TENSOR CALCULUS with applications to Differential Theory of Surfaces and Dynamics Nielsen, Søren R. K.

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1 Aalborg Universitet TENSOR CALCULUS with applications to Differential Theory of Surfaces and Dynamics Nielsen, Søren R. K. Publication date: 2018 Document Version Publisher's PDF, also known as Version of record Link to publication from Aalborg University Citation for published version (APA): Nielsen, S. R. K. (2018). TENSOR CALCULUS with applications to Differential Theory of Surfaces and Dynamics. Aalborg: Department of Civil Engineering, Aalborg University. DCE Technical Reports, No. 242 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.? You may not further distribute the material or use it for any profit-making activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: september 02, 2018

2 TENSOR CALCULUS with applications to Differential Theory of Surfaces and Dynamics Søren R. K. Nielsen a) b) 1 θ 2 x 3 f ω a x 2 1 θ 1 a x 1 Aalborg University Department of Civil Engineering ISSN X DCE Technical Report No. 242

3 2 Published 2018 by: Aalborg University Department of Civil Engineering Thomas Manns Vej 23 DK-9220 Aalborg East, Denmark Printed in Aalborg at Aalborg University ISSN X DCE Technical Report No. 242

4 Contents 1 Tensor Calculus Vectors, curvilinear coordinates, covariant and contravariant bases Tensors, dyads and polyads Gradient, covariant and contravariant derivatives Riemann-Christoffel tensor Geodesics Exercises Differential Theory of Surfaces Differential geometry of surfaces, first fundamental form Principal curvatures, second fundamental form Codazzi s equations Surface area elements Exercises Dynamics Equation of motion of a mass particle Nonlinear multi-degree-of-freedom systems Exercises Index 61 8 Bibliography 62 3

5 4 Contents

6 Preface The present outline on tensor calculus with special application to differential theory of surfaces and dynamics represents a modified and extended version of a lecture note written by the author as an introduction to a course on shell theory given together with Ph.D. Jesper Winther Stærdahl and Professor Lars Vabbersgaard Andersen in 2007, based on the book of (Niordson, 1985). The text is written with inspiration from both mathematical based texts on tensor calculus, such as the books of (Spain, 1965) and (Synge and Schild, 1966), and the more geometrical based interpretation often used in continuum mechanics, (Malvern, 1969). Chapter 1 introduces the concept of vectors and tensors in a Riemann space, and their components in covariant and contravariant vector and tensor bases. Next, the concepts of gradient of vectors and tensors, as well as co- and contravariant derivatives are introduced. Finally, the Riemann tensor and the concept of geodesics curves in Riemann space is treated. Chapter 2 deals with the differential theory of a surface in the three dimension Euclidean space, as described by its first and second fundamental form. Further, the Bianchi identities for the surface Riemann-Christoffel tensor, and the Codazzi equation for the second fundamental form are derived. Chapter 3 deals with the description of the motion of a mass particle in curvilinear coordinates and of a non-linear multi-degree-of-freedom dynamic system, which conveniently may be formulated in tensor notation. Thanks to Ph.D. student Tao Sun for preparing the figures. Aalborg, May 2018 Søren R.K. Nielsen 5

7 6 Contents

8 CHAPTER 1 Tensor Calculus 1.1 Vectors, curvilinear coordinates, covariant and contravariant bases x 3 s 3 g 3 θ 1 s 2 g 2 i 3 θ 3 i 1 i 2 g 1 x 2 θ 2 x 1 s 1 Fig. 1 1 Spherical coordinate system. Fig. 1-1 shows a Cartesian (x 1,x 2,x 3 )-coordinate system, as well as a spherical (θ 1,θ 2,θ 3 )- coordinate system. θ 1 is the zenith angle, θ 2 is the azimuth angle, and θ 3 is the radial distance. Notice that superscripts are use for the identification of the coordinates, which should not be confused with power raising. Instead, this will be indicated by parentheses, so if x 1 specifies the first Cartesian coordinate,(x 1 ) 2 indicates the corresponding coordinate raised to the second power. With the restrictions θ 1 [0,π], θ 2 ]0,2π] and θ 3 0 a one-to-one correspondence between the coordinates of the two systems exists except for points at the line x 1 = x 2 = 0. These represent the singular points of the mapping. For regular points the relations become, see e.g. (Zill and Cullen, 2005): 7

