The Infinitesimal Circulation Theorem - Gauge-institute.org

Gauge Institute Journal,
H. Vic Dannon
The Circulation Theorem
H. Vic Dannon
vic0@comcast.net
January, 2012
Abstract The Curl Theorem involves 3 projected areas in
3-space, 6 in 4-space, 10 in 5-space, 15 in 6-space, etc.
Infinitesimal Calculus yields the Circulation Theorem. A
version of the Curl Theorem that depends on 3 areas in 3-
space, 4 areas in 4-space, 5 in 5-space, etc.
The Circulation Theorem is easier to state, and comprehend,
than the Curl Theorem.
It requires Infinitesimals, and does not exist in the Calculus
of Limits, where infinitesimals are not defined.
Thus, The Circulation Theorem is another demonstration of
the power of the Infinitesimal Calculus, and its advantage
over the Calculus of Limits.
Keywords: Infinitesimal, Infinite-Hyper-real, Hyper-real,
Cardinal, Infinity. Non-Archimedean, Calculus, Limit,
Continuity, Derivative, Integral, Gradient, Divergence, Curl,
1

Gauge Institute Journal,
H. Vic Dannon
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17;
03H15;46S20; 97I40; 97I30.
2

Gauge Institute Journal,
H. Vic Dannon
Introduction
The Curl Theorem over an infinitesimal rectangle in the
Plane says that the circulation of ⎡Pxy
(, ), Qxy (, ) ⎤
⎢⎣
⎥ around the
⎦
rectangle depends on one area
∫
infinitesimal
rectangle
∂x
∂y
P(, x y) dx + Q(, x y)
dy = dxdy
P Q
The Curl Theorem Over an infinitesimal 3-box says that the
circulation of ⎡Pxyz (,, ), Qxyz (,, ), Rxyz (,, ) ⎤
⎢⎣
⎥⎦
sides of the 3-box depends on 3 areas.
around the
∫ Pxyzdx (,, ) + Qxyzdy (,, ) + Rxyzdz (,, ) =
sides of
infinitesimal 3-box
∂x ∂y ∂z ∂x ∂y ∂z
= dxdy + dzdx + dydz
P Q R Q Q R
The Curl Theorem Over an infinitesimal 4-box says that the
circulation of
⎡Pxyzt (,,,), Qxyzt (,,,), Rxyzt (,,,), Sxyzt (,,,) ⎤
⎢⎣
⎥⎦
around the sides of the 3-box rectangle depend on 6 areas.
∫ Pxyztdx (,,,) + Qxyztdy (,,,) + Rxyztdz (,,,) + Sxyztdt (,,,) =
sides of
infinitesimal 4-box
3

Gauge Institute Journal,
H. Vic Dannon
∂z ∂t ∂t ∂x ∂x ∂z
= dzdt + dtdx + dxdz +
R S S P P R
∂x ∂y ∂t ∂y ∂y ∂z
+ dxdy + dtdy + dydz
P Q S Q Q R
The Curl Theorem Over an infinitesimal 5-box says that the
circulation of
⎡Pxyztu (,,,, ), Qxyztu (,,,, ), Rxyztu (,,,, ), Sxyztu (,,,, ), Txyztu (,,,, ) ⎤
⎢⎣
⎥⎦
around the sides of the 5-box rectangle depends on 10 areas.
The number of different areas is the number of combinations
of pairs in the space,
⎛2⎞ ⎜
=
2
⎜⎝ ⎠ ⎟
⎛3⎞ 1, ⎛ 4⎞ = 3, ⎛ 5
⎜
, , ,…
2
= 6
⎞ ⎛ 6⎞ ⎜⎝ ⎠ ⎟ ⎜2
= 10
⎜⎝ ⎠ ⎟ ⎜2
15
⎜⎝ ⎟ ⎜
=
⎠ 2
⎜⎝ ⎟ ⎠
The false belief that the 6 areas in 4-space are all
independent, lead to the definition of a 6 dimensional Curl in
4-space. But as we pointed out in [Dan4], the 4-space Curl is
4 dimensional.
While the number of circulation terms in the Curl Theorem
equals the space dimension, the number of areas depends on
the space dimension indirectly.
Here, we seek, and obtain a Circulation Theorem in which
the number of areas involved is the dimension of the space.
4

