Tensors third printing petr v anicek department of surveying engineering. An an inner product, a 2form or metric tensor is an example of a tensor of type 0. I do not include results relying on a metric or hermitian metric. Lecture notes on vector and tensor algebra and analysis ilyal. An introduction to tensors for students of physics and engineering joseph c. Given two metric tensors g ij and eg, when can they be transformed into each other by a change of coordinates. At this point if we were going to discuss general relativity we would have to learn what a manifold 16. Technically, a manifold is a coordinate system that may be curved but which is. For example as discussed in last chapter, if i consider this page, it is flat.
The ubiquitous problem of tensor decomposition is illustrated with two examples. Scalars, vectors, second rank tensors sometimes referred to loosely as tensors, and higher rank tensors can all be represented in tensor notation. You will be able to use the notion of a tensor to describe what happens. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. I know the author as a research scholar who has worked with me for several years. An introduction to the riemann curvature tensor and. It is called the metric tensor because it defines the way length is measured. In cartesian space they must be an orthogonal norm preserving matrix. What is a tensor and can any examples of their use be given. Using simple terms, what are tensors and how are they used in. The most pertinent parts are at the beginning and end. Reynolds, hyperbolic geometry on a hyperboloid, amer.
Examples of tensors february 3, 20 wewilldevelopanumberoftensorsasweprogress,butthereareafewthatwecandescribeimmediately. Also describe its matrix in terms of bases as in i and the corresponding matrices of the tjs. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. There are many other examples of tensors appearing in physics. Matthew allen answer tensors are most easily understood by discussing the progression of tensor ranks. Wn, and discuss its behavior with respect to composites with linear maps wj uj. The end result does not of course change but it does create confusion. An example is the stress on a material, such as a construction beam in a bridge. For example, the metric tensor g has two free indices, so it is a rank two tensor. As a result, it is possible for such courses to become terminal mathematics courses rather than. Beware that there are other, less relevant, pages with metric in their titles. Years ago, he made the statement that in working with tensors, one must learn to findand feelthe rhythm inherent in the indices. Metric tensor of spacetime in general relativity written as a matrix. Udayanandan associate professor department of physics nehru arts and science college, kanhangad 1.
In order to do this however it will be necessary to expand the notion of \ tensor to include a larger class of functions. An ndimensional vector eld is described by a onetoone correspondence between nnumbers and a point. It gives me great pleasure to write the foreword to dr. As a freshman in college, i once asked my physics professor what a tensor was. A manifold equipped with a positivedefinite metric tensor. Because it is often denoted without a symbol between the two vectors, it is also referred to. It is this chapter that provides the foundations for tensor applications in physics. These topics are usually encountered in fundamental mathematics courses. The tensor product is just another example of a product like this. Introduction to tensor calculus for general relativity mit. The main problem for us about the metric tensor is the following. He certainly felt that rhythm, and his ability to do so made a major difference in his approach to teaching the material and enabling his students to. Physics 221a fall 2019 appendixe introduction to tensor.
The tensor product of two vectors represents a dyad, which is a linear vector transformation. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. For simplicity the definitions are sated here in terms of euclidean point spaces only even though they can be defined for metric spaces in general. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler, which characterizes simplified approaches to a problem as the poor mans way to do x. Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. If two tensors of the same type have all their components equal in. For example, if the metric tensor gmn is constant in some coordinate system, then gs are all 0 and so the covariant derivative reduces to the usual partial derivative. In general relativity, the metric tensor in this context often abbreviated to simply the metric is the fundamental object of study.
For example, in the newtonian theory of gravity the gravitational force depends only on the current distance between two interacting bodies. In preparing this two volume work our intention is to present to engineering and science students a modern introduction to vectors and tensors. A good starting point for discussion the tensor product is the. We investigate results due to the distribution of the vector or tensor values of the field on its domain. The length of a physical vector must have physical units. So if one body changes its position then the other one \feels this immediately, no matter how great the distance between them is. Nazrul islams book entitled tensors and their applications. If v 1 and v 2 are any two vector spaces over a eld f, the tensor product is a bilinear map. This paper is a more detailed version of my talk given at the mathphysics symposium at unbc on february 25, 2016. It is called the metric tensor because it defines the way length is measured at this point if we were going to discuss general relativity we would have to learn what a manifold 16. Metric tensor and the line element phys 500 southern illinois university september 22, 2016. In order to do this however it will be necessary to expand the notion of \tensor to include a larger class of functions.
Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. The poor mans introduction to tensors physics the university of. Another physical example of a tensor is the polarizability tensor relating the electric dipole moment. Introduction these notes contain an introduction to tensor analysis as it is commonly used in physics, but mostly limited to the needs of this course. Introduction to tensor calculus for general relativity. Unsubscribe from physics videos by eugene khutoryansky. A metric tensor geon real vector space is a nonsingular, symmetric 0 2 tensor that maps each pair of vectors u.
Foundations of tensor analysis for students of physics and. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. A gentle introduction to tensors washington university in. It is useful to extend the definition of a field to onevalued real functions of a.
There are also natural operations like tensor products, and additions of tensors of the same type. Typically we are given the metric tensor of equation 1. These tensors can be classified according to the following order. A scalar quantity is simply a number it has only magnitude. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. Course notes tensor calculus and differential geometry. Tensor algebra operations for making new tensors from old tensors 1. Note that the metric tensor may be a function of position in the space. Generally when one talks about tensors, though, one is referring to tensors of rank two or higher.
A metric tensor is called positivedefinite if it assigns a positive value gv, v 0 to every nonzero vector v. However, in a more thorough and indepth treatment of mechanics, it is. The definition in 3 is the most popular in hep and relativity books and this is what we will follow throughout this course. The advantage of the covariant derivative is that it gives rise to tensors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. A tensor field has a tensor corresponding to each point space. Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3. A dyad is a special tensor to be discussed later, which explains the name of this product. General relativity, however, requires tensor algebra in a general curvilinear coordinate system. When a stress is applied, the crystal will distort slightly and this relation of strain to stress is, for small stress, a linear one. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems.
An introduction to tensors for students of physics and. The images have not been converted to searchable text. For this let us nd out what is the exact nature of moment of inertia. For all of these basic questions have been pretty much solved. Looking forward an introduction to the riemann curvature tensor and di. The stress at a point of an elastic solid is an example of a. When proofs are not included, references for them are given. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. In addition tothis type of manipulations, one canalso. In minkowski space and under poincare transformations, tensors are lorentz tensors, or, 4tensors. I have 3 more videos planned for the noncalculus videos. Zero tensor 0 has all its components zero in all coord systems.
Roughly speaking this can be thought of as a multidimensional array. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor. A tensor is what you get when you divide two vectors. Similarly, it turns out that there is an electromagnetic tensor, usually represented f note that im changing and to. Metric tensor if fe mg m is a basis for a metric space, the components of the metric tensor are g jk gee j. Linear algebra forms the skeleton of tensor calculus and differential geometry. The presentation is based on how various quantities transform under coordinate transformations, and is fairly standard. Here i will present the basic theory for the riemannian metrics. Introduction to tensor calculus a scalar eld describes a onetoone correspondence between a single scalar number and a point.
For example, the metric is a bilinear operator which takes two vectors. There are several concepts from the theory of metric spaces which we need to summarize. Introduction to the tensor product james c hateley in mathematics, a tensor refers to objects that have multiple indices. Introduction to tensor notation tensor notation provides a convenient and uni ed system for describing physical quantities. While we do not discuss general differentiable manifolds, we do include a chapter on vector and tensor fields defined on hypersurfaces in a euclidean manifold. For example, the components of the 0, 2 metric tensor, the 2, 0 inverse metric tensor and the 1, 1 identity tensor are. In ndimensional space a tensor of rank n has nn components. In the simple case in which, for example, the basis vector e1.
Several books including perkins use several different metric definitions. Transformations of the metric and the unit vector basis. In this chapter we will learn the language of tensors to help us interpret physical phenomena. V is an elementary tensor if and only if cijci0j0 cij0ci0j for all i,i0,j,j0. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries. The matrix g with components g jk is a real, symmetric matrix.
Here it is just as an example of the power of the index notation. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in di. Nasatp20052115 1 foundations of tensor analysis for students of physics and engineering with an introduction to the theory of relativity. Covariant, contravariant, rank physics videos by eugene khutoryansky. Lecture notes on vector and tensor algebra and analysis. But tensors also show up in all sorts of places that dont connect to an anisotropic material property, in fact even spherical harmonics are tensors. I feel the way im editing videos is really inefficient. It follows at once that scalars are tensors of rank. Looking forward a quick example of what i mean by intrinsic. Arc length and riemannian metric geometry references. The quality of the images varies depending on the quality of the originals. He leaned back, paused for what im sure he thought was a dramatic moment, and said. Consider our action on this expansion we observe that dimv dimv.