Introduction to Tensor Calculus and Continuum Mechanics by J.H. Heinbockel Department of Mathematics and Statistics Old Dominion University 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 introduc- tion 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, cur- vature 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 electromag- netic 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 Syst`eme International d’Unit`es 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. INTRODUCTION TO TENSOR CALCULUS AND CONTINUUM MECHANICS PART 1: INTRODUCTION TO TENSOR CALCULUS §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 2: INTRODUCTION TO CONTINUUM MECHANICS §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 B CHRISTOFFEL SYM BOLS OF SECOND KIND 355 APPENDIX C VECTOR IDENTITIES 362 INDEX 363 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 n- dimensional 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  A = A 1  e 1 + A 2  e 2 + A 3  e 3 and  B = B 1  e 1 + B 2  e 2 + B 3  e 3 , where  e 1 ,  e 2 and  e 3 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 =(A 1 ,A 2 ,A 3 )and  B =(B 1 ,B 2 ,B 3 ) where it is understood that only the components of the vectors  A and  B are given. The unit vectors would be represented  e 1 =(1, 0, 0),  e 2 =(0, 1, 0),  e 3 =(0, 0, 1). A still shorter notation, depicting the vectors  A and  B is the index or indicial notation. In the index notation, the quantities A i ,i=1, 2, 3andB p ,p=1, 2, 3 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 A i refers to all of the components of the vector  A simultaneously. The dummy subscript i can have any of the integer values 1, 2or3. For i = 1 we focus attention on the A 1 component of the vector  A. Setting i =2focuses attention on the second component A 2 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. 2 It is also convenient at this time to mention that higher dimensional vectors may be defined as ordered n−tuples. For example, the vector  X =(X 1 ,X 2 , ,X N ) with components X i ,i=1, 2, ,N is called a N−dimensional vector. Another notation used to represent this vector is  X = X 1  e 1 + X 2  e 2 + ···+ X N  e N where  e 1 ,  e 2 , ,  e N 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 A k ij e ijk δ ij δ j i A i B j a ij . 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, A i jk and B m st , (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 A i jk and C mn p 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)bythesymbols(x 1 ,x 2 ,x 3 ), then we are letting y = x 2 where x 2 is a variable and not x raised to a power. Similarly, the substitution z = x 3 is the replacement of z by the variable x 3 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, (x 2 ) 3 is the variable x 2 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, 3 the Kronecker delta symbol δ ij , defined by δ ij =1ifi = j and δ ij =0fori = j,withi, j ranging over the values 1,2,3, represents the 9 quantities δ 11 =1 δ 21 =0 δ 31 =0 δ 12 =0 δ 22 =1 δ 32 =0 δ 13 =0 δ 23 =0 δ 33 =1. The symbol δ ij refers to all of the components of the system simultaneously. As another example, consider the equation  e m ·  e n = δ mn m, n =1, 2, 3(1.1.1) 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, 2or3 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  e 1 ·  e 1 =1  e 2 ·  e 1 =0  e 3 ·  e 1 =0  e 1 ·  e 2 =0  e 2 ·  e 2 =1  e 3 ·  e 2 =0  e 1 ·  e 3 =0  e 2 ·  e 3 =0  e 3 ·  e 3 =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. For example, the third order system T ijk is symmetric in the indices i and k if T ijk = T kji 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 T ijkl is skew-symmetric in the indices i and l if T ijkl = −T ljki for all values of ijk and l. As another example, consider the third order system a prs , p,r,s =1, 2, 3 which is completely skew- symmetric in all of its indices. We would then have a prs = −a psr = a spr = −a srp = a rsp = −a rps . 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. 4 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 y 1 = a 11 x 1 + a 12 x 2 y 2 = a 21 x 1 + a 22 x 2 can be represented as one equation by introducing a dummy index, say k, and expressing the above equations as y k = a k1 x 1 + a k2 x 2 ,k=1, 2. 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 y k = 2  i=1 a ki x i = a k1 x 1 + a k2 x 2 where i is the dummy summation index. When the summation sign is removed and the summation convention is adopted we have y k = a ki x i i, k =1, 2. 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 y n = a nm x m n, m =1, 2 where m is the summation index and n is the free index. Summing on m produces y n = a n1 x 1 + a n2 x 2 and letting the free index n take on the values of 1 and 2 we produce the original two equations. 5 EXAMPLE 1.1-2. For y i = a ij x j ,i,j=1, 2, 3andx i = b ij z j ,i,j=1, 2, 3solveforthey variables in terms of the z variables. Solution: In matrix form the given equations can be expressed:   y 1 y 2 y 3   =   a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33     x 1 x 2 x 3   and   x 1 x 2 x 3   =   b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33     z 1 z 2 z 3   . Now solve for the y variables in terms of the z variables and obtain   y 1 y 2 y 3   =   a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33     b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33     z 1 z 2 z 3   . The index notation employs indices that are dummy indices and so we can write y n = a nm x m ,n,m=1, 2, 3andx m = b mj z j ,m,j=1, 2, 3. Here we have purposely changed the indices so that when we substitute for x m , 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 y n = a nm b mj z j , m,n,j =1, 2, 3 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 A q ,q=1, 2, 3andB j ,j=1, 2, 3 can be represented with the index notation by the product A i B i = AB cos θi=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 A 1 B 1 + A 2 B 2 + A 3 B 3 = AB cos θ. 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, 2and3. 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. 