Gauge Institute Journal,
H. Vic Dannon
The Circulation Theorem is easier to state, and comprehend,
than the Curl Theorem.
For space dimension greater than 3, the Circulation
Theorem holds, while the Curl Theorem holds only if one
accepts an erroneous definition of the Curl.
For instance, in 4 dimensions the Curl is 4 dimensional, and
not 6-dimensional
The derivation, and the statement of the Circulation
Theorem requires Infinitesimals.
Consequently, the Circulation Theorem does not exist in the
Calculus of Limits, where infinitesimals are not defined.
Thus, the Circulation Theorem is another demonstration of
the power of the Infinitesimal Calculus, and its advantage
over the Calculus of Limits.
We need to use Infinitesimals, Infinitesimal Calculus, and
Infinitesimal Vector Calculus.
We have constructed infinitesimals in [Dan1], established
the Infinitesimal Calculus in [Dan2], and derived the
Infinitesimal Vector Calculus in [Dan3]. The Hyper-real
Plane, Hyper-real Vector Functions, Continuity, Derivatives,
and Hyper-real Integration are presented in [Dan3].
5

Gauge Institute Journal,
H. Vic Dannon
1.
Hyper-real Plane
We present the Hyper-real 2-Space, which is a cross product
of two Hyper-real lines.
Each 2-vector of real numbers ( αβ) , can be represented by a
Cauchy sequence of rational numbers,
so that ( rn, qn ) → ( αβ , ) .
( r1, q1),( r2, q2),( r3, q3)...
The constant sequence ( αβ , ),( αβ , ),( αβ , )... is a constant
hyper-real 2-vector.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically
decreasing to (0, 0) sequences of 2-vectors
( ι1, ο1)( ι2, ο2),( ι3, ο3)...
constitutes a family of
infinitesimal hyper-real 2-vectors.
2. The infinitesimal 2-vectors are smaller than any real
2-vector, yet strictly greater than the zero 2-vector.
3. Their reciprocals
1 1 1 1 1 1
( , ),( , ),( , ),...
ι ο ι ο ι ο
1 1 2 2 3 3
are the infinite
hyper-real 2-vectors.
6

Gauge Institute Journal,
H. Vic Dannon
4. The infinite hyper-real 2-vectors are greater than any
real 2-vector, yet strictly smaller than the infinity 2-
vector.
5. The infinite hyper-real 2-vectors with negative signs
are smaller than any real 2-vector, yet strictly greater
than ( −∞, −∞) .
6. The sum of a real 2-vector with an infinitesimal 2-
vector is a non-constant hyper-real 2-vector.
7. The Hyper-real 2-vectors are the totality of
constant hyper-real 2-vectors,
a family of infinitesimal 2-vectors, with signs that
may be ( ++ , ), ( +− , ), ( −+ , ), or ( −− , ),
a family of infinite hyper-real 2-vectors with signs
that may be ( ++ , ), ( +− , ), ( −+ , ), or ( −− , ), and
non-constant hyper-real 2-vectors.
8. The hyper-real 2-vectors constitute the Hyper-real
Plane.
9. That plane includes the real 2-vectors separated by the
non-constant hyper-real 2-vectors. Each real 2-vector is
the center of a disk of infinitesimal radius of hyper-real
2-vectors, that includes no other real 2-vector.
7

Gauge Institute Journal,
H. Vic Dannon
10. In particular, the zero 2-vector is separated from
any real 2-vector by infinitesimal 2-vectors that lie in a
disk of infinitesimal radius around the zero.
11. The Zero 2-vector is not an infinitesimal 2-vector,
because zero is not strictly greater than zero.
12. We do not add the infinity 2-vector to the hyperreal
Plane.
13. The infinitesimal 2-vectors, and the infinite
hyper-real 2-vectors, are semi-groups with respect to
addition. Neither set includes zero.
14. The hyper-real Plane is embedded in × ,
and is not homeomorphic to the real Plane. There is no
bi-continuous one-one mapping from the hyper-real
Plane onto the real plane.
15. In particular, there are no points in the real
Plane that can be assigned uniquely to the
infinitesimal hyper-real 2-vectors, or to the infinite
hyper-real 2-vectorss, or to the non-constant hyper-real
2-vectors.
∞
∞
8

