Generalized Functions in Mathematical Physics: Main Ideas by A. S. Demidov

By A. S. Demidov

This crucial ebook offers an interconnected presentation of a few uncomplicated principles, recommendations, result of the idea of generalised features (first of all, within the framework of the idea of distributions) and equations of mathematical physics. part of the cloth is given in response to the scheme: definition -- theorem -- evidence. This scheme is handy for proposing leads to transparent and focused shape. despite the fact that, it sort of feels average to provide a scholar the prospect not just to check a priori given definitions and proofs of theorems, but additionally to find them whereas contemplating the issues concerned. a sequence of sections serve this objective. additionally, part of the fabric is given as routines and difficulties.

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The Ostrogradsky–Gauss formula. The Green formulae and the Green function Let Ω be a bounded domain in Rn with a smooth (n − 1)dimensional boundary ∂Ω. Let f = (f1 , . . , fn ) be a vector-function ¯ and ∂fk /∂xk ∈ P C(Ω) ∀k. 1) ∂Ω k=1 holds, where αk = αk (x) is the cosine of the angle between the outward normal ν to Γ = ∂Ω at the point x ∈ Γ and kth coordinate axis, and dΓ is the “area element” of Γ. 1) becomes the Newton–Leibniz formula. 1) is a special case of the important Stokes theorem on integration of differential forms on manifolds with boundary (see, for Rinstance, R[63, 72]), which can be represented by the Poincar´e formula: dω = ∂Ω ω.

P. 2) depends continuously in L1loc (R2+ ) on the initial function f ∈ L1loc (R). P. P. 7, if f (x) = θ(x). In the exercises below, we assume Q = {(x, t) ∈ R2 | x > 0, t > 0}. P. 12) u x=0 = h(t), t > 0. 13). 13) has a (unique) solution u ∈ C 1 (Q) 1 ¯ 1 ¯ and only if f ∈ C (R+ ), h ∈ C (R+ ), ans f (0) = h(0), f (0) = −h (0). P. 15) 54 1. INTRODUCTION TO PROBLEMS OF MATHEMATICAL PHYSICS ¯ + ). 13). 14) are constant. P. 18) ¯ + ). where f , g, and h are functions from C 1 (R (1) Draw the level lines of the functions u ± (1/ρc)p.

8). 10). Nevertheless, it is helpful to use the dimension arguments. Firstly, they allow us to test the correctness of involving some parameters when formulating the problem: both sides of any equality used in the problem should have consistent dimensions. Secondly, the reasoning of dimension allows to find the necessary change of variables (not necessarily connected only with scaling coefficients). All these facts allow automatically (hence, easily) to get rid of “redundant” parameters and so to simplify the analysis as well as the calculations2) .

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