**Introdução ao cálculo tensorial**

Introdução ao cálculo tensorial

(Parte **1** de 5)

Introduction to

Tensor Calculus and Continuum Mechanics by J.H. Heinbockel

Department of Mathematics and Statistics Old Dominion University

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 (i) 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 A contains units of measurements from the Systeme International d’Unites along with some selected physical constants. 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.

J.H. Heinbockel, 1996

Copyright c©1996 by J.H. Heinbockel. All rights reserved.

Reproduction and distribution of these notes is allowable provided it is for non-proﬁt purposes only.

§1.1 INDEX NOTATION | 1 |

Exercise 1.1 | 28 |

§1.2 TENSOR CONCEPTS AND TRANSFORMATIONS | 35 |

Exercise 1.2 | 54 |

§1.3 SPECIAL TENSORS | 65 |

Exercise 1.3 | 101 |

§1.4 DERIVATIVE OF A TENSOR | 108 |

Exercise 1.4 | 123 |

§1.5 DIFFERENTIAL GEOMETRY AND RELATIVITY | 129 |

Exercise 1.5 | 162 |

PART 1: INTRODUCTION TO TENSOR CALCULUS

§2.1 TENSOR NOTATION FOR VECTOR QUANTITIES | 171 |

Exercise 2.1 | 182 |

§2.2 DYNAMICS | 187 |

Exercise 2.2 | 206 |

§2.3 BASIC EQUATIONS OF CONTINUUM MECHANICS | 211 |

Exercise 2.3 | 238 |

§2.4 CONTINUUM MECHANICS (SOLIDS) | 243 |

Exercise 2.4 | 272 |

§2.5 CONTINUUM MECHANICS (FLUIDS) | 282 |

Exercise 2.5 | 317 |

§2.6 ELECTRIC AND MAGNETIC FIELDS | 325 |

Exercise 2.6 | 347 |

BIBLIOGRAPHY | 352 |

APPENDIX A UNITS OF MEASUREMENT | 353 |

APPENDIX C VECTOR IDENTITIES | 362 |

1 PART 1: INTRODUCTION TO TENSOR CALCULUS

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.

§1.1 INDEX NOTATION Two vectors A and B can be expressed in the component form

where e1, e2 and e3 are orthogonal unit basis vectors. Often when no confusion arises, the vectors A and B are expressed for brevity sake as number triples. For example, we can write

A =( A1,A 2,A 3)a nd B =( B1,B 2,B 3) where it is understood that only the components of the vectors A and B are given. The unit vectors would

A still shorter notation, depicting the vectors A and B is the index or indicial notation. In the index notation, the quantities represent the components of the vectors A and B. This notation focuses attention only on the components of the vectors and employs a dummy subscript whose range over the integers is speciﬁed. The symbol Ai refers to all of the components of the vector A simultaneously. The dummy subscript i can have any of the integer values 1,2o r3 . For i = 1 we focus attention on the A1 component of the vector A. Setting i =2 focuses attention on the second component A2 of the vector A and similarly when i = 3 we can focus attention on the third component of A. The subscript i is a dummy subscript and may be replaced by another letter, say p, so long as one speciﬁes the integer values that this dummy subscript can have.

It is also convenient at this time to mention that higher dimensional vectors may be deﬁned as ordered n−tuples. For example, the vector

with components Xi,i =1 ,2, | ,N is called a N−dimensional vector. Another notation used to represent |

this vector is

where e1, e2, | , eN |

are linearly independent unit base vectors. 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 eijk δij δji Ai Bj aij.

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. 2. 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 Nth 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.

For example, Aijk and Bmst, (all indices range 1 to N), are of the same type because they have the same number of subscripts and superscripts. In contrast, the systems Aijk and Cmnp 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. For example, if we replace the independent variables (x,y,z)b y the symbols (x1,x 2,x 3), then we are letting y = x2 where x2 is a variable and not x raised to a power. Similarly, the substitution z = x3 is the replacement of z by the variable x3 and this should not be confused with x raised to a power. 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,... ,N where N is a speciﬁed integer. For example, the Kronecker delta symbol δij, deﬁned by δij =1 if i = j and δij =0 for i = j,w ith i,j ranging over the values 1,2,3, represents the 9 quantities

The symbol δij refers to all of the components of the system simultaneously. As another example, consider

the subscripts m, n occur unrepeated on the left side of the equation and hence must also occur on the right hand side of the equation. These indices are called “free ”indices and can take on any of the values 1,2o r3 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 (1.1.1) represents nine equations simultaneously. These nine equations are

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. For example, the third order system Tijk is symmetric in the indices i and k if

Tijk = Tkji for all values of i, j and k.

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. For example, the fourth order system Tijkl is skew-symmetric in the indices i and l if

Tijkl = −Tljki for all values of ijk and l.

As another example, consider the third order system aprs, p,r,s =1 ,2,3 which is completely skewsymmetric in all of its indices. We would then have

It is left as an exercise to show this completely skew- symmetric systems has 27 elements, 21 of which are zero. The 6 nonzero elements are all related to one another thru the above equations when (p,r,s)= (1,2,3). This is expressed as saying that the above system has only one independent component.

Summation Convention

The summation convention states that whenever there arises an expression where there is an index which occurs twice on the same side of any equation, or term within an equation, it is understood to represent a summation on these repeated indices. 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.

EXAMPLE 1.1-1 The two equations

can be represented as one equation by introducing a dummy index, say k, and expressing the above equations as

The range convention states that k is free to have any one of the values 1 or 2, (k is a free index). This equation can now be written in the form

where i is the dummy summation index. When the summation sign is removed and the summation convention is adopted we have

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. For example, we can write where m is the summation index and n is the free index. Summing on m produces yn = an1x1 + an2x2 and letting the free index n take on the values of 1 and 2 we produce the original two equations.

EXAMPLE 1.1-2. For yi = aijxj,i ,j =1 ,2,3a nd xi = bijzj,i ,j =1 ,2,3s olve for the y variables in terms of the z variables.

Solution: In matrix form the given equations can be expressed: y1 y2

Now solve for the y variables in terms of the z variables and obtain y1 y2

The index notation employs indices that are dummy indices and so we can write

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. Substituting we ﬁnd the indicial form of the above matrix equation as where n is the free index and m,j are the dummy summation indices. 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.

EXAMPLE 1.1-3. The dot product of two vectors Aq,q =1 ,2,3a nd Bj,j =1 ,2,3 can be represented with the index notation by the product AiBi = AB cosθi =1 ,2,3,A = | A|,B = | B|. Since the subscript i is repeated it is understood to represent a summation index. Summing on i over the range speciﬁed, there results

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 1,2a nd 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.

Addition, Multiplication and Contraction

The algebraic operation of addition or subtraction applies to systems of the same type and order. That is we can add or subtract like components in systems. For example, the sum of Aijk and Bijk is again a system of the same type and is denoted by Cijk = Aijk + Bijk, where like components are added. 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. For example, if Aij is a second order system and Bmnl is a third order system, with all indices having the range 1 to N, then the product system is ﬁfth order and is denoted Cimnlj = AijBmnl. The product system represents N5 terms constructed from all possible products of the components from Aij with the components from Bmnl. The operation of contraction occurs when a lower index is set equal to an upper index and the summation convention is invoked. For example, if we have a ﬁfth order system Cimnlj and we set i = j and sum, then we form the system

(Parte **1** de 5)