Gauge Institute Journal,
H. Vic Dannon
16. No neighbourhood of a hyper-real 2-vector is
n × n
homeomorphic to an ball. Therefore, the
hyper-real plane is not a manifold.
17. The hyper-real plane is not spanned by two
elements, and it is not two-dimensional.
9

Gauge Institute Journal,
H. Vic Dannon
2.
Hyper-real Vector Function
2.1 Definition of a hyper-real function
f (, xy ) is a hyper-real function, iff it is from the hyper-real 2-
vectors into the hyper-reals.
This means that any number in the domain, or in the range
of a hyper-real f (, xy ) is either one of the following
Clearly,
real vector
real vector + infinitesimal vector
infinitesimal vector
infinite hyper-real vector
2.2 Every function from the real plane into the reals is a
hyper-real function.
10

Gauge Institute Journal,
H. Vic Dannon
3.
2-Space Path Integral
The definition of Path Integration extends to a vector of two
Hyper-real functions
3.1 2-space Path Integral Definition
the Hyper-real Path Integral of ⎡Pxy
(, ), Qxy (, ) ⎤
⎢⎣
⎥ over a path
⎦
((),()), xt yt τ ∈ [ α, β]
, is the sum of the areas
∑
t∈[ αβ , ]
{ P( x(), t y()) t dx() t + Q((), x t y()) t dy()
t }
If for any infinitesimal dt , the Integration Sum equals the
same hyper-real number, then ⎡Pxy
(, ), Qxy (, ) ⎤
⎢⎣
⎥⎦
Real Integrable over the path γ(,)
ab .
is Hyper-
Then, we call the Integration Sum the Hyper-Real Path
Integral of ⎡Pxy
(, ), Qxy (, ) ⎤ ⎥ over γ(,)
ab , and denote it by
⎢⎣
⎦
∫ Pxydx (, ) + Qxydy (, ) .
γ(,)
ab
Since there are countably many real numbers in [ αβ] , ,
5.2 The Integration Sum has countably many terms.
11

Gauge Institute Journal,
H. Vic Dannon
5.3 Continuous ⎡Pxy
(, ), Qxy (, ) ⎤ is Path-Integrable
⎢⎣
⎥⎦
If Pxy (, ), Qxy (, ),are Continuous on a domain D
Then
⎡Pxy
(, ), Qxy (, ) ⎤
⎢⎣
⎥⎦
is Path-Integrable in D
12

Gauge Institute Journal,
H. Vic Dannon
4.
Hyper-real Area Integral
4.1 Area Integral of Pxy (, ) Definition
Let
Pxy (, )
be a hyper-real function, defined on a bounded
domain in the Hyper-Real Plane.
Pxy (, ) may take infinite hyper-real values.
An area element is
dA
= dxdy ,
For each ( xy) , , there is an infinitesimal rectangular box with
base area dA = dxdy , height Pxy (, ), and volume Pxydxdy (, ) .
We form the Double Sum of all the volumes that are
enclosed between the surface of
Pxy (, ), and the Pxy (, )
domain in the plane
y= b x=
a
2 2
∑∑ Pxydxdy (, ) .
y= b x=
a
1 1
If for any infinitesimals dx , and dy , the Double Sum equals
the same hyper-real number, then
Integrable over the plain domain.
Pxy (, )
is Hyper-Real
13

