation gives
x x x x x x
,
1
2
1 2
uv
⋅
u
= Γ12 u
⋅
u
+ Γ12
v
⋅
u
∂v( xu⋅xu)
g11 g12
g11
34

Gauge Institute Journal,
H. Vic Dannon
x x x x x x
,
1
2
1 2
uv
⋅
v
= Γ12 u
⋅
v
+ Γ12
v
⋅
v
∂u( xv⋅xv)
g12 g22
g22
7.7
7.8
g
1
2 11, v 12
1
g
1 2 22, u
g 22 1
12 2 21
g11g22 − ( g12)
Γ = = Γ
g
1
11 2 11, v
g
1
2 12
g
2 22, u 2
12 2 21
g11g22 − ( g12)
Γ = = Γ
g
g
The Fourth Equation gives
,
1 2
xvv ⋅ xu = Γ22 xu ⋅ xu + Γ22
xv ⋅xu
( )
1
∂v xu⋅xv − ∂ ( )
g
2 u
xv⋅xv
11
g12
g12 g22
x x x x x x
,
1
2
1 2
vv
⋅
v
= Γ22 u
⋅
v
+ Γ22
v
⋅
v
∂v( xv⋅xv)
g12 g22
g22
7.9
g − g g
1
12, v 2 22, u 12
1
g
1
2 22, v
g 22
22 2
g11g22 − ( g12)
Γ =
35

Gauge Institute Journal,
H. Vic Dannon
7.10
g g − g
1
11 12, v 2 22, u
g
1
2 12
g
2 22, v
22 2
g11g22 − ( g12)
Γ =
36

Gauge Institute Journal,
H. Vic Dannon
8.
Space-time Normals Equations,
and Curvatures
Since the Space-time Vector Functions
are in
N u
, N v
,
Span{ xu, xv, W
}, we have
N (,) u v =Ω x +Ω x + MW
8.1
u 11 u 12 v 1
N u v x x M W
8.2
v(,)
=Ω
21 u
+Ω
22 v
+
2
Multiplying each equation by x , and by x ,
The First Equation gives
Nu ⋅ xu = Ω11( xu ⋅ xu) + Ω12(
xv ⋅xu)
8.3
g
u
11 12
N ⋅ x = Ω ( x ⋅ x ) + Ω ( x ⋅x
)
u v 11 u v 12 v v
g
g
12 22
Nu
⋅ xu
g
N ⋅ x g
12
u v 22
11 2
g11g22 − ( g12)
Ω =
v
g
37

Gauge Institute Journal,
H. Vic Dannon
8.4
g11
Nu
⋅ x
g N ⋅ x
12 u v
12 2
g11g22 − ( g12)
Ω =
u
The Second Equation gives
Nv ⋅ xu = Ω21( xu ⋅ xu) + Ω22(
xv ⋅xu)
8.5
8.6
g
11 12
Nv ⋅ xv = Ω21( xu ⋅ xv) + Ω22(
xv ⋅xv)
g
12 22
Nv
⋅ xu
g
N ⋅ x g
12
v v 22
21 2
g11g22 − ( g12)
Ω =
g11
Nv
⋅ x
g N ⋅ x
12 v v
22 2
g11g22 − ( g12)
Ω =
u
g
g
Since the Space-time Vector Functions
are in
W u
, W v
,
Span{ x
, x
, N
} , we have
u
v
W (,) u v =Λ x +Λ x −M N
8.7
u 11 u 12 v 1
W (,) u v =Λ x +Λ x −M N
8.8
v 21 u 22 v 2
38

Gauge Institute Journal,
H. Vic Dannon
The First Equation gives
Wu ⋅ xu = Λ11( xu ⋅ xu) + Λ12(
xv ⋅xu)
8.9
8.10
g
11 12
Wu ⋅ xv = Λ11( xu ⋅ xv) + Λ12(
xv ⋅xv)
g
12 22
Wu
⋅ xu
g
W ⋅ x g
12
u v 22
11 2
g11g22 − ( g12)
Λ =
g11
Wu
⋅ x
g W ⋅ x
12 u v
12 2
g11g22 − ( g12)
Λ =
The Second Equation gives
Wv ⋅ xu = Λ21( xu ⋅ xu) + Λ22(
xv ⋅xu)
8.11
g
u
11 12
Wv ⋅ xv = Λ21( xu ⋅ xv) + Λ22(
xv ⋅xv)
g
12 22
Wv
⋅ xu
g
W ⋅ x g
12
v v 22
21 2
g11g22 − ( g12)
Λ =
g
g
g
g
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Gauge Institute Journal,
H. Vic Dannon
8.12
g11
Wv
⋅ x
g W ⋅ x
12 v v
22 2
g11g22 − ( g12)
Λ =
u
8.13 Differential Equations of a Space-Time Surface
1 2
⎡
xuu(,)
u v ⎤ ⎡
Γ11 Γ11
−xu ⋅Nu −xu ⋅W
⎤
u
1 2
xuv(,)
u v
Γ12 Γ12
−xu ⋅Nv −xu ⋅W
v
x (,) 1 2
vu
u v
Γ21 Γ21
−xv ⋅Nu −xv ⋅W
⎡
u
xu
⎤
x (,) 1 2
vv
u v
Γ22 Γ22
−xv ⋅Nv −xv
⋅W
x
v
=
v
Nu(,)
u v
11 12
0 Nu
W
N
Ω Ω ⋅ Nv(,)
u v
21 22
0 N W
v
W
⎥
Ω Ω ⋅ ⎥⎣⎢ ⎥⎦
Wu
(,) u v
Λ11 Λ12
−Nu
⋅W
0
Wv
(,) u v
⎢⎣
⎥⎦
⎢Λ21 Λ22
−Nv
⋅W
0 ⎥
⎣
⎦
40

Gauge Institute Journal,
H. Vic Dannon
References
[Dan] Victor Dannon, “Integral Characterizations and the Theory of
Curves”, Proceedings of the American Mathematical Society, Volume
81, Number 4, April 1981;
[Dan1] H. Vic Dannon, “Infinitesimals” in Gauge Institute Journal
Vol.6 No 4, November 2010;
[Dan2] H. Vic Dannon, “Infinitesimal Calculus” in Gauge Institute
Journal Vol.7 No 4, November 2011;
[Dan3] H. Vic Dannon, “Infinitesimal Vector Calculus” posted to
www.gauge-institute.org December, 2011;
[Dan4] H. Vic Dannon, “4-space Curl is a 4-Vector” posted to
www.gauge-institute.org, January, 2012;
[Dan5] H. Vic Dannon, “Space-time Electrodynamics, and Magnetic
Monopoles” posted to www.gauge-institute.org, June, 2013;
[Gauss] Karl Friedrich Gauss, “General Investigations of Curved
Surfaces” Raven Press, 1965.
[Kuhnel] Wolfgang Kuhnel, “Differential Geometry Curves-Surfaces-
Manifolds” AMS, 2002.
[Lipschutz] Martin Lipschutz, “Differential Geometry” Schaum’s, 1969.
[Spiegel] Murray R. Spiegel, “Mathematical Handbook of Formulas
and Tables”, Schaum’s, 1968.
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