The secondorder and higherorder tensors are symbolized with a boldface letter. Oct 15, 2019 mgsu msc mathematics lecture series,quotient law, tensors analysis,bikaner quotient law of tensor pdf. A tensor of rank 2 is necessary to represent a magnetic permeability in complex materials, or the stresses in a material object or in a field, and so on. Vector algebra and calculus using cartesian tensors. Harris, in mathematics for physical science and engineering, 2014. Thus, if and are tensors, then is a tensor of the same type. Derivative generalizations differentiation notation.
On vectors and tensors, expressed in cartesian coordinates. Tensor techniques in physics learning development institute. Objects that tensors may map between include vectors which are often, but not always, understood as arrows with length that point in a direction and scalars which are often familiar numbers such as the real numbers, and. Examples of tensors february 3, 20 wewilldevelopanumberoftensorsasweprogress,butthereareafewthatwecandescribeimmediately. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Metric tensors, covariant and contravariant tensors, simple applications to general theory of relativity and klein gordon and dirac equations in relativistic quantum mechanics 07 6. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space.
In the first part of this chapter we will study several tools to manage tensors scalars. Chapter 10 vectors and tensors georgia institute of. Mis an abelian group which is closed under the scaling operation. A submodule of a module is very much like an ideal of a ring. Tensors in index notation, kronecker and levi civita tensors, inner and outer products, contraction, symmetric and antisymmetric tensors, quotient law. This wellwritten text gives the origin and nature of the tensor along with the scope of the tensor.
A basic knowledge of vectors, matrices, and physics is assumed. Tensor calculus while presenting the concepts and techniques begins with a brief introduction and history of tensors, followed by the study of systems of different orders, einstein summation convention, kronecker symbol leading to the concepts of tensor algebra and tensor calculus. This is the transformation rule for a tensor of the form aqip. The representation of a tensor in a particular coordinate system is an array of elements of the form. This can be done by explicitly writing down the maps. The relation between the potential a and the fields e and b given in section 4. If you have have visited this website previously its possible you may have a mixture of incompatible files. Browse other questions tagged differentialgeometry riemannian. Tensor which depends on two directions one normal to the area and other that of the force on it.
Apr 23, 2009 quotient theorem tensors thread starter latentcorpse. It follows at once that scalars are tensors of rank 0,0, vectors are tensors of rank 1,0 and oneforms are tensors of. Once such a law holds in one coordinate system the same form of the law holds in any other coordinate system. This gives the name to a particularly useful rule for recognizing tensors, the quotient rule. This book provides a conceptual exposition of the fundamental results in the theory of tensors. Typical situations for which the quotient rule establishes the tensor nature of s are equations such as the following all written using the summation convention. Because x has two indices, we identify it as a tensor of rank 2. A gentle introduction to tensors washington university in. This book has been presented in such a clear and easy way that the students will have no difficulty.
In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Here is a brief history of tensors and tensor products. It follows at once that scalars are tensors of rank. As a start, the freshman university physics student learns that in ordinary cartesian coordinates, newtons second law, p i f. Tensor analysisintroductiondefinitiondefinition of different rank tensors contraction and direct productquotient rulepseudo tensors. Pdf linear algebra and tensor analysis notes free download. Objects that tensors may map between include vectors which are often, but not always, understood as arrows with length that point in a direction and scalars which are often familiar numbers such as the real numbers, and, recursively, even.
In order to read the online edition of the feynman lectures on physics, javascript must be supported by your browser and enabled. Finally, in section7we describe the notation used for tensors in physics. Note that the sum of tensors at different points in space is not a tensor if the s are position dependent. The addition and quotient theorems in tensor analysis. Quantities that have magnitude and two directions, e. We have already established that a product of two cartesian tensors either a direct product or one involving contraction yields as its result a tensor. 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. For an analogy, retreat to the case of a real valued function. By this law, we can test a given quantity is a tensor or not.
A tensor of rank 1 is required to represent the electric field surrounding a point charge in space or the gravitational field of a massive object. It is intended to serve as a bridge from the point where most undergraduate students leave off in their studies of mathematics to the place where most texts on tensor analysis begin. Leta bea coordinate vector attached to the point p that has coordinates. Scalars, vectors and tensors a scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. In rowvector notation, the basis vectors themselves are just i ex 1,0,0 j ey 0,1,0 k ez 0,0,1 1. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. Association of antisymmetric tensor of order two and vectors. In 1822 cauchy introduced the cauchy stress tensor. The quotient theorem for tensors consider an array of the form ap,qi where p and qi are sequences of indices and suppose the inner product of ap,qi with an arbitrary contravariant tensor of rank one a vector. Rank2 tensors may be called dyads although this, in common use, may be restricted to the outer product of two vectors and hence is a special case of rank2 tensors assuming it meets the requirements of a tensor and hence transforms as a tensor.