Gauge Institute Journal,
H. Vic Dannon
Then, we call the Double Integration Sum the Hyper-Real
Area Integral of
Pxy (, )
over the Domain, and denote it by
y= b x=
a
2 2
∫ ∫ Pxydxdy (, ) .
y= b x=
a
1 1
If the number is an infinite hyper-real, then it equals
y= b x=
a
2 2
∫ ∫
y= b x=
a
1 1
Pxydxdy (, ) .
If the number is a finite hyper-real, then its constant part
y= b x=
a
2 2
∫ ∫
equals Pxydxdy (, ) .
y= b x=
a
1 1
The Integration Sum may take infinite hyper-real values,
such as
1
( dx )( dy )
, but may not equal to ∞ .
Since there are countably many real numbers in the plane,
4.2 The Integration Sum is countable.
4.3 Continuous Pxy (, ) is Area-Integrable
14

Gauge Institute Journal,
H. Vic Dannon
5.
3-Space Surface Integral
5.1 The Surface Area Element
A point on a surface is determined by two parameters u , and
v , so that in the xyz , , coordinate system,
At the point ruv (,),
⎡xuv
(,) ⎤
ruv (,) = yuv (,)
.
zuv (,)
⎢⎣
⎥⎦
the tangent to the surface in the direction of u is
∂r
,
∂u
the tangent to the surface in the direction of v is
∂r
,
∂v
and the normal to the tangent plane is
⎡ 1x 1y 1 ⎤
z
∂r
∂r
× = ∂ux ∂uy ∂uz
∂u
∂v
∂vx ∂vy ∂vz
⎢⎣
⎥⎦
⎛ (, yz) (, zx) (, xy)
⎞
= ∂ ,
∂ ∂ ⎜⎝∂(,) uv ∂(,) uv ∂(,)
uv ⎠ ⎟
.
15

Gauge Institute Journal,
H. Vic Dannon
The unit normal is
∂r ∂r
×
1 u v
n
= ∂ ∂
, ∂ r ∂ r
×
∂u
∂v
⎡(1 , 1 ) ⎤
x n
= (1
y, 1
n)
,
⎢(1 z, 1
n)
⎣ ⎥⎦
⎡cos(1 ,1 ) ⎤
x n
= cos(1
y, 1
n)
.
⎢ cos(1
z, 1
n)
⎣ ⎥⎦
The surface area element is
∂r
∂r
dS = ( du) × ( dv)
∂u
∂v
∂r
∂r ∂r
∂r
= × dudv = × 1
∂u
∂v
n
dudv
∂u
∂v
⎡∂(, yz)
⎤
dudv
∂(,)
uv
∂(, xz)
= dudv
∂(,)
uv
(, xy)
∂ dudv
∂(,)
uv
⎢⎣
⎥⎦
⎡dydz
⎤ ⎡
dS ⎤ ⎡1
x
x
dS ⎤
⋅
= dxdz
= dS
y
= 1y
dS
⋅
dxdy
⎢⎣
⎥ dS
⎦ ⎢⎣
z ⎥⎦
⎢ 1z
⋅ dS
⎣ ⎥⎦
5.2 Surface Integral of ϕ(,, xyz)
Definition
Let
16

Gauge Institute Journal,
H. Vic Dannon
ϕ(,, xyz) = ϕ((, xuv),(, yuv),(, zuv))
= ϕ(,),
uv
be hyper-real function, defined on a bounded surface
ruv (,)
in the Hyper-Real 3-space.
ϕ (,) uv may take infinite hyper-real values.
For each ( uv) , , there is an infinitesimal volume
ϕ(,)1 uv ⋅ dS = ϕ(,,)
xyzdydz.
We form the Double Sum
x
z= c y=
b
2 2
∑∑ ϕ(,, xyzdydz ) .
z= c y=
b
1 1
If for any infinitesimals dy , and dz , the Double Sum equals
the same hyper-real number, then
Integrable over the surface.
ϕ(,, xyz)
is Hyper-Real
Then, we call the Double Integration Sum the Hyper-Real
Surface Integral of
ϕ(,)
uv
over the surface, and denote it by
z= c y=
b
2 2
∫ ∫ ϕ(,, xyzdydz ) .
z= c y=
b
1 1
If the number is an infinite hyper-real, then it equals
z= c y=
b
2 2
∫ ∫
z= c y=
b
1 1
ϕ(,,)
xyzdydz.
17

