Multivector fields. (Поливекторные поля.) (Russian) Zbl 0671.35076
Tashkent (USSR): Fan. 184 p. R. 0.80 (1986).
The theory of multivectors and multivector fields of \(K\)-th order in \(m\)-dimensional Riemann and Euclidean spaces and its applications to some multidimensional problems of mathematical physics are presented.
In Chapter 1 the algebra of multivectors in Riemann space is constructed. Different kinds of multiplication of vector, multivector, arbitrary tensor of the second rank by a multivector of \(K\)-th order are defined and their properties are studied. Chapter 2 is devoted to the foundations of multivector analysis. Integral theorems are proved and some types of multivector fields (potential, solenoidal, harmonic) are considered. In Chapter 3 the theory is applied to multivector Poisson equations, Lamé equations, multidimensional elastic theory and to the basic problem of multivector field theory, the determining of a multivector field by its gradient and cogradient.
In Chapter 1 the algebra of multivectors in Riemann space is constructed. Different kinds of multiplication of vector, multivector, arbitrary tensor of the second rank by a multivector of \(K\)-th order are defined and their properties are studied. Chapter 2 is devoted to the foundations of multivector analysis. Integral theorems are proved and some types of multivector fields (potential, solenoidal, harmonic) are considered. In Chapter 3 the theory is applied to multivector Poisson equations, Lamé equations, multidimensional elastic theory and to the basic problem of multivector field theory, the determining of a multivector field by its gradient and cogradient.
Reviewer: A.Borisov
MSC:
35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
53A45 | Differential geometric aspects in vector and tensor analysis |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |
26B12 | Calculus of vector functions |