By Arkadii V. Kim

The version introduces a brand new type of invariant derivatives and exhibits their relationships with different derivatives, equivalent to the Sobolev generalized spinoff and the generalized by-product of the distribution thought. this can be a new course in mathematics.

*i*-Smooth research is the department of sensible research that considers the idea and purposes of the invariant derivatives of services and functionals. the real path of i-smooth research is the research of the relation of invariant derivatives with the Sobolev generalized by-product and the generalized spinoff of distribution theory.

Until now, *i*-smooth research has been built mostly to use to the speculation of useful differential equations, and the aim of this publication is to give i-smooth research as a department of sensible analysis. The thought of the invariant spinoff (i-derivative) of nonlinear functionals has been brought in arithmetic, and this in flip built the corresponding *i*-smooth calculus of functionals and confirmed that for linear non-stop functionals the invariant spinoff coincides with the generalized spinoff of the distribution theory. This booklet intends to introduce this thought to the overall arithmetic, engineering, and physicist communities.

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**Example text**

7) is the quadratic functional (regular homogeneous functional of the second degree, [38], p. 8) −τ −τ where γ[s, ν] (s, ν ∈ [−τ, 0]) is a continuous n × n matrix with continuous elements. In general case one can construct regular functionals, described by m-multiple integrals. However in practice integrals of order m > 3 are applied in rare cases. 3. In general, singular functionals can be deﬁned in a similar manner. 8 (singular functional) Let P ∗ [·, . . , ·] : Rn × . . × Rn → R be a continuous function and τ1 , .

It easy to verify that ≺u , φ = a [(f, φ)](f , φ), etc. The set of analytical on D functionals [15] is the subspace in SD. 4 Directivity (uα )α∈A ⊂ SD converges to u ∈ SD, if for all φ ∈ D and j ∈ N the sequence (j) (≺uα , φ )α∈A converges to ≺u(j) , φ . Multiplication in SD The space SD is the algebra with respect to the operation of pointwise multiplication (denoting by ) deﬁned by the 30 i-Smooth Analysis rule: for v, w ∈ SD ≺v w, φ = ≺v, φ ≺w, φ , φ∈D . 11) Obviously this product is associative, commutative and the Leibniz diﬀerentiation principle is valid: (v w) = v w+v w .

D = D (a, b) is the set of linear continuous on D = D(a, b) functionals (the space of distributions (generalized functions), and (f, φ) denotes the value of a distribution f ∈ D at a test function φ ∈ D. 1) (f , φ) = −(f, φ ) , φ ∈ D . Emphasize the important fact that the generalized derivative can be also deﬁned on the basis of the (left) shift operator, [Ty φ](x) ≡ φ(x − y) , a ≤ x ≤ b. 2) (see for example [8]) as the limit, (f , φ) = lim y→0 (f, Ty φ) − (f, φ) . 3) Below in section Ty is the left translation operator.