9 8 Chapter 1 Tensor Calculus x 1 x 2 x 3 = θ 3 sinθ 1 cosθ 2 θ 3 sinθ 1 sinθ 2 θ 3 cosθ 1 θ 1 θ 2 θ 3 = cos 1 x 3 (x1 ) 2 +(x 2 ) 2 +(x 3 ) 2 1 x2 tan x 1 (x1 ) 2 +(x 2 ) 2 +(x 3 ) 2 (1 1) We shall refer to θ l, l = 1,2,3, as curvilinear coordinates. Generally, the relation between the Cartesian and curvilinear coordinates are given by relations of the type: x j = f j (θ l ) (1 2) The Jacobian of the mapping (1-2) is defined as: [ ] f j J = det θ k (1 3) Points, where J = 0, represent singular points of the mapping. In any regular point, where J 0, the inverse mapping exists locally, given as: θ j = h j (x l ) (1 4) Forθ 1 = c 1 (constant), (1-2) defines a surface in space, defined by the parametric description: x j = f j (c 1,θ 2,θ 3 ) (1 5) Similar parametric description of surfaces arise, whenθ 2 orθ 3 are kept constant, and the remaining two coordinates are varied independently. The indicated three surfaces intersect pairwise along three curves s j, j = 1,2,3, at which two of the curvilinear coordinates are constant, e.g. the intersection curve s 1 is determined from the parametric description x j = f j (θ 1,c 2,c 3 ). All three surfaces intersect at the point P with the Cartesian coordinates x j = f j (c 1,c 2,c 3 ). Locally, at this point an additional curvilinear (s 1,s 2,s 3 )-coordinate system may be defined with axes made up of the said intersection curves as shown on Fig We shall refer to these coordinates as the arc length coordinates. The position vectorxfrom the origin of the Cartesian coordinate system to the pointp has the vector representation: x = x 1 i 1 +x 2 i 2 +x 3 i 3 = x j i j (1 6) where i j, j = 1,2,3, signify the orthonormal base vectors of the Cartesian coordinate system. In the last statement the summation convention over dummy indices has been used. This

10 1.1 Vectors, curvilinear coordinates, covariant and contravariant bases 9 convention will extensively be used in what follows. The rules is that dummy Latin indices involves summation over the range j = 1,2,3, whereas dummy Greek indices merely involves summation over the range α = 1,2. As an example a j b j = a 1 b 1 + a 2 b 2 + a 3 b 3, whereas a α b α = a 1 b 1 + a 2 b 2. The summation convention is abolished, if the dummy indices are surrounded by parentheses, i.e. a (j) b (j) merely means the product of thejth componentsa j andb j. Let dx j denote an infinitesimal increment of the jth coordinate, whereas the other coordinates are kept constant. From (1-6) follows that this induces a change of the position vector given as dx = i (j) dx (j). Hence: i j = x x j (1 7) A corresponding independent infinitisimal increment dθ j of the jth curvilinear coordinate induce a change of the position vector given as dx = g (j) dθ (j), so: g j = x θ j (1 8) g j is tangential to the curve s j at the point P, see Fig In any regular point the vectors g j, j = 1, 2, 3, may be used as base vector for the arc length coordinate system at P. The indicated vector base vectors is referred to as the covariant vector base, andg j are denoted the covariant base vectors. Especially, if the motion is described in arc length coordinates,g j becomes equal to the unit tangent vectorst j, i.e.: t j = x s j Use of the chain rule of partial differentiation provides: (1 9) g j = x θ j = x x k x k θ j = c k j i k (1 10) i j = x x j = x θ k θ k x j = d k j g k (1 11) where: c k j = xk = fk θ j θ j d k j = θk = hk x j x j (1 12) c k j specify the Cartesian components of the covariant base vector g j. Similarly, d k j specifies the components in the covariant base of the Cartesian base vector i j. Obviously, the following relation is valid:

