On a former page looking for a mathematical context facilitating the resolution of the issue of deformed tensor product decompositions, I explored Taylor expansions up to the second order of a numerical function depending on several real variables and mentioned the formal possibility of linking them to the question posed; please visit this page.
Here:
1) I suggest the possibility of linking the first-order Taylor expansions of a vector function F(x) depending on several real variables with question (E);
2) I then study a simple problem summarized by the following question:
"Can the calculation of ordinary first-order variations of the vector function F be replaced by the calculation of a classical cross product?"
More precisely: does there exist an element q of E(3, R) allowing one to write:
dF(u) = q ∧ dx + 0(2) ?
I would like to recall a simple but important fact that justifies why this question is raised within the broader scope of this study: tensor products deformed by antisymmetric cubes are deformed Lie products and, in particular within three-dimensional vector spaces, they are deformed cross products.
The first-order Taylor expansion of a single numerical function depending on several variables is written as:
df(u) = < ∇uf(u), du > + 0(2)
Limiting the order of the expansion (p = 1) does not prevent us from generalizing the points made on the page regarding the Taylor expansions up to the second order inclusive for these functions.
To do this, it is necessary to consider the set F(Fm(V, K) ; Km) of m numerical functions depending on D real variables: F(V ; K) dans Km.
The discussion then concerns m variations:
dfa(u) = < ∇ufa(u), du > + 0a(2)
a = 1, 2, …, m ∈ N
It allows for the introduction of a vector function F such that:
F(u) ∈ E(m, K)
≡ (…, fa(u), …) ∈ Km
≡ | F(u) > = | fa(u) > ∈ MK(m, 1)
... and the of writing:
| dF(u) > = T2(o)(∇u, F(u)). | du > + | 0(2) >
Please note that the vector has D components and the process involves m functions fa (a = 1, …, m).
Therefore, in general, the Pythagorean table appearing there is an element of the set M(m x D ; K) of matrices whose entries are real and which have m rows but D columns. D and m are not necessarily equal. Their potential equality constitutes an interesting special case.
The formalism of the variation of F(u) also evokes the way in which I generically defined the decompositions of a family of deformed tensor products. This formal analogy therefore also suggests the question of the existence of a pair (A', a') allowing the following relation to be written:
∃ ? (A’, a’) :
| dF(u) >
= | ⊗A’(a’, du) >
= T2(o)(∇uF(u)). | du > + | 0(2) >
With, to ensure the connection with question (E):
[P] = T2(o)(∇u, F(u))
b = du
z = 0(2)
By restricting the previous considerations to the case of a three-dimensional vector space (D = 3) and to sets of three numerical functions of three real variables (m = 3), it becomes possible in principle to begin discussing the decompositions of vector products in detail.
This will be explained on another page.
Up to now, you return to the French part of this website.