Harry Lass Vector And Tensor Analysis Pdf
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- vector and tensor analysis
- Harry Lass-Vector and Tensor Analysis.pdf
- Tensor-based derivation of standard vector identities
vector and tensor analysis
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All rights reserved. The student totally unfamiliar with vector analysis can peruse Chapters 1, 2, and 4 to gain familiarity with the algebra and calculus of vectors. These chapters cover the ordinary one-semester course in vector analysis. Numerous examples in the fields of differential geometry, electricity, mechanics, hydrodynamics, and elasticity can be found in Chapters 3, 5, 6, and 7, respectively.
Those already acquainted with vector analysis who feel that they would like to become better acquainted with the applications of vectors can read the above-mentioned chapters with little difficulty: only a most rudimentary knowledge of these fields is necessary in order that the reader be capable of following their contents, which are fairly complete from an elementary viewpoint.
A knowledge of these chapters should enable the reader to further digest the more comprehensive treatises dealing with these subjects, some of which are listed in the reference section. It is hoped that these chapters will give the mathematician a brief introduction to elementary theoretical physics.
Finally, the author feels that Chapters 8 and 9 deal sufficiently with tensor analysis and Riemannian geometry to enable the reader to study the theory of relativity with a minimum of effort as far as the mathematics involved is concerned.
In order to cover such a wide range of topics the treatment has necessarily been brief. It is hoped, however, that nothing has been sacrificed in the way of clearness of ideas. The author has attempted to be as rigorous as is possible in a work of this nature. Numerous examples have been worked out fully in the text. The teacher who plans on using this book as a text can surely arrange the topics to suit his needs for a one-, two-, or even threesemester course.
If the book is successful, it is due in no small measure to the composite efforts of those men who have invented and who have vii viii PREFACE applied the vector and tensor analysis.
The excellent works listed in the reference section have been of great aid. Finally, I wish to thank Professor Charles de Prima of the California Institute of Technology for his kind interest in the development of this text.
Definition of a vector 2. Equality of vectors 3. Multipli- cation by a scalar 4. Addition of vectors 5. Subtraction of 6. Linear functions 7. Coordinate systems 8. Scalar, or dot, product 9. Applications of the scalar product to space geometry Vector, or cross, product The distributive law for the vector product Examples of the vector product The triple scalar product The triple vector product Differentiation of vectors Differentiation rules The gradient The vector operator del, V The divergence of a vector The curl of a vector Recapitulation Frenet-Serret formulas Fundamental planes Intrinsic equations of a curve Involutes Evolutes Spherical indicatrices Envelopes Surfaces and curvilinear coordinates Length of arc on a surface Surface curves Normal to a surface The second fundamental form Geometrical significance of the second fundamental form Principal directions Conjugate directions Asymptotic lines Point-set theory Uniform continuity Some properties of continuous functions Cauchy criterion for sequences Regular area in the plane Jordan curves Functions of bounded variation Arc length Connected and simply connected regions The line inte Stokes's theorem Line integral continued Examples of Stokes's theorem The divergence theorem Gauss Electrostatic forces Gauss's law Poisson's formula Dielectrics Energy of the electrostatic field Discontinuities of D and E Green's reciprocity theorem Method of images Conjugate harmonic functions Integration of Laplace's equation Solution of Laplace's equation in spherical coordinates Applications Integration of Poisson's equation Decomposition of a vector into a sum of solenoidal and irrotational vectors Dipoles Electric polarization Magnetostatics Solid angle Moving charges, or currents Magnetic effect of currents Oersted Mutual induction and action of two circuits Law of induction Faraday Maxwell's equations Solution of Maxwell's equations for electrically free space Poynting's theorem Lorentz's electron theory
Harry Lass-Vector and Tensor Analysis.pdf
Last reviewed: The systematic study of tensors which led to an extension and generalization of vectors, begun in by two Italian mathematicians, G. Ricci and T. Levi-Civita, following G. Riemann's proposal concerning a generalization of Euclidean geometry. The principal aim of the tensor calculus absolute differential calculus is to construct relationships which are generally covariant in the sense that these relationships or laws remain valid in all coordinate systems.
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Tensor-based derivation of standard vector identities
International series in pure and applied mathematics william ted martin, consulting editor vector and tensor analysis. Roughly speaking this can be thought of as a multidimensional array. Chapters range from elementary operations and applications of geometry, to application of vectors to mechanics, partial differentiation, integration, and tensor analysis.
In geometry and algebra , the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product also called the mixed product , box product , or triple scalar product is defined as the dot product of one of the vectors with the cross product of the other two.
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Matrix and tensor analysis in electrical network theory. Skip to main Skip to similar items. HathiTrust Digital Library. Search full-text index.
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These differentials in the form of infinitesimalshave been around since the beginning days of the invention of calculus. Theyare still widely used in applications of mathematics.