Nlinear vector spaces and cartesian tensors pdf files

Vector analysis and vector identities by means of cartesian tensors kenneth h. H110 vector spaces, bases, and dual spaces november 4, 1998 w1. Chapter 3 deals with linear transformations on finite dimensional real euclidean spaces i. Notes on locally convex topological vector spaces 5 ordered family of. The vector space of tensors of type kis denoted tkv. How quaternions can be used to describe vector rotations in 3d space. The idea of a linear transformation of a vector space into itself is introduced and explored in chapter 2. Carpenter august 29, 2001 1 the cartesian tensor concept 1. In addition, no solutions manual is available for the problems at the end of the 14 chapters. Most of the readings and all practice problems are from the course text.

Cartesian tensors 7 rank tensor and is a vector first rank tensor. Tensorsbases, components, and dual spaces wikiversity. Abstract vector spaces, linear transformations, and their. Their calculus tensor functions of timelike variable curves in space. In these equations one main condition for these physical quantities is they must be independent of the reference system, i. Of course, the vector v, which is invariant by definition, is neither contravariant or covariant. A vector space also called a linear space is a collection of objects called vectors, which may be added together and multiplied scaled by numbers, called scalars. The location of any point can now be quantified by its position vector, the vector extending from the origin to the point in question. We begin by choosing a single point to serve as the origin. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The linear structures are genuine vector spaces or vector spaces with additional binary operations.

As with the vector, a higher order tensor is a mathematical object which represents many physical phenomena and which exists independently of any coordinate system. Their algebra definition of a vector examples of vectors scalar multiplication addition of vectors coplanar vectors unit vectors a basis of noncoplanar vectors scalar product orthogonality directional cosines for coordinate transformation vector product velocity due to rigid body rotations. If all of the vectors in a vector space can be expressed as a linear combination of a set of vectors, that set is said to span the vector space. Such vectors belong to the foundation vector space rn of all vector spaces. There are many physical laws for which a secondorder tensor is an oper. Vectors and vector spaces e1 1,0 e2 0,1 1,0 0,1 0,0 1 2 e graphical representation of e1 and e2 in the usual two dimensional plane. For an arbitrary point space the function f is called the point difference, and it is customary to use the suggestive notation f, x y x.

Consider our action on this expansion we observe that dimv dimv. Linear transformations 18 references and problems 36 3. Finitedimensional euclidean spaces and cartesian tensors 42 references and problems 59 4. However, for matters of convenience, when solving problems, we need to express the tensor in a given. Cross product between a vector and a 2nd order tensor. Without matlab the reader will be at a disadvantage. Then v is union of vector spaces v a associated with all points a in space. Linear vector spaces and cartesian tensors is primarily concerned with the theory of finite dimensional euclidian spaces. If a tensor is a linear combination of monomials all of the same length k, the. Our previous discussion of vectors, tensors, and dyads can be generalized to these nonorthogonal coordinates, as long as extreme care is taken in keeping track of the contravariant and covariant components.

It makes use of the more familiar methods and notation of matrices to make this introduction. Chapter 10 vectors and tensors georgia institute of. W of two vector spaces v and w over the same field is itself a vector space, endowed with the operation of bilinear composition, denoted by. The rest of the course is divided into three parts. There are considerable algebraic simplifications, the matrix transpose is the inverse from the. Hi, i have a problem understanding the difference between cartesian product of vector spaces and tensor product. Readings special topics in mathematics with applications. What is the difference between cartesian and tensor. A discussion of linear vector spaces is given in appendix 4.

For a general vector x x 1,x 2,x 3 we shall refer to x i, the ith component of x. Chapter 10 vectors and tensors in this chapter we explain how a vector space v gives rise to a family of associated tensor spaces, and how mathematical objects such as linear maps or quadratic forms should be understood as being elements of these spaces. That is, a classical vector is the difference of two points ine 3. No basis vector b j is a linear combination of the others. Kicking off the linear algebra lessons, lets make sure were all on the same page about how specifically to think about vectors in this context. A euclidean point space is not a vector space but a vector space with inner product is made a euclidean point space by defining f, vv v v12 1 2. However, in a more thorough and indepth treatment of mechanics, it is. Abstract vector spaces, linear transformations, and their coordinate representations contents.

V1 x v2 is cartesian product and v1 xc v2 is tensor product xc for x circled. Vector spaces, cartesian bases, handedness of basis indices and the summation convention, the kronecker delta and levicevita epsilon symbols, product of two epsilons rotations of bases, orthogonal transformations, proper and improper transformations, transformation of vectors and scalars. If w is another vector space over f, then a mapping t. Similarly, if is a third rank tensor, then is a vector. It makes a careful distinction between real and complex spaces, with an emphasis on real spaces, and focuses on those elements of the theory that are especially important in applications to continuum mechanics. In what follows, a cartesian coordinate system is used to describe tensors.

We allow k 0, in which case the tensor is just a real number. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. It is a quotient of the free vector space with basis the cartesian product. Vector and tensor analysis is a new work, first published by dover publications, inc. This is an introduction to the concepts and procedures of tensor analysis. Linear vector spaces and cartesian tensors james k. I have been searching for quite a long time, and havent been able to find any good reference about the cross product between a vector and a tensor. The tensor product of two vector spaces is not a quotient of the cartesian product of those spaces. A convenient way to measure space is to assign to each point a label consisting of three numbers, one for each dimension. These notes are aimed at students in the course linear algebra ii mat 3141 at the university of ottawa. Vectors, tensors and fields university of edinburgh.

A gentle introduction to tensors boaz porat department of electrical engineering. It is straightforward to show that has the relevant transformation properties. The index i may take any of the values 1, 2 or 3, and we refer to the. These topics are usually encountered in fundamental mathematics courses. Free vectors, taken as they are, do not form a linear vector space.

Use of cartesian tensors occurs in physics and engineering, such as with the cauchy stress tensor and the moment of inertia tensor. Applications 78 references and problems 94 appendix i. It needs to be pointed out that the concepts being discussed here, and tensor algebra in general, do not depend on the dimension being 3, do not depend on the basis being cartesian or orthogonal, and. The rst three chapters contain a revision of basic notions covered in the prerequisite course linear algebra i mat 2141. Linear vector spaces i references and problems 14 2. Again the relevant tr4ansformation properties are easy to prove.