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Tensor Algebra and Tensor Calculus Appendix A Tensor Algebra and Tensor Calculus We collect here some notions of geometry and analysis used in the previous chapters. This appendix is no more than a brief summary containing a few defi- nitions and statements of results without proof. The conscientious reader unfamiliar with the topics collected here should take from these pages some directions for further reading in appropriate textbooks. A.1 Vector Spaces Definition 1. AsetV is called a vector space (or a linear space) over a field F if it is endowed with two operations, CWV V ! V and ıWV F ! V, called respectively addition and multiplication, such that – addition is commutative, i.e., v C w D w C v for every pair v;w 2 V, – addition is associative, i.e., .v Cw/Cz D v C.wCz/ for every triple v;w; z 2 V, – addition admits a unique neutral element 0 2 V such that v C 0 D v;8v 2 V, – with ˛v denoting the product ˛ ıv, for every ˛; ˇ 2 F and v 2 V, multiplication is associative, i.e., ˛.ˇv/ D .˛ˇ/v, where ˛ˇ is the product in F, – multiplication admits a unique neutral element 1 2 F such that 1v D v;8v 2 V, – multiplication is distributive with respect to addition, i.e., ˛.v C w/ D ˛v C ˛w for every pair v;w 2 V, – multiplication is distributive with respect to addition in F, i.e., for every pair ˛; ˇ 2 F and every v 2 V, .˛ C ˇ/v D ˛v C ˇv: We do not recall the abstract definition of a field in the algebraic sense, because we consider here only the special case of F equal to the set of real numbers R. © Springer Science+Business Media New York 2015 385 P.M. Mariano, L. Galano, Fundamentals of the Mechanics of Solids, DOI 10.1007/978-1-4939-3133-0 386 A Tensor Algebra and Tensor Calculus Definition 2. Elements v1;:::;vn of a linear space are said to be linearly depen- dent if there exist scalars ˛1;:::;˛n not all equal to zero such that Xn ˛ivi D 0: iD1 In contrast, when the previous equation holds if and only if ˛i D 0 for every i,the elements v1;:::;vn are said to be linearly independent. From now on, we shall use Einstein notation, and we shall write ˛ivi instead of Xn ˛ivi, implying summation over repeated indices. iD1 Given v1;:::;vn 2 V, we denote by span fv1;:::;vng the subset of V containing v1;:::;vn and all their linear combinations. Definition 3. A basis (or a coordinate system)inV is a set of linearly independent elements of V,sayv1;:::;vn, such that span fv1;:::;vng D V: In this case, n is the dimension of V. In particular, when n < C1, we say that V has finite dimension, which will be the case throughout this book. The dimension of a linear space V is independent of the selection of a basis for it. Every set of m < n linearly independent elements of an n-dimensional linear space V can be extended to a basis of V. Let e1;:::;en be a basis of a vector space V over the field of real numbers, in short a real vector space. Every v 2 V admits a representation i v D v ei with vi 2 R the ith component of v in that basis. In other words, v is independent of the basis, while its generic component vi exists with respect to a given basis. The distinction is essential, because we want to write the laws of physics independently of specific frames of reference (coordinate systems), i.e., of the observers. Consider the n-dimensional point space E. We can construct a linear space V over it by taking the difference between points, namely y x with x and y in E, and considering V the set of all possible differences between pairs of points in E. Then V is a linear space with dimension n. We call it the translation space over E. Once we select a point in E, call it O, we can express all differences of points in reference to it, by writing y x D .y O/ .x O/ D .y O/ C .1/.x O/: A Tensor Algebra and Tensor Calculus 387 By assigning a basis in V consisting of elements expressed in reference to O, which becomes the origin of the coordinate system, we identify V with Rn. There is, then, a hierarchy among E, V, and Rn,then-dimensional Euclidean point space, the related translational space, and finally, Rn, determined by an increasing richness of geometric structure. We call the elements of a given vector space (a linear space) vectors, although they can be far from the standard visualization of vectors as arrows in the three- dimensional ambient space or in the plane. For example, the set of nm real matrices is a linear space, and so is the set of all polynomials with complex coefficients in a variable t, once we identify F with the set of complex numbers and consider addition and multiplication as the ordinary addition of two polynomials and the multiplication of a polynomial by a complex number. Several other examples of even greater complexity could be given, which are more intricate and difficult to visualize then those we have given. Definition 4. The scalar (inner) product over V is a function h; i W V V ! R that is – commutative, i.e., hv;wi D hw;vi, – distributive with respect to vector addition, i.e., hv;w C zi D hv;wi C hv;zi, – associative with respect to multiplication by elements of F, i.e., ˛ hv;wi D h˛v;wi D hv;˛wi, 8˛ 2 F;v;w 2 V, – nonnegative definite, i.e., hv;vi 0, 8v 2 V, and equality holds only when v D 0. When V is endowed with a scalar product, we say that it is a Euclidean space. We call a point space E Euclidean if its translational space is endowed with a scalar product. Let n be the dimension of V with respect to E. In this case, we shall write En, saying that it is an n-dimensional Euclidean point space. Definition 5. Let fe1;:::;eng be a basis in a Euclidean vector space V. We say that it is orthonormal if ˝ ˛ ei; ej D ıij with ıij the Kronecker delta: 1 if i D j; ı D ij 0 if i ¤ j: ıij ıi We shall also write and j for the components of the Kronecker delta, with the same numerical values but with different meaning of the indices, as we shall make precise later. 388 A Tensor Algebra and Tensor Calculus i j Take two vectors v D v ei, w D w ej in V, expressed in terms of the basis fe1;:::;eng, with n finite, which is not necessarily orthonormal. In terms of components, we write ˝ ˛ ˝ ˛ i j i j i j i j hv;wi D v ei; w ej D v w ei; ej D v gijw D v w gij; where ˝ ˛ gij WD ei; ej and gij D gji; due to the commutativity of the scalar product. In particular, when the basis is orthonormal with respect to the scalar product considered, we have i j i hv;wi D v ıijw D v wi; j where wi D ıijw and the position of the index i has a geometric meaning that we explain below. Before going into details, however, we will say a bit about changes of frames in V. We call gij the ijth component of the metric tensor over V.The terminology recalls that gij determines in the basis fe1;:::;eng the length jvj of v 2 V, defined by the square root of the scalar product of v with itself. The meaning of the word tensor is defined below. We compute 2 i j jvj WD hv;vi D v gijv : In particular, when gij D ıij, we say in this case that the metric is flat,wehave Xn 2 i j i i 2 jvj D v ıijv D v vi D .v / : iD1 Assume V with finite dimension, as we shall do from now on. Take two bases fe1;:::;eng and fi1;:::;ing. Assume that the latter is orthonormal. With respect to such a basis, we have ki ; ei D Ai k k with Ai appropriate proportionality factors. In turn, we have also i N i ; k D Akei N i with Ak other factors. Consequently, we obtain A Tensor Algebra and Tensor Calculus 389 i N i N i li ; k D Akei D AkAi l so that since fi1;:::;ing is orthonormal, we have ˝ ˛ ˝ ˛ ˝ ˛ ı i ; i N r l i ; i N r l i ; i N r l ı N r j : ij D i j D Ai Ar l j D Ai Ar l j D Ai Ar lj D Ai Ar We specialize the˚ analysis to«Rn to clarify further the nature of the proportionality 1 factors. Let z D z ;:::;zn be˚ coordinates« in reference to an orthonormal 1 n basis fi1;:::;ing, and let x D x ;:::;x be another system of coordinates associated with a different basis, namely fe1;:::;eng, and assume the possibility of a coordinate change zj 7! xi.zj/; which is one-to-one and differentiable, with differentiable inverse mapping xi 7! zj.xi/: Consider two smooth curves s 7! x.s/ and s 7!x O .s/ intersecting at a point xN, and compute the derivatives with respect to s at xN: dxi dxOj v D .xN/e ; w D .xN/e ; ds i ds j where ei and ej are the elements of the generic basis mentioned above as a counterpart to the orthonormal one fi1;:::;ing.
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