This paper develops the theory of differential forms introduced by E. Cartan. We consider the integral of a completely skew symmetric contravariant tensor
T
ab. . . s
of rank
p
over a
q
-dimensional differentiable manifold
M
in a Riemannian space
R
n
of
n
dimensions (
n
=
p
+
q
). We prove that the integral can be expressed in the form ∫
T
ab. . . s
χ
ab. . . sʼ
, where
χ
ab. . . s
= ∂(
α
1
,
α
2
, . . . ,
α
p
)/∂(
x
a
,
x
b
, . . . ,
x
s
) (─1)
p
, and
α
1
,
α
2
, . . . ,
α
p
are the characteristic functions of
p
(
n
-dimensional) domains
A
1
,
A
2
, . . . ,
A
p
, whose boundaries intersect in
M
. The integral is taken over the whole of the space
R
n
. The tensor
χ
ab . . . s
is the ‘characteristic tensor’ of
M
. In the representation by a Grassmann algebra
χ
ab. . . s
is expressed by a ‘characteristic form’
χ
= (— 1)
p
d
α
1
⋀d
α
2
⋀. . . ⋀d
α
p
, and the dual of
T
ab. . . s
by a form
U
of rank
q
. If d denotes the exterior derivative,
γ
the characteristic form of
C
and
ω
the characteristic form of Ω ≡ ∂
C
, then
ω
= -d
γ
. Stokes’s theorem is proved in the form — ∫d
γ
⋀
U
= ∫
γ
⋀d
U
. In the case of three-dimensional Euclidean space very simple proofs can be given by these results which form the basic theorems of ‘continuous vector analysis’ as introduced by Weyl (1940) and H. Cartan (1949).
Cauchy’s Flux Theorem in Light of Geometric Integration Theory