i: (3) The summation convention implies that repeated indices appearing exactly twice in a single term (by term we mean any single vector or tensor quantity or any product of vector and/or tensor quantities) are to be summed from 1 through d(where dis the number of dimensions), i.e., for example in three dimensions, v= v ig i= v 1g 1+ v 2g 2+ v 3g Tensor Algebra II Continuum Mechanics: Fall 2007 Reading: Gurtin, Section 2 Definition. A number ω is an eigenvalue of a tensor S if there exists a unit vector e such that Se = ωe. Definition. The characteristic space of S corresponding to ω is {v : Sv = ωv}. Spectral Theorem. Let S ∈ Sym. Then, there exists an orthonormal basis CHAPTER XII MULTILINEAR ALGEBRA 1. Tensor and Symmetric Algebra Let kbe a commutative ring.By a k-algebra, we mean a ring homomorphism ˚: k!Asuch that each element of Im˚commutes with each element of A.(IfAis a ring, we de ne its center to be the subring Z(A)=fa2Ajax= xa;for all x2Ag.So this can also be abbreviated Im˚ Z(A).)) Such a k-algebra may be viewed as a k-module by de ning ax= ˚(a 'Tensors' were introduced by Professor Gregorio Ricci of University of Padua (Italy) in 1887 primarily as extension of vectors. A quantity having magnitude only is called Scalar and a quantity with tensor elds of rank or order one. Closely associated with tensor calculus is the indicial or index notation. In section 1 the indicial notation is de ned and illustrated. We also de ne and investigate scalar, vector and tensor elds when they are subjected to various coordinate transformations. It turns out that tensors have certain properties which 1.Prerequisites from Linear Algebra Linear algebra forms the skeleton of tensor calculus and differential geometry. We recall a few basic definitions from linear algebra, which will play a pivotal role throughout this course. scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The very brief recapitulation oi' vector algebra ana analysis as taught in the undergraduate courses. Particular attention is paid to the appli- Tensor Field, Tensor write more documents of the same kind. I chose tensors as a first topic for two reasons. First, tensors appear everywhere in physics, including classi-cal mechanics, relativistic mechanics, electrodynamics, particle physics, and more. Second, tensor theory, at the most elementary level, requires only linear algebra and some calculus as The addition of two tensors of B. Vector Computations valence [ pq ] yields a tensor of valence [ pq ]: A + B = C. Vector calculus is the simplest case of tensor calculus (a vector is a tensor of valence one). The N -dimensional vec- tor an is represented as a set of components n = 1, N , de- pending on the basis, and a linear law of the The tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. In general, there are two Vectors and tensors algebra Algebra is concerned with operations de ned in sets with certain properties. ensorT and vec-tor algebra deals with properties and operations in the set of tensors and vectors. Through-out this section together with algebraic aspects, we also consider geometry of tensors to obtain further insight. 1.1 Scalars and vectors Lecture Notes on Vector and Tensor Algebra and Analysis IlyaL.Shapiro Departamento de F´ısica - Institu