8,841 research outputs found
Vector Functions and their Differentiation Formulas in 3-dimensional Euclidean Spaces
In this article, we first extend several basic theorems of the
operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors:
e1, e2, e3 and the definition of vector function in the same spaces are introduced.
By dint of unit vector the main operation properties as well as the differentiation
formulas of vector function are shown [12].Liang Xiquan - Qingdao University of Science and Technology, ChinaZhao Piqing - Qingdao University of Science and Technology, ChinaBai Ou - University of Science and Technology of China, Hefei, ChinaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703-709, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Murray R. Spiegel. Vector Analysis and an Introduction to Tensor Analysis. McGraw-Hill Book Company, New York, 1959.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990
The Gauss-Green theorem in stratified groups
We lay the foundations for a theory of divergence-measure fields in
noncommutative stratified nilpotent Lie groups. Such vector fields form a new
family of function spaces, which generalize in a sense the fields. They
provide the most general setting to establish Gauss-Green formulas for vector
fields of low regularity on sets of finite perimeter. We show several
properties of divergence-measure fields in stratified groups, ultimately
achieving the related Gauss-Green theorem.Comment: 69 page
Absolute Time Derivatives
A four dimensional treatment of nonrelativistic space-time gives a natural
frame to deal with objective time derivatives. In this framework some well
known objective time derivatives of continuum mechanics appear as
Lie-derivatives. Their coordinatized forms depends on the tensorial properties
of the relevant physical quantities. We calculate the particular forms of
objective time derivatives for scalars, vectors, covectors and different second
order tensors from the point of view of a rotating observer. The relation of
substantial, material and objective time derivatives is treated.Comment: 26 pages, 4 figures (minor revision
A new approach on helices in pseudo-Riemannian manifolds
In this paper, we give a definition of harmonic curvature functions in terms
of V_{n} and define a new kind of slant helix which is called V_{n}-slant helix
in n-dimensional pseudo-Riemannian manifold. Also, we give important
characterizations about the helix.Comment: arXiv admin note: substantial text overlap with arXiv:1305.704
The Paley-Wiener Theorem and the Local Huygens' Principle for Compact Symmetric Spaces
We prove a Paley-Wiener Theorem for a class of symmetric spaces of the
compact type, in which all root multiplicities are even. This theorem
characterizes functions of small support in terms of holomorphic extendability
and exponential type of their (discrete) Fourier transforms. We also provide
three independent new proofs of the strong Huygens' principle for a suitable
constant shift of the wave equation on odd-dimensional spaces from our class.Comment: 26 pages, 1 figur
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