11 10 Chapter 1 Tensor Calculus c k ld l j = xk θ l θ l x j = δk j (1 13) δ k j denotes Kronecker s delta in the applied notation, defined as: δ k j = { 0, j k 1, j = k (1 14) s 3 V g 3 g 1 s 1 g 3 s 2 g 2 P Fig. 1 2 Covariant and contravariant base vectors. Generally, the covariant base vectors g j are neither orthogonal nor normalized to the length 1, as is the case for the Cartesian base vectors i j. In order to perform similar operations on components of vectors and tensors as for an orthonormal vector base, a so-called contravariant vector base or dual vector base is introduced. The corresponding contravariant base vectorsg j are defined from: g j g k = δ k j (1 15) where indicates a scalar product. (1-15) implies that g 3 is orthogonal to g 1 and g 2. Further, the angle between g 3 and g 3 is acute in order that g 3 g 3 = +1, see Fig Generally, the contravariant base vectors can be determined from the covariant base vectors by means of the vector products: g 1 = 1 V g 2 g 3 g 2 = 1 V g 3 g 1 g i = 1 V eijk g j g k (1 16) g 3 = 1 V g 1 g 2 wherev denotes the volume of the parallelepiped spanned by the covariant base vectorsg j, see Fig This is given as:

12 1.1 Vectors, curvilinear coordinates, covariant and contravariant bases 11 V = g 1 (g 2 g 3 ) = g 2 (g 3 g 1 ) = g 3 (g 1 g 2 ) (1 17) e ijk is the permutation symbol defined as: e ijk = 1, (i,j,k) = (1,2,3), (2,3,1), (3,1,2) 1, (i,j,k) = (1,3,2), (3,2,1), (2,1,3) 0, else (1 18) The permutation symbol does not indicate the components of a 3rd order tensor in any coordinate system, and should merely be considered as an indexed sequence of numbers. For this reason we shall not make distinction between subscript and superscript indices, so we may writee ijk = e ijk. The permutation symbol and the Kronecker delta are related by the following so-called e δ relation: e ijk e klm = δ i l δj m δi m δj l (1 19) In the Cartesian coordinate system we have i j = i j, i.e. the dual vector basis is identical to original. Further, the Cartesian base vectors are constant throughout the space, whereas the covariant and contravariant base vector are locally attached to each point in the space, and change from point to point. The previous theory merely applies to a 3-dimensional Euclidean spaces. In the following this is generalized to a Riemann space of arbitrary dimension N. A Riemann space is a manifold related with a distance measure, which is defined between any two points in the space. A curve and surface in the 3-dimensional Euclidian space are examples of Riemann spaces of dimension N = 1 and N = 2. The space-time manifold in relativity theory is a Riemann space of dimensionn = 4. Similar to the concept in the 2- and 3-dimensional Euclidian spaces a vector in Riemann space is envisioned as a geometric quantity (an arrow with a given length and orientation). From this interpretation it follows that a vector is independent of any coordinate system used for its specification. Actually, infinitely many coordinate systems can be used for the representation (or decomposition) of one and the same vector. In the Cartesian, the covariant and the contravariant bases a given vectorvcan be represented in the following ways: v = v j i j = v j g j = v j g j (1 20) where v j,v j andv j denotes the Cartesian vector components, the covariant vector components, and the contravariant vector components of the vector v. Dummy indices now indicates summation over the range 1,...,N. Generally, Cartesian components of vectors and tensors will be indicated by a bar. Use of (1-15), and scalar multiplication of the two last relations in (1-20) with g k and g k, respectively, provides the following relations between the covariant and contravariant vector components:

13 12 Chapter 1 Tensor Calculus v j = g jk v k v j = g jk v k (1 21) where: g jk = g j g k = g kj g jk = g j g k = g kj (1 22) The indicated symmetry property of the quantities g jk and g jk follows from the commutativity of the involved scalar products. From (1-21) follows: v j = g jl v l = g jl g lk v k g jl g lk = δ k j (1 23) By the use of (1-20), (1-21) and (1-22) the following relations between the covariant and the contravariant base vectors may be derived: v j g j = v j g j = g jk v k g j = v j g jk g k v j g j = v j g j = g jk v k g j = v j g jk g k { gj = g jk g k g j = g jk g k (1 24) As seen g jk represent the components of g j in the contravariant vector base. Similarly g jk signify the components of g j in the covariant vector base. Use of (1-10) and (1-11) provides the following relation between the Cartesian and the covariant vector components: v j i j = v j g j = v j c k j i k = c j k vk i j v j g j = v j i j = v j d k j g k = d j k vk g j { v j = c j k vk v j = d j k vk (1 25) Finally, using (1-15) and (1-21) the scalar product of two vectors u and v can be evaluated in the following alternative ways: u v = {ūj v j = u j v j = g jk u j v k ū j v j = u j v j = g jk u j v k (1 26) where it has been used thatū j = ū j. 1.2 Tensors, dyads and polyads A second order tensortis defined as a linear mapping of a vectorvonto a vectoruby means of a scalar product, i.e.: u = T v (1 27)

