IX – Multivariate Differential and Integral
Calculus
§ 1. Classical Differential Calculus – § 2. Differential Forms of Degree 1 – § 3. Integration of Differential Forms – § 4. Differential
Manifolds
§ 1. Classical Differential Calculus
The aim of this § is to present the differential calculus of multivariate functions in the framework of finite dimensional real vector spaces; the case of a
2-dimensional space having been dealt with in Chap. III, § 5, we will mostly
generalize the results, the proofs being the same as in dimension 2. Nonetheless, this § also contains considerations about tensors that are not found
everywhere.
1 – Linear Algebra and Tensors
(i) Finite-dimensional vector spaces .
1 The elements of such a space E are,
depending on the context, called “ points ” or “ vectors ”, the numbers – real or
complex depending on needs of the analysis – by which vectors are multiplied
being “ scalars ” ; in what follows, the letter K will indifferently denote R, C
or any other field in which the scalars vary. A basis for an n-dimensional
vector space E is a family (a i ) of n linearly independent vectors, i.e. such
that any h ∈ E can be written in a unique way as h =
h i a i , the scalars h i
being the “ components ” or “ coordinates ” of h with respect to the basis
considered.
2
1 For details and proofs, see for example sections §§ 10 to 24 in Cours d’alg` ebre
(Hermann, 1966 or 1997) by the author, or else the somewhat condensed thirteen
pages of the Annex to El´ ements d’analyse, vol. 1, by Dieudonn´ e, or Serge Lang,
Linear Algebra (Springer, 1987), etc.
2 The use of the letter h instead of x to denote vectors is due to the fact that in
analysis, vectors occur mostly as increases of a variable, like in the notion of a
differential defined later.
© Springer International Publishing Switzerland 2015
133
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_2
Calculus
§ 1. Classical Differential Calculus – § 2. Differential Forms of Degree 1 – § 3. Integration of Differential Forms – § 4. Differential
Manifolds
§ 1. Classical Differential Calculus
The aim of this § is to present the differential calculus of multivariate functions in the framework of finite dimensional real vector spaces; the case of a
2-dimensional space having been dealt with in Chap. III, § 5, we will mostly
generalize the results, the proofs being the same as in dimension 2. Nonetheless, this § also contains considerations about tensors that are not found
everywhere.
1 – Linear Algebra and Tensors
(i) Finite-dimensional vector spaces .
1 The elements of such a space E are,
depending on the context, called “ points ” or “ vectors ”, the numbers – real or
complex depending on needs of the analysis – by which vectors are multiplied
being “ scalars ” ; in what follows, the letter K will indifferently denote R, C
or any other field in which the scalars vary. A basis for an n-dimensional
vector space E is a family (a i ) of n linearly independent vectors, i.e. such
that any h ∈ E can be written in a unique way as h =
h i a i , the scalars h i
being the “ components ” or “ coordinates ” of h with respect to the basis
considered.
2
1 For details and proofs, see for example sections §§ 10 to 24 in Cours d’alg` ebre
(Hermann, 1966 or 1997) by the author, or else the somewhat condensed thirteen
pages of the Annex to El´ ements d’analyse, vol. 1, by Dieudonn´ e, or Serge Lang,
Linear Algebra (Springer, 1987), etc.
2 The use of the letter h instead of x to denote vectors is due to the fact that in
analysis, vectors occur mostly as increases of a variable, like in the notion of a
differential defined later.
© Springer International Publishing Switzerland 2015
133
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_2