Gauge Institute Journal,
H. Vic Dannon
If the number is a finite hyper-real, then its constant part
z= c y=
b
2 2
∫ ∫
equals ϕ(,,)
xyzdydz.
z= c y=
b
1 1
The Integration Sum may take infinite hyper-real values,
such as
1
( dy)( dz )
, but may not equal to ∞ .
Since there are countably many real numbers in the plane,
5.3 The Integration Sum is countable.
5.4 Continuous ϕ(,, xyz)
is Surface-Integrable.
18

Gauge Institute Journal,
H. Vic Dannon
6.
Plane Curl
Let Pxy (, ), and Qxy (, ), be hyper-real differentiable
functions, defined on a rectangle with vertex at
sides dx , dy .
( x0,
y
0)
and
6.1 The Area Curl
The circulation of ⎡Pxy
(, ), Qxy (, ) ⎤ along the rectangle is
⎢⎣
⎥⎦
Pxy (, )1x ⋅ 1
ldl+ Qxy (, )1y ⋅1ldl.
∫
rectangle
We define the area density of that circulation multiplied by
⎡Pxy
(, ) ⎤
1 z
as the Curl of
Qxy (, )
⎢⎣
⎥⎦
⎡Pxy
(, ) ⎤ 1
Curl
= 1
z
Pxy ( , )1x ⋅ 1
ldl+
Qxy ( , )1 1 dl
Qxy (, ) dxdy
∫ y
⋅
l
∇× ⎢⎣
⎥⎦
rectangle
19

Gauge Institute Journal,
H. Vic Dannon
⎡Pxy
(, ) ⎤ ∂x
∂
6.2 Curl y
1
Qxy (, ) =
Pxy (, ) Qxy (, )
∇× ⎢⎣
⎥⎦
x , y x , y
0 0 0 0
z
Proof:
Since Pxy (, )1 ⋅ 1dl
x
l
is nonzero when 1 is along the x axis,
l
the circulation path starts at
( x 0, y 0)
, and ends at ( x 0 , y 0 + dy)
∫
( x , y + dy)
0 0
Pxy (, )1 ⋅ 1 dl= Pxy (, )1 ⋅1dl
x l x l
rectangle ( x , y )
∫
0 0
= Px ( , y)1 ⋅ 1 dx+ Px ( , y + y)1 ⋅( −1 ) dx
0 0 x x 0 0 x x
{ }
= Px ( , y) − Px ( , y + y)
dx.
0 0 0 0
⎧
∂P
= ⎪
⎨−
∂y
⎪⎩
x , y
0 0
⎫
dy ⎪
⎬dx
.
⎪⎭
Since Qxy (, )1 ⋅ 1dl
y
l
is nonzero when 1 is along the y axis,
l
20

Gauge Institute Journal,
H. Vic Dannon
the circulation path starts at
( x + dx,
y0 ) , and ends at
0
( x , y )
0 0
∫
( x , y )
0 0
Qxy (, )1 ⋅ 1 dl= Qxy (, )1 ⋅1dl
y l x l
rectangle ( x + dx, y )
∫
0 0
⎧⎪
Q
= ⎪∂
⎨ ∂x
⎪⎩
Therefore,
= Q( x + dx, y )1 ⋅ 1 dy + P( x , y )1 ⋅( −1
) dy
0 0 y y 0 0 y y
{ }
= Qx ( + dxy , ) −Qx ( , y)
dy.
0 0 0 0
x , y
0 0
⎫
dx ⎪
⎬dy
.
⎪
⎭
⎡Pxy
(, ) ⎤ 1
Curl
= 1
z
Pxy ( , )1x ⋅ 1
ldl+
Qxy ( , )1 1 dl
Qxy (, ) dxdy
∫ y
⋅
l
∇× ⎢⎣
⎥⎦
rectangle
1
⎧
⎪ ∂P
∂Q
= − +
⎪⎩
= ( ∂ Q −∂ P)
.
1 z ⎨
dxdy ⎪ ∂y x , y
∂x
x , y
x y
1z
0 0 0 0
⎫
⎪
⎬dxdy
⎪⎭
21

