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- Heinbockel J.H. Introduction to tensor calculus and continuum mechanics
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PREFACE This is an introductory text which presents fundamental concepts from the subject areas of tensor calculus, differential geometry and continuum mechanics. The material presented is suitable for a two semester course in applied mathematics and is flexible enough to be presented to either upper level undergraduate or beginning graduate students majoring in applied mathematics, engineering or physics. The presentation assumes the students have some knowledge from the areas of matrix theory, linear algebra and advanced calculus.

Each section includes many illustrative worked examples. At the end of each section there is a large collection of exercises which range in difficulty. Many new ideas are presented in the exercises and so the students should be encouraged to read all the exercises. The purpose of preparing these notes is to condense into an introductory text the basic definitions and techniques arising in tensor calculus, differential geometry and continuum mechanics.

In particular, the material is presented to i develop a physical understanding of the mathematical concepts associated with tensor calculus and ii develop the basic equations of tensor calculus, differential geometry and continuum mechanics which arise in engineering applications.

From these basic equations one can go on to develop more sophisticated models of applied mathematics. The material is presented in an informal manner and uses mathematics which minimizes excessive formalism. The material has been divided into two parts. The first part deals with an introduction to tensor calculus and differential geometry which covers such things as the indicial notation, tensor algebra, covariant differentiation, dual tensors, bilinear and multilinear forms, special tensors, the Riemann Christoffel tensor, space curves, surface curves, curvature and fundamental quadratic forms.

The second part emphasizes the application of tensor algebra and calculus to a wide variety of applied areas from engineering and physics. The selected applications are from the areas of dynamics, elasticity, fluids and electromagnetic theory. The continuum mechanics portion focuses on an introduction of the basic concepts from linear elasticity and fluids. The Appendix B contains a listing of Christoffel symbols of the second kind associated with various coordinate systems.

The Appendix C is a summary of useful vector identities. All rights reserved. Reproduction and distribution of these notes is allowable provided it is for non-profit purposes only.

Exercise 1. Exercise 2. A scalar field describes a one-to-one correspondence between a single scalar number and a point. An ndimensional vector field is described by a one-to-one correspondence between n-numbers and a point. Let us generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single point. When these numbers obey certain transformation laws they become examples of tensor fields.

In general, scalar fields are referred to as tensor fields of rank or order zero whereas vector fields are called tensor fields of rank or order one. Closely associated with tensor calculus is the indicial or index notation. In section 1 the indicial notation is defined and illustrated. We also define and investigate scalar, vector and tensor fields when they are subjected to various coordinate transformations. It turns out that tensors have certain properties which are independent of the coordinate system used to describe the tensor.

Because of these useful properties, we can use tensors to represent various fundamental laws occurring in physics, engineering, science and mathematics. These representations are extremely useful as they are independent of the coordinate systems considered.

In the index notation, A still shorter notation, depicting the vectors A the quantities Ai ,. Note that many of the operations that occur in the use of the index notation apply not only for three dimensional vectors, but also for N dimensional vectors. In future sections it is necessary to define quantities which can be represented by a letter with subscripts or superscripts attached. Such quantities are referred to as systems. When these quantities obey certain transformation laws they are referred to as tensor systems.

For example, quantities like Akij. The subscripts or superscripts are referred to as indices or suffixes. When such quantities arise, the indices must conform to the following rules: 1. They are lower case Latin or Greek letters. The letters at the end of the alphabet u, v, w, x, y, z are never employed as indices. The number of subscripts and superscripts determines the order of the system. A system with one index is a first order system. A system with two indices is called a second order system.

In general, a system with N indices is called a N th order system. A system with no indices is called a scalar or zeroth order system. The type of system depends upon the number of subscripts or superscripts occurring in an expression. In contrast, the systems Aijk and Cpmn are not of the same type because one system has two superscripts and the other system has only one superscript.

For certain systems the number of subscripts and superscripts is important. In other systems it is not of importance. The meaning and importance attached to sub- and superscripts will be addressed later in this section. In the use of superscripts one must not confuse powers of a quantity with the superscripts. In order to write a superscript quantity to a power, use parentheses.

For example, x2 3 is the variable x2 cubed. One of the reasons for introducing the superscript variables is that many equations of mathematics and physics can be made to take on a concise and compact form. There is a range convention associated with the indices. This convention states that whenever there is an expression where the indices occur unrepeated it is to be understood that each of the subscripts or superscripts can take on any of the integer values 1, 2,.