14 1.2 Tensors, dyads and polyads 13 Since the vectors u and v are coordinate independent quantities, the 2nd order tensor T must also be independent of any selected coordinate system chosen for the specification of the relation (1-27). Equations in continuum mechanics and physics are independent of the chosen coordinate system for which reason these are basically formulated as tensor equations. A dyad (or outer product or tensor product) of two vectors a and b is denoted as ab. An alternatively often applied notation, which will not be used here, is a b. The tensor product of more than two vectors is denoted a polyad. The polyad abc formed by the three vectors a, b and c is denoted a triad, and the polyad abcd formed by the four vectors a, b, c and d is denoted a tetrad. For dyads and triads the following associative rules apply: m(ab) = (ma)b = a(mb) = (ab)m abc = (ab)c = a(bc) (ma)(nb) = mnab (1 28) wheremandnare arbitrary constants. Further, the following distributive rules are valid: a(b+c) = ab+ac (1 29) (a+b)c = ab+bc No commutative rule is valid for dyads formed by two vectorsaand b. Hence, in general: ab ba (1 30) If the outer product of two vectors entering a polyad is replaced by a scalar product of the same vectors, a polyad of an order two smaller is obtained. This operation is known as contraction. Contraction of a triad is possible in the following two ways: a bc = (a b)c (1 31) ab c = (b c)a The scalar product of two dyads, the so-called double contraction, can be defined in two ways: ab cd = (a c)(b d) (1 32) ab cd = (a d)(b c) Hence, the symbol defines scalar multiplication between the two first and the two last vectors in the two duads, whereas defines scalar multiplication between the first and the last vector, and the last and the first vectors in the two dyads. Because of the commutativity of the scalar product of two vectors follows: (a c)(b d) = (b d)(a c) = (c a)(d b) = (d b)(c a) (1 33) (a d)(b c) = (b c)(a d) = (c b)(d a) = (d a)(c b)

15 14 Chapter 1 Tensor Calculus Use of (1-32) and (1-33) provides the following identities for double contraction of two dyads: ab cd = ba dc = cd ab = dc ba ab cd = ba dc = cd ab = dc ba (1 34) From the second relation of (1-31) follows that the dyad ab/(b c) is mapping the vector c onto the vector a via a scalar multiplication. From the definition (1-27) follows that this makes a dyad a second order tensor. Now, it can be proved that any second order tensor can be represented as a linear combination of nine dyads, formed as outer product of three arbitrary linearly independent base vectors. Hence, we have following alternative representations of a second order tensortin the Cartesian, a covariant base and its contravariant base: T = T jk i j i k = T jk g j g k = T jk g j g k (1 35) The dyads i j i k, g j g k and g j g k form so-called tensor bases. Obviously, (1-35) represents the generalization to second order tensors of (1-20) for the decomposition of a vector in the corresponding vector bases. Tjk, T jk and T jk denotes the Cartesian components, the covariant components, and the contravariant components of the second order tensort. Using (1-10), (1-11), (1-13) and the associate rules (1-28) the following relations between the dyads and tensor components related to the considered three tensor bases may be derived: i j i k = d l j dm k g lg m g j g k = c l j cm k i li m g j g k = g jl g km g l g m = g jl g km g l g m (1 36) T jk = c j l ck mt lm T jk = g jl g km T lm, T jk = d j l dk m T lm, T jk = g jl g km T lm (1 37) Hence, c l j cm k and g jlg km specify the Cartesian and contravariant tensor components of the dyad g j g k, whereas g jl g km denotes the covariant tensor components of g j g k. (1-36) and (1-37) represent the equivalent to the relations (1-23) and (1-24) for base vectors and vector components. In some outlines of tensor calculus the transformation rules in (1-37) between Cartesian and curvilinear tensor components are used as a defining property of the tensorial character of a doubled indexed quantity, (Spain, 1965), (Synge and Schild, 1966). Alternatively, T may be decomposed after a tensor base with dyads of mixed covariant and contravariant base vectors, i.e.: T = T j k g jg k = T j k gk g j (1 38) T j k and T j k represent the mixed covariant and contravariant tensor components. In general, T j k T j k as a consequence of g jg k g k g j, i.e. the relative horizontal position of the upper