Gauge Institute Journal,
H. Vic Dannon
7.
Plane Curl Theorem
Let Pxy (, ), and Qxy (, ), be hyper-real differentiable
functions, defined on a plane domain S , enclosed in the loop
∂S .
7.1 Green’s Curl Theorem
∂S
∂x
∂y
P(, x y)1x ⋅ 1
ldl + Q(, x y)1y ⋅ 1ldl = ∫∫
dxdy
Pxy (, ) Qxy (, )
∫
Proof: The sum of the { ∂ Q −∂ P dxdy over the
infinitesimal rectangles enclosed in a plane area S ,
equals the sum of the circulations
Pxy (, )1 ⋅ 1 dl+ Qxy (, )1 ⋅1dl
∫
rectangle
x
S
y
}
x l y l
over the sides of the infinitesimal rectangles enclosed in the
area S ,
Q P dxdy P (, x y
)1 1 dl Q (, x y
∂ −∂ = ⋅ + )1 ⋅1
dl
y = b x = a y = b x = a
2 2 2 2
∑∑ { } ∑ ∑
x y x l y l
y= b x= a y= b x=
a
1 1 1 1
The sum over the areas is
∫∫ { ∂xQ(, x y) −∂yP(, x y)
} dxdy .
S
22

Gauge Institute Journal,
H. Vic Dannon
The Interior path integrals of
Pxy (, )1 ⋅ 1dl, and
∫
rectangle
x
l
∫
rectangle
Qxy (, )1 ⋅ 1dl,
appear in pairs of opposite signs and cancel, leaving the
Integral over the boundary line,
P(, x y)1 ⋅ 1 dl + Q(, x y)1 ⋅ 1dl = ∂ Q −∂ P dxdy
∫ x l y l ∫∫ { x y } .
∂S
S
y
l
23

Gauge Institute Journal,
H. Vic Dannon
8.
The Infinitesimal Circulation
Theorem
Let Pxyz (,, ), Qxyz (,, ), Rxyz (,, ) be hyper-real differentiable
functions, defined on an infinitesimal area element dS .
8.1 Infinitesimal Circulation Theorem in 3-Space
∫ ∫∫
∂( dS )
P(,, x y z) dx + Q(,, x y z) dy + R(,, x y z)
dz = dPdx + dQdy + dRdz
Proof:
A point on dS is determined by two parameters u , and v ,
that define a uv plane
dS
Hence,
x
= x(,)
u v , y = y(,)
u v , z = z(,)
u v
Pxyz (,, ) = Pxu ((),(),()) yu zu = Puv (, )
Qxyz (,, ) = Qxu ((),(),()) yu zu = Quv (, )
Rxyz (,, ) = Rxu ((),(),()) yu zu = Ruv (, )
Thus,
Pxyzdx (,, ) + Qxyzdy (,, ) + Rxyzdz (,, )
=
24

Gauge Institute Journal,
H. Vic Dannon
= P(,)[ u v x du + x dv] + Q(,)[ u v y du + y dv] + R(,)[ u v z du + z dv]
u v u v u v
{ (,) (,) (,) } { (,) (,) (,) }
= Puvx + Quvy + Ruvz du+ Puvx + Quvy + Ruvz dv
Therefore,
∂( dS )
u u u v v v
∫ Pxyzdx (,, ) + Qxyzdy (,, ) + Rxyzdz (,, ) =
∫
∂( dS )
{ (,) (,) (,) } { (,) (,) (,) }
= Puvx + Quvy + Ruvz du+ Puvx + Quvy + Ruvz dv
u u u v v v
By the Curl Theorem in the plane,
=
∫∫
dS
∂
u
Puvx (,) + Quvy (,) + Ruvz (,) Puvx (,) + Quvy (,) + Ruvz (,)
u u u v v v
∂
v
dudv
Now,
∂
u
Px + Qy + Rz P(,)
u v x + Qy + Rz
u u u v v v
∂
v
dudv
=
{ ⎡ Px u v
Px vu
Qy u v
Qy vu
Rz u v
Rz vu
=
⎣
+ + + + +
− ⎡Px v u
Pxuv Qy
v u
Qyuv Rz
v u
Rz ⎤
⎣
+ + + + +
uv ⎦
dudv =
= ( P x − P x ) dudv + ( Q y − Q y ) dudv + ( R z −R z ) dudv
u v v u u v v u u v v u
∂( Px , ) ∂( Qy , ) ∂( Rz , )
∂( uv , ) ∂( uv , ) ∂( uv , )
∂( Px , ) ∂( Qy , ) ∂( Rz , )
= dudv + dudv + dudv
∂(,) uv ∂(,) uv ∂(,)
uv
= dPdx + dQdy + dRdz .
Consequently,
⎤
⎦
}
25