For example,. The symbol ij refers to all of the components of the system simultaneously. These indices are called free indices and can take on any of the values 1, 2 or 3 as specified by the range.

Since there are three choices for the value for m and three choices for a value of n we find that equation 1. Symmetric and Skew-Symmetric Systems A system defined by subscripts and superscripts ranging over a set of values is said to be symmetric in two of its indices if the components are unchanged when the indices are interchanged.

A system defined by subscripts and superscripts is said to be skew-symmetric in two of its indices if the components change sign when the indices are interchanged. It is left as an exercise to show this completely skew- symmetric systems has 27 elements, 21 of which are zero. This is expressed as saying that the above system has only one independent component. The summation being over the integer values specified by the range. A repeated index is called a summation index, while an unrepeated index is called a free index.

The summation convention requires that one must never allow a summation index to appear more than twice in any given expression. Because of this rule it is sometimes necessary to replace one dummy summation symbol by some other dummy symbol in order to avoid having three or more indices occurring on the same side of the equation.

The index notation is a very powerful notation and can be used to concisely represent many complex equations. For the remainder of this section there is presented additional definitions and examples to illustrated the power of the indicial notation. This notation is then employed to define tensor components and associated operations with tensors. The range convention states that k is free to have any one of the values 1 or 2, k is a free index.

Since the subscript i repeats itself, the summation convention requires that a summation be performed by letting the summation subscript take on the values specified by the range and then summing the results. The index k which appears only once on the left and only once on the right hand side of the equation is called a free index.

It should be noted that both k and i are dummy subscripts and can be replaced by other letters. Here we have purposely changed the indices so that when we substitute for xm , from one equation into the other, a summation index does not repeat itself more than twice.

It is left as an exercise to expand both the matrix equation and the indicial equation and verify that they are different ways of representing the same thing. Observe that the index notation employs dummy indices.

At times these indices are altered in order to conform to the above summation rules, without attention being brought to the change. As in this example, the indices q and j are dummy indices and can be changed to other letters if one desires.

Also, in the future, if the range of the indices is not stated it is assumed that the range is over the integer values 1, 2 and 3. To systems containing subscripts and superscripts one can apply certain algebraic operations. We present in an informal way the operations of addition, multiplication and contraction.

That i is again a is we can add or subtract like components in systems. The product of two systems is obtained by multiplying each component of the first system with each component of the second system. Such a product is called an outer product. The order of the resulting product system is the sum of the orders of the two systems involved in forming the product. The product system represents N 5 terms constructed from all possible products of the components from Aij with the components from B mnl.

The operation of contraction occurs when a lower index is set equal to an upper index and the summation convention is invoked. Here the symbol C mnl is used to represent the third order system that results when the contraction is performed. Whenever a contraction is performed, the resulting system is always of order 2 less than the original system.

Under certain special conditions it is permissible to perform a contraction on two lower case indices. These special conditions will be considered later in the section. The above operations will be more formally defined after we have explained what tensors are.

The e-permutation symbol and Kronecker delta Two symbols that are used quite frequently with the indicial notation are the e-permutation symbol and the Kronecker delta. The e-permutation symbol is sometimes referred to as the alternating tensor.

The e-permutation symbol, as the name suggests, deals with permutations.

PREFACE This is an introductory text which presents fundamental concepts from the subject areas of tensor calculus, differential geometry and continuum mechanics. The material presented is suitable for a two semester course in applied mathematics and is flexible enough to be presented to either upper level undergraduate or beginning graduate students majoring in applied mathematics, engineering or physics. The presentation assumes the students have some knowledge from the areas of matrix theory, linear algebra and advanced calculus. Each section includes many illustrative worked examples. At the end of each section there is a large collection of exercises which range in difficulty. Many new ideas are presented in the exercises and so the students should be encouraged to read all the exercises.

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Vector and Tensor Calculus. Typically, the vectors and tensors used in continuum mechanics will be functions of position (as well as time).

Skip to main content Skip to table of contents. Advertisement Hide. This service is more advanced with JavaScript available. Tensor Analysis and Continuum Mechanics. Front Matter Pages i-vii.

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A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra.

Talpaert, , Y. January ; 57 1 : B1. Tensor Analysis and Continuum Mechanics. Kluwer Acad Publ, Dordrecht, Netherlands. ISBN

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