184
IX – Multivariate Differential and Integral Calculus
varying vectors h, k ∈ E, called a differential form of degree 2 and class C
p on
an open subset G of a Cartesian space E, which, for given x, is an alternating
bilinear form
31
ω(x) : (h, k) −→ ω(x; h, k)
in h, k and which, for given h, k is a C
p function of x. The case p = 2 is
sufficient for what follows.
Let B(h, k) be an alternating bilinear form (a purely algebraic notion)
and (a i ) a basis for E. Since h = h
i a i and k = k
j a j ,
B(h, k) = h
i B (a i , k) = h
i k
j B (a i , a j ) = b ij h
i k
j
with antisymmetric coefficients
b ij = B (a i , a j ) = −b ji ,
that are, therefore zero for i = j. This can be also written
B(h, k) = b ij h
i k
j =
i
h
i k
j
− h
j k
i
=
1
2
b ij
h
i k
j
− h
j k
i
.
The factor 1/2 corrects the fact that each term is written twice. In this form,
the antisymmetry is highlighted.
Applying this calculation to B = ω(x), we, therefore, get the relation
ω(x; h, k) = p ij (x)h
i k
j =
i
h
i k
j
− h
j k
i
=
(8.1)
=
1
2
p ij (x)
h
i k
j
− h
j k
i
with coefficients
p ij (x) = ω (x; a i , a j ) = −p ji (x)
(8.2)
depending on the basis (a i ) used . Note that in (1), we have been forced to
abandon Einstein’s convention in the first sum. When ω = d(p i dx
i ), p ij =
D i p j − D j p i .
In degree 2, expressions similar to p i (x)dx
i can be used. For this, given
two linear functionals u(h) = u i h
i and v(h) = v i h
i on the vector space
considered, the alternating bilinear form
u ∧ v : (h, k) −→ u(h)v(k) − u(k)v(h) = u i v j
h
i k
j
− h
j k
i
(8.3)
= (u i v j − u j v i ) h
i k
j
is called the exterior product of u and v. This is a purely algebraic notion.
31 See for example Cours d’Alg` ebre by the author, §§ 21 to 24. A differential form
of degree 2 is, therefore, an “ antisymmetric ” tensor field of type (0, 2).
IX – Multivariate Differential and Integral Calculus
varying vectors h, k ∈ E, called a differential form of degree 2 and class C
p on
an open subset G of a Cartesian space E, which, for given x, is an alternating
bilinear form
31
ω(x) : (h, k) −→ ω(x; h, k)
in h, k and which, for given h, k is a C
p function of x. The case p = 2 is
sufficient for what follows.
Let B(h, k) be an alternating bilinear form (a purely algebraic notion)
and (a i ) a basis for E. Since h = h
i a i and k = k
j a j ,
B(h, k) = h
i B (a i , k) = h
i k
j B (a i , a j ) = b ij h
i k
j
with antisymmetric coefficients
b ij = B (a i , a j ) = −b ji ,
that are, therefore zero for i = j. This can be also written
B(h, k) = b ij h
i k
j =
i
h
i k
j
− h
j k
i
=
1
2
b ij
h
i k
j
− h
j k
i
.
The factor 1/2 corrects the fact that each term is written twice. In this form,
the antisymmetry is highlighted.
Applying this calculation to B = ω(x), we, therefore, get the relation
ω(x; h, k) = p ij (x)h
i k
j =
i
h
i k
j
− h
j k
i
=
(8.1)
=
1
2
p ij (x)
h
i k
j
− h
j k
i
with coefficients
p ij (x) = ω (x; a i , a j ) = −p ji (x)
(8.2)
depending on the basis (a i ) used . Note that in (1), we have been forced to
abandon Einstein’s convention in the first sum. When ω = d(p i dx
i ), p ij =
D i p j − D j p i .
In degree 2, expressions similar to p i (x)dx
i can be used. For this, given
two linear functionals u(h) = u i h
i and v(h) = v i h
i on the vector space
considered, the alternating bilinear form
u ∧ v : (h, k) −→ u(h)v(k) − u(k)v(h) = u i v j
h
i k
j
− h
j k
i
(8.3)
= (u i v j − u j v i ) h
i k
j
is called the exterior product of u and v. This is a purely algebraic notion.
31 See for example Cours d’Alg` ebre by the author, §§ 21 to 24. A differential form
of degree 2 is, therefore, an “ antisymmetric ” tensor field of type (0, 2).