If an index shows up once on the left hand side lhs of sign, it must show up once and only once on the right hand side rhs of sign. Mgsu msc mathematics lecture series, quotient law, tensors analysis,bikaner quotient law of tensor pdf. Vectors and tensors as thenotationimplies, thematrices ofcoe cients a anda 1 are inverses of each other. This note is for i year post graduate classes of kannur university india.
Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Pdf advanced mathematical physics notes free download. The direct method requires us to nd out if the given quantity obeys the transformation law or not. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. The syllabus is tensor analysisintroductiondefinitiondefinition of different rank tensorscontraction and direct productquotient rulepseudo tensors general. Bibliography includes bibliographical references p.
From this, we see that the vector athat points from p to q is just the position vector of the last point minus that of the. Dear reader, there are several reasons you might be seeing this page. Tensor comes from the latin tendere, which means \to stretch. In tensor analysis it is often necessary to ascertain whether a given quantity is tensor or not and if it is tensor we have to nd its rank. However, under linear coordinate transformations the s are constant, so the sum of tensors at different points behaves as a tensor. Tensors of the same type can be added or subtracted to form new tensors. Tensor analysis in euclidean space james emery edited. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. An introduction to tensors for students of physics and. It also illustrates the applications of tensors to differential geometry, mechanics and relativity. Cauchys law cauchys law states that there exists a cauchy stress tensor.
The transformation, a and its covariant counterpart, b, can be any linear transformation on the coordinates such as the general coordinate transformations we considered in chapter ii. Since the curvature tensor depends on a connectionnot metric, is it the relevant quantity to characterize the curvature of riemannian manifolds. A vector is a bookkeeping tool to keep track of two pieces of information typically magnitude and direction for a physical quantity. A secondorder tensor is one that has two basis vectors standing next to each other, and they satisfy the same rules as those of a vector hence, mathematically, tensors are also called vectors. Tensors with respect to the general functional transformation r 32 part ii. The generalization of the case that we analyzed, the quotient rule, is a tool. Another way to write the last equation is v v 1,v 2,v 3, which is another expression of the same vector v, this time in terms of its components in the new. It is the output of the function iafter the independent variable. It is linear in vc, hence may be shown by the quotient theorem to be a tensor.
Syllabus tensor analysisintroductionde nitionde nition of di erent rank tensors contraction and direct product quotient rulepseudo tensors general tensors metric tensors. A primer on tensor calculus 1 introduction in physics, there is an overwhelming need to formulate the basic laws in a socalled invariant form. Since vectors are higher order quantities than scalars, the physical realities they correspond to are typically more complex than those represented by scalars. Quotient theorem tensors thread starter latentcorpse.
Kronecker and alternating tensors, association of antisymmetric tensor of order two and vectors. Then the quotient rule permits us to conclude that the coefficient array x ij must also be a tensor. If an index shows up once on the left hand side lhs of sign, it must show up once and only once on the right hand side rhs of. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Now, i can see that the rcdab is a tensor by construction based on the lhs of the ricci identity. The riemann curvature tensor rcdab is given by the ricci identity. Introduction systems of different orders sumination convention kronecker symbols some results of determinant differentiation of a determinant linear equations, cramers rule examples exercises tensor algebra. Contraction, quotient law of tensors, symmetric and antisymmetric tensors, invariant tensors. For algebra on vectors and tensors, an index must show up twice and only twice.
Quotient rule if a and b are tensors, and if the expression a bt is invariant under coordinate transformation, then t is a tensor. The quotient rule mctyquotient20091 a special rule, thequotientrule, exists for di. Although we cannot usefully divide tensors, one by another, an entity like in the equation, where and are tensors, can be formally regarded as the quotient of and. Stress balance principles 03 the cauchy stress tensor. Scalars, vectors and tensors pennsylvania state university. Textbook of tensor calculus and differential geometry. Law of universal gravitation, and of the electromagnetic field in classical electrodynamics maxwells four equations are all given in vector form. Introduction to tensor calculus for general relativity. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about tensors in general.