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).