Gauge Institute Journal,
H. Vic Dannon
P(,, x y z) dx + Q(,, x y z) dy + R(,, x y z)
dz = dPdx + dQdy + dRdz .
∫
∂( dS )
∫∫
dS
The Proof shows that the Circulation Theorem holds for any
number of any dimensions in this form.
That is,
8.2 Infinitesimal Circulation Theorem in n-Space
∫
∂( dS )
Pi( x1,... xn)
dxi = ∫∫dPdx
i i
, for any n ≥ 2
dS
26

Gauge Institute Journal,
H. Vic Dannon
9.
Meaning of
∫∫
dS
dPdx
+ dQdy
Under positive right hand orientation, the infinitesimals are
anti-commutative. That is,
And
dxdy =−dydx .
dxdx = 0 , dydy = 0 .
dPdx + dQdy = ( P dx + P dy) dx + ( Q dx + Q dy)
dy
x y x y
= Pxdxdx + Pydydx
+ Qxdxdy + Qydydy
x
0 −dxdy
0
= Qdxdy−Pdxdy
y
27

Gauge Institute Journal,
H. Vic Dannon
10.
Meaning of
∫∫
dS
dPdx + dQdy + dRdz
Under positive right hand orientation, the infinitesimals are
anti-commutative. That is,
And
dydz
=− dzdy , dzdx =−dxdz
, dxdy =−dydx
.
dxdx = 0 , dydy = 0 , dzdz = 0.
dPdx = ( P dx + P dy + P dz)
dx
x y z
That is,
10.1 Pdy + Pdz is the diagonal in the rectangle with sides
y
z
Pdy
y
, and Pdz. That diagonal is perpendicular to dx
z
dP dx is the area of the rectangle with sides dx , and
Pdy
y
+ Pdz.
z
28

Gauge Institute Journal,
H. Vic Dannon
10.2 The Infinitesimal 3-Space Circulation Theorem implies
the Infinitesimal 3-Space Curl Theorem.
Proof:
dPdx + dQdy + dRdz = ( P dx + P dy + P dz)
dx +
x y z
+ ( Qdx+ Qdy+
Qdzdy )
x y z
+
+ ( Rdx+ Rdy+
Rdzdz )
x y z
= Pxdxdx + Pydydx
+ Pzdzdx
0
−dxdy
+ Qxdxdy + Qydydy
+ Qzdzdy
0
−dydz
+
+
+ R dxdz + R dydz + R dzdz
x y z
−dzdx
0
= ( Q − P ) dxdy + ( P − R ) dzdx + ( R −Q ) dydz
x y z x y z
29

Gauge Institute Journal,
H. Vic Dannon
11.
Meaning of
∫∫
dS
dPdx + dQdy + dRdz + dSdt
Under positive right hand orientation, the infinitesimals are
anti-commutative. That is,
dydz
dzdy
=− dzdy , dzdx =−dxdz
, dxdy =−dydx
,
=− dydz , dz dt =−dtdz
, dtdx =−dxdt
.
Therefore,
dxdx = 0 , dydy = 0 , dzdz = 0,
dtdt = 0
dPdx = ( P dx + P dy + P dz + Pdt)
dx
x y z t
11.1 Pdy Pdz Pdt is the diagonal in the box with sides
y
z
+ + t
Pdy
y
, Pdz
z
, and Pdt
t
. That diagonal is perpendicular
to dx
dP dx is the area of the rectangle with sides dx , and
Pdy y
+ Pdz z
+ Pdt t .
11.2 The Infinitesimal 4-Space Circulation Theorem implies
the Infinitesimal 4-Space Curl Theorem.
Proof:
30

