148
IX – Multivariate Differential and Integral Calculus
where the scalars h
i are placed on the right of the vectors D i f (x) contrary
to tradition,
14
(b) df (x) exists if and only the partial derivatives D i f exist and are
continuous in the neighbourhood
15 of x (Chap. III, § 5, n
◦ 20).
(6) reduces to numerical expressions by choosing a basis (b j ) for F and
setting f (x) = f
j (x)b j , whence
D i f (x) = D i f
j (x)b j
(2.7)
since the b j are independent of x ; as a result,
f
(x)h = df (x; h) = D i f
j (x)h
i b j .
(2.8)
The coordinates of the vector df (x; h) ∈ F are, therefore, the numbers
df (x; h)
j = D i f
j (x)h
i = df
j (x; h) .
(2.9)
This is the value at the vector h of the differential in x of the function f
j .
When the D i f (x) exist and are continuous on U , f is said to be of class
C
1 on U , etc.
D i D j f (x) = D j D i f (x)
(2.10)
if f is of class C
2 and in fact, using weaker assumptions (Chap. III, § 5, n
◦ 23).
The n × p matrix whose entries are the D i f
j (x) is the Jacobian matrix
of f at x. It is that of the linear map f
(x) with respect to the two chosen
bases of E and F since
f
(x)a i = D i f
j (x)b j .
The rank (n
◦ 1, (i)) of this linear map is the rank of f at x . As the determinants of the square sub-matrices of the Jacobian matrix are continuous
functions of x if f is C
1 , if f is of rank r at x, then its rank is clearly ≥ r in
the neighbourhood of x. In the case of map from E to itself, its determinant
can be associated to f
(x). This is the Jacobian
J f (x) = det f
(x) = det
D i f
j (x)
(2.11)
of f at the point x ; it was earlier called the functional determinant of the f j
at x and was written D(f
1 , . . . , f
n )/D(x
1 , . . . , x
n ) or D(f )/D(x) for short.
Using the differential notation df = f
(x)dx as in the case of a real variable is possible and in practice very useful. Clearly, the differential of the
14 It suffices to set once and for all that th = ht if h is a vector and t a scalar : there
is no problem since the field R is commutative.
15 In a topological space, a relation involving a variable x holds in the neighbourhood
of a if there is an open subset U containing a such that it holds for all x ∈ U
(Chap. II, § 1, n
◦ 3 for the case of R or C).
IX – Multivariate Differential and Integral Calculus
where the scalars h
i are placed on the right of the vectors D i f (x) contrary
to tradition,
14
(b) df (x) exists if and only the partial derivatives D i f exist and are
continuous in the neighbourhood
15 of x (Chap. III, § 5, n
◦ 20).
(6) reduces to numerical expressions by choosing a basis (b j ) for F and
setting f (x) = f
j (x)b j , whence
D i f (x) = D i f
j (x)b j
(2.7)
since the b j are independent of x ; as a result,
f
(x)h = df (x; h) = D i f
j (x)h
i b j .
(2.8)
The coordinates of the vector df (x; h) ∈ F are, therefore, the numbers
df (x; h)
j = D i f
j (x)h
i = df
j (x; h) .
(2.9)
This is the value at the vector h of the differential in x of the function f
j .
When the D i f (x) exist and are continuous on U , f is said to be of class
C
1 on U , etc.
D i D j f (x) = D j D i f (x)
(2.10)
if f is of class C
2 and in fact, using weaker assumptions (Chap. III, § 5, n
◦ 23).
The n × p matrix whose entries are the D i f
j (x) is the Jacobian matrix
of f at x. It is that of the linear map f
(x) with respect to the two chosen
bases of E and F since
f
(x)a i = D i f
j (x)b j .
The rank (n
◦ 1, (i)) of this linear map is the rank of f at x . As the determinants of the square sub-matrices of the Jacobian matrix are continuous
functions of x if f is C
1 , if f is of rank r at x, then its rank is clearly ≥ r in
the neighbourhood of x. In the case of map from E to itself, its determinant
can be associated to f
(x). This is the Jacobian
J f (x) = det f
(x) = det
D i f
j (x)
(2.11)
of f at the point x ; it was earlier called the functional determinant of the f j
at x and was written D(f
1 , . . . , f
n )/D(x
1 , . . . , x
n ) or D(f )/D(x) for short.
Using the differential notation df = f
(x)dx as in the case of a real variable is possible and in practice very useful. Clearly, the differential of the
14 It suffices to set once and for all that th = ht if h is a vector and t a scalar : there
is no problem since the field R is commutative.
15 In a topological space, a relation involving a variable x holds in the neighbourhood
of a if there is an open subset U containing a such that it holds for all x ∈ U
(Chap. II, § 1, n
◦ 3 for the case of R or C).