134
IX – Multivariate Differential and Integral Calculus
If E and F are vector spaces over the same field K, their Cartesian product
E × F can be regarded as a vector space over K by defining the fundamental
operations by
(x
, y
) + (x
, y
) = (x
+ x
, y
+ y
) , t(x, y) = (tx, ty) .
With the same assumptions, a map u : E −→ F is said to be linear if
u(x + y) = u(x) + u(y) and u(tx) = tu(x) for any vector x and y and any
scalar t ; more generally,
u
t i x i
=
t i u (x i )
for all vectors x i and scalars t i . If (a i ) forms a basis for the initial space E,
then for any vector h =
h
i a i ,
u(h) =
h
i u i where u i = u (a i ) ∈ F .
(1.1)
If (b j ) is a basis for F , setting u(a i ) =
u
j
i b j , the table of coefficients u
j
i is
the matrix of u with respect to the chosen bases of E and F .
For F = K, the term linear functionals or, sometimes that of covectors is
used. The u i ∈ K are the coefficients of u. The set of these forms, equipped
with the obvious algebraic operations (sum of two forms, scalar product), is
the dual space of E, written E
∗ ; the following notation is often used:
h, u = u(h) for h ∈ E , u ∈ E
∗ ,
similarly to a scalar product. In analysis, the case K = R and F = C is
constantly needed ; the term complex linear functionals is then used ; they
are given by (1) with coefficients u i ∈ C and their set, the dual complex space
E
∗
C , is a complex vector space whose dimension over C is equal to that of E
over R. Each basis (a i ) of E has an associated dual basis (a
i ) of E
∗ over R
or of E
∗
C over C consisting of linear functionals
a
i : h −→ h
i
(1.2)
on E.
Each linear map A : E −→ F is associated to its transpose
t A or A
:
F
∗
−→ E
∗ , given by the relation
A(h), u =
h,
t A(u)
;
this definition is justified by the fact that for given u, the left hand side is a
linear function of h ∈ E.
t (BA) =
t A
t B holds for all linear maps A : E −→ F
and B : F −→ G.
Classical results on determinants will also be needed. If dim(E) = n,
then, up to a constant factor, there is a unique function D(h 1 , . . . , h n ) ∈ K
of n variables h i ∈ E satisfying the following two properties: it is multilinear,
IX – Multivariate Differential and Integral Calculus
If E and F are vector spaces over the same field K, their Cartesian product
E × F can be regarded as a vector space over K by defining the fundamental
operations by
(x
, y
) + (x
, y
) = (x
+ x
, y
+ y
) , t(x, y) = (tx, ty) .
With the same assumptions, a map u : E −→ F is said to be linear if
u(x + y) = u(x) + u(y) and u(tx) = tu(x) for any vector x and y and any
scalar t ; more generally,
u
t i x i
=
t i u (x i )
for all vectors x i and scalars t i . If (a i ) forms a basis for the initial space E,
then for any vector h =
h
i a i ,
u(h) =
h
i u i where u i = u (a i ) ∈ F .
(1.1)
If (b j ) is a basis for F , setting u(a i ) =
u
j
i b j , the table of coefficients u
j
i is
the matrix of u with respect to the chosen bases of E and F .
For F = K, the term linear functionals or, sometimes that of covectors is
used. The u i ∈ K are the coefficients of u. The set of these forms, equipped
with the obvious algebraic operations (sum of two forms, scalar product), is
the dual space of E, written E
∗ ; the following notation is often used:
h, u = u(h) for h ∈ E , u ∈ E
∗ ,
similarly to a scalar product. In analysis, the case K = R and F = C is
constantly needed ; the term complex linear functionals is then used ; they
are given by (1) with coefficients u i ∈ C and their set, the dual complex space
E
∗
C , is a complex vector space whose dimension over C is equal to that of E
over R. Each basis (a i ) of E has an associated dual basis (a
i ) of E
∗ over R
or of E
∗
C over C consisting of linear functionals
a
i : h −→ h
i
(1.2)
on E.
Each linear map A : E −→ F is associated to its transpose
t A or A
:
F
∗
−→ E
∗ , given by the relation
A(h), u =
h,
t A(u)
;
this definition is justified by the fact that for given u, the left hand side is a
linear function of h ∈ E.
t (BA) =
t A
t B holds for all linear maps A : E −→ F
and B : F −→ G.
Classical results on determinants will also be needed. If dim(E) = n,
then, up to a constant factor, there is a unique function D(h 1 , . . . , h n ) ∈ K
of n variables h i ∈ E satisfying the following two properties: it is multilinear,