Gauge Institute Journal,
H. Vic Dannon
dPdx + dQdy + dRdz + dSdt = ( P dx + P dy + P dz + Pdt)
dx +
x y z t
+ ( Q dx + Q dy + Q dz + Q dt)
dy
x y z t
+ ( R dx + R dy + R dz + R dt)
dz
x y z t
+
+
+ ( S dx + S dy + S dz + S dt)
dt
x y z t
= Pxdxdx + Pydydx
+ Pzdzdx + Pdtdx
t
0
−dxdy
+
+ Qxdxdy + Qydydy
+ Qzdzdy
+ Qtdtdy
+
0
−dydz
−dydt
+ R dxdz + R dydz + R dzdz + R dtdz +
x y z t
−dzdx
0 −dzdt
+ S dxdt + S dydt + S dzdt + S dtdt
x y z t
−dtdx
0
= ( Q − P ) dxdy + ( P − R ) dzdx + ( R −Q ) dydz
x y z x y z
+ ( P − S ) dtdx + ( S − Q ) dydt + ( S −R ) dzdt
t x y t z t
31

Gauge Institute Journal,
H. Vic Dannon
12.
Circulation Theorem
We shall state, and prove the theorem for
n = 3. The proof
extends to any
n ≥ 2
Let Pxyz (,, ), Qxyz (,, ), Rxyz (,, ), be hyper-real differentiable
functions, defined on a surface Σ , with closed boundary ∂Σ.
8.1 3-Space Circulation Theorem
∫ ∫∫
∂Σ
P(,,) x y z dx + Q(,,) x y z dy + R(,,)
x y z dz = dPdx + dQdy + dRdz
Proof: The sum of the dP dx + dQdy + dRdz over the
infinitesimal rectangles enclosed in the plane area Σ ,
equals the sum of the circulations
That is,
z= c y= b x=
a
∫ P (,,) x y z dx + Q (,,) x y z dy + R (,,) x y z dz .
sides of rectangles
bounding 3-box
2 2 2
∑∑∑ dPdx + dQdy + dRdz =
z= c1 y= b1 x=
a1
Σ
32

Gauge Institute Journal,
H. Vic Dannon
x = a y = b z = c
2 2 2
∑ ∫ ∑ ∫ ∑ ∫ .
= Pxyzdx (,, ) + Qxyzdy (,, ) +
Rxyzdz (,, )
x= a1 sides of y= b1 sides of z=
c1
sides of
rectangles rectangles rectangles
The sum over the areas is
∫∫ dPdx + dQdy + dRdz .
The Interior path integrals of
Σ
x = a y = b z = c
2 2 2
∑ ∫ ∑ ∫ ∑ ∫
Pxyzdx ( , , ) + Qxyzdy ( , , ) +
Rxyzdz ( , , )
x= a1 sides of y= b1 sides of z=
c1
sides of
rectangles rectangles rectangles
appear in pairs of opposite signs and cancel, leaving the
Integral over the boundary line,
∫ ∫∫
∂Σ
P(,,) x y z dx + Q(,,) x y z dy + R(,,)
x y z dz = dPdx + dQdy + dRdz
Σ
References
[Dan1] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal
Vol.6 No 4, November 2010;
[Dan2] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute
Journal Vol.7 No 4, November 2011;
[Dan3] Dannon, H. Vic, “Infinitesimal Vector Calculus” in Gauge
Institute Journal, December, 2011;
[Dan4] Dannon, H. Vic, “4-Space Curl is a 4-Vector” in Gauge Institute
Journal, January 2012;
33