186
IX – Multivariate Differential and Integral Calculus
dω(x; h, k) = dp i (x; h)k
i
− dp i (x; k)h
i =
= dp i (x; h)dx
i (k) − dp i (x; k)dx
i (h) =
= dp i (x) ∧ dx
i (h, k) ,
a relation involving the value at (h, k) of the exterior product of the linear
functionals h → dp i (x; h) and h → dx
i (h) = h
i .
ω = p i dx
i =⇒ dω = dp i ∧ dx
i
(8.7)
for short.
Exercise 1. Using the definition of dω, show that
d(fω) = df ∧ ω + fdω
(8.8)
if f is a function and ω a form of degree 1. Deduce (7).
In dimension 3 and in rectangular coordinates, but only in this case, (5’)
can be identified to the vector field H(x) with coordinates p, q, r, though the
vector with coordinates
h
2 k
3
− h
3 k
2 , h
3 k
1
− h
1 k
3 , h
1 k
2
− h
2 k
1
is called the vector product (or, which is better, exterior product) of the
vectors h and k, and is written h × k or h ∧ k. Then
ω(x; h, k) = (H(x)|h ∧ k) ,
the scalar product of the vectors H(x) and h ∧ k. This does not substitute
for the theory of alternating bilinear forms.
(iii) Forms of degree p. All this can be generalized
33 and differential forms
of arbitrary degree p as well as an operation d taking a form of degree p to
a form of degree p + 1 can be defined. A form of degree p is a tensor field of
type (0, p), i.e. a function
ω (x; h 1 , . . . , h p )
multilinear in h i for given x, which is also alternating, i.e. multiplied by −1
when two variables h are permuted. It can easily be shown that
34 ω = 0 if p
is greater than the dimension n of E and that when p = n,
ω (x; h 1 , . . . , h n ) = p(x) det (h 1 , . . . , h n ) ,
33 See for example Henri Cartan, Calcul diff´ erentiel.
34 If p > n, the vectors h1, . . . , hp are never linearly independent; hence one of
them can be expressed as the linear combination of the others. Substituting in
a p-linear alternating form, we get a linear combination of its values for vectors
that are not pairwise distinct, hence that are zero because of the antisymmetry
of the form considered. See Cours d’alg` ebre, § 23, also for determinant theory.
IX – Multivariate Differential and Integral Calculus
dω(x; h, k) = dp i (x; h)k
i
− dp i (x; k)h
i =
= dp i (x; h)dx
i (k) − dp i (x; k)dx
i (h) =
= dp i (x) ∧ dx
i (h, k) ,
a relation involving the value at (h, k) of the exterior product of the linear
functionals h → dp i (x; h) and h → dx
i (h) = h
i .
ω = p i dx
i =⇒ dω = dp i ∧ dx
i
(8.7)
for short.
Exercise 1. Using the definition of dω, show that
d(fω) = df ∧ ω + fdω
(8.8)
if f is a function and ω a form of degree 1. Deduce (7).
In dimension 3 and in rectangular coordinates, but only in this case, (5’)
can be identified to the vector field H(x) with coordinates p, q, r, though the
vector with coordinates
h
2 k
3
− h
3 k
2 , h
3 k
1
− h
1 k
3 , h
1 k
2
− h
2 k
1
is called the vector product (or, which is better, exterior product) of the
vectors h and k, and is written h × k or h ∧ k. Then
ω(x; h, k) = (H(x)|h ∧ k) ,
the scalar product of the vectors H(x) and h ∧ k. This does not substitute
for the theory of alternating bilinear forms.
(iii) Forms of degree p. All this can be generalized
33 and differential forms
of arbitrary degree p as well as an operation d taking a form of degree p to
a form of degree p + 1 can be defined. A form of degree p is a tensor field of
type (0, p), i.e. a function
ω (x; h 1 , . . . , h p )
multilinear in h i for given x, which is also alternating, i.e. multiplied by −1
when two variables h are permuted. It can easily be shown that
34 ω = 0 if p
is greater than the dimension n of E and that when p = n,
ω (x; h 1 , . . . , h n ) = p(x) det (h 1 , . . . , h n ) ,
33 See for example Henri Cartan, Calcul diff´ erentiel.
34 If p > n, the vectors h1, . . . , hp are never linearly independent; hence one of
them can be expressed as the linear combination of the others. Substituting in
a p-linear alternating form, we get a linear combination of its values for vectors
that are not pairwise distinct, hence that are zero because of the antisymmetry
of the form considered. See Cours d’alg` ebre, § 23, also for determinant theory.