6 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 A i jk and B i jk is again a system of the same type and is denoted by C i jk = A i jk + B i jk , 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 A i j is a second order system and B mnl is a third order system, with all indices having the range 1 to N, then the product system is fifth order and is denoted C imnl j = A i j B mnl . The product system represents N 5 terms constructed from all possible products of the components from A i j 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. For example, if we have a fifth order system C imnl j and we set i = j and sum, then we form the system C mnl = C jmnl j = C 1mnl 1 + C 2mnl 2 + ···+ C Nmnl N . 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. A permutation is an arrangement of things. When the order of the arrangement is changed, a new permutation results. A transposition is an interchange of two consecutive terms in an arrangement. As an example, let us change the digits 1 2 3 to 3 2 1 by making a sequence of transpositions. Starting with the digits in the order 1 2 3 we interchange 2 and 3 (first transposition) to obtain 1 3 2. Next, interchange the digits 1 and 3 ( second transposition) to obtain 312. Finally, interchange the digits 1 and 2 (third transposition) to achieve 3 2 1. Here the total number of transpositions of 1 2 3 to 3 2 1 is three, an odd number. Other transpositions of 1 2 3 to 3 2 1 can also be written. However, these are also an odd number of transpositions. [...]... between the vector notation and index notation Observe that a vector A can be represented A = Ai ei or its components can be represented A · ei = Ai , i = 1, 2, 3 Do not set a vector equal to a scalar That is, do not make the mistake of writing A = Ai as this is a misuse of the equal sign It is not possible for a vector to equal a scalar because they are two entirely different quantities A vector has both... indices is to put parenthesis about the indices which are not to be summed For example, a(k)j δ(k)(k) = akj , since δ(k)(k) represents a single term and the parentheses indicate that no summation is to be performed At any time we may employ either the underscore notation, the capital letter notation or the parenthesis notation to denote that no summation of the indices is to be performed To avoid confusion... surface integrals are to be extended over the range specified by the indices This suggests that the divergence theorem can be applied to vectors in n−dimensional spaces The vector form and indicial notation for the Stokes theorem are ( F · dr × F ) · n dσ = S Fi dxi eijk Fk,j ni dσ = C S i, j, k = 1, 2, 3 (1.1.17) C and the Green’s theorem in the plane, which is a special case of the Stoke’s theorem, can... They are not involved in any summations and their values, whatever you choose to assign them, are fixed Let us assign a value of j and k to the values of j and k The underscore is to remind you that these values for j and k are fixed and not to be summed When we perform the summation over the summation index i we assign values to i from the range and then sum over these values Performing the indicated... the dummy summation index Curl To represent the vector B = curl A = × A in Cartesian coordinates, we note that the index notation focuses attention only on the components of this vector The components Bi , i = 1, 2, 3 of B can be represented Bi = ei · curl A = eijk Ak,j , for i, j, k = 1, 2, 3 where eijk is the permutation symbol introduced earlier and Ak,j = ∂Ak ∂xj To verify this representation of... and j EXAMPLE 1.1-12 Given the vectors Ap , p = 1, 2, 3 and Bp , p = 1, 2, 3 the cross product of these two vectors is a vector Cp , p = 1, 2, 3 with components Ci = eijk Aj Bk , i, j, k = 1, 2, 3 (1.1.2) The quantities Ci represent the components of the cross product vector C = A × B = C1 e1 + C2 e2 + C3 e3 The equation (1.1.2), which defines the components of C, is to be summed over each of the indices... A = Ai is an incorrect notation because a vector can not equal a scalar The notation A · ei = Ai should be used to express the ith component of a vector (a) A · (B × C) (c) B(A · C) (b) A × (B × C) (d) B(A · C) − C(A · B) eijk = ejki = ekij (b) eijk = −ejik = −eikj = −ekji 4 Show the e permutation symbol satisfies: (a) 5 Use index notation to verify the vector identity A × (B × C) = B(A · C) − C(A ·... elements of an N × N matrix Here i and j are allowed to range over the integer values 1, 2, , N Consider the product aij δik where the range of i, j, k is 1, 2, , N The index i is repeated and therefore it is understood to represent a summation over the range The index i is called a summation index The other indices j and k are free indices They are free to be assigned any values from the range of the... noncoplaner vectors A, B, C The absolute value is needed because sometimes the triple scalar product is negative This physical interpretation can be obtained from an analysis of the figure 1.1-4 Figure 1.1-4 Triple scalar product and volume 17 In figure 1.1-4 observe that: (i) |B × C| is the area of the parallelogram P QRS (ii) the unit vector en = B×C |B × C| is normal to the plane containing the vectors B... form where we can use the e − δ identity applied to the term in parentheses to produce Gq = (δqn δip − δqp δin )eijk Aj Bk Cn Dp Simplifying this expression we have: Gq = eijk [(Dp δip )(Cn δqn )Aj Bk − (Dp δqp )(Cn δin )Aj Bk ] = eijk [Di Cq Aj Bk − Dq Ci Aj Bk ] = Cq [Di eijk Aj Bk ] − Dq [Ci eijk Aj Bk ] which are the vector components of the vector C(D · A × B) − D(C · A × B) 18 Transformation . for non-profit purposes only. INTRODUCTION TO TENSOR CALCULUS AND CONTINUUM MECHANICS PART 1: INTRODUCTION TO TENSOR CALCULUS §1.1 INDEX NOTATION 1 Exercise 1.1 28 §1.2 TENSOR CONCEPTS AND TRANSFORMATIONS. 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. 353 APPENDIX B CHRISTOFFEL SYM BOLS OF SECOND KIND 355 APPENDIX C VECTOR IDENTITIES 362 INDEX 363 1 PART 1: INTRODUCTION TO TENSOR CALCULUS A scalar field describes a one -to- one correspondence