16 1.2 Tensors, dyads and polyads 15 and lower indices of the tensor components is important, and specify the sequence of covariant and contravariant base vector in the dyads of the tensor base. Tensor bases with dyads of mixed Cartesian and curvilinear base vectors may also be introduced. However, in what follows only the mixed curvilinear dyads in (1-38) will be considered. The following identities may be derived from (1-35) and (1-38) by the use of (1-24): T = T jk g j g k = T j l g jg l = T j l glk g j g k T jk = g lk T j l T = T jk g j g k = T k l g l g k = T k l g lj g j g k T = T jk g j g k = T l kg l g k = T l kg lj g j g k T = T jk g j g k = Tj l gj g l = Tj l g lkg j g k T jk = g jl T k l T jk = g jl T l k T jk = g lk Tj l (1 39) Use of (1-23) and (1-39) provides the following representations of the mixed components in terms of covariant and contravariant tensor components: T k j = g jl T kl T k j = g jl T lk T k j = g kl T lj T k j = g kl T jl (1 40) It follows from (1-40) that if T jk = T kj then T k j = T j k and T jk = T kj. A second order tensor for which the covariant components fulfill the indicated symmetry property is denoted a symmetric tensor. Let T jk denote the covariant components of a second order tensor T. The related so-called transposed tensort T is defined as the tensor with the covariant componentst kj, i.e.: T T = T kj g j g k (1 41) A symmetric tensor is defined by the symmetry of components T jk = T kj for which reason T T = T. The identity tensor is defined as the tensor with the covariant and contravariant componentsg jk and g jk. Use of (1-23) and (1-24) provides: g = g jk g j g k = g jk g j g k = δ k j g j g k (1 42) The mixed components ofgfollows from (1-23) and (1-40): g k j = g jl g kl = δj k g k j = g jl g lk = δ k j (1 43) Hence, the mixed components are equal to the Kronecker s delta, which explains the last statement for the mixed representation in (1-42). The designation identity tensor stems from the fact thatg maps any vectorvonto itself. Actually:

17 16 Chapter 1 Tensor Calculus g v = g jk g j g k v l g l = g jk v l g j δ l k = g jk v k g j = v j g j = v (1 44) The length of a vectorv is determined from: v 2 = v v = v g v = g jk v j v k = g jk v j v k (1 45) Because of its relationship to the length of a vector the identity tensor is also designated the metric tensor or the fundamental tensor. For an ordinary Riemann space g is positive definite, so v 2 is always positive. In relativity theory the metric tensor is indefinite, so the left hand side in Eq. (1-45) may be negative. A manifold related with an indefinite metric tensor is referred to as a pseudo-riemann space. The increment dx of the position vector x with Cartesian components d x j and covariant componentsdθ j is given as, cf. (1-25): dx = d x j i j = dθ j g j (1 46) Then, the lengthds = dx dx of the incremental position vector becomes, cf. (1-45): ds 2 = d x j d x j = g jk dθ j dθ k (1 47) The inverse tensort 1 related to the tensortis defined from the equation: T 1 T = T T 1 = g (1 48) LetT 1 jl denote the contravariant components oft 1, andt mk the covariant components oft. Then, cf. (1-42): g = δ k j gj g k = T 1 T = T 1 jl g j g l T mk g m g k = T 1 jl T mk δ l m gj g k = T 1 jl T lk g j g k T 1 jl T lk = δ k j (1 49) In matrix notation this means that the N-dimensional matrix, which stores the components T 1 jl, is the inverse of the matrix, which stores the components T lk. Similarly, the covariant components (T 1 ) jl of T 1 and the contravariant components T lk of T are stored in inverse matrices. In contrast, the matrices which store the contravariant components T 1 jl and T lk will not be mutual inverse. From (1-15) follows that: g j T g k = g j (T lm g l g m ) g k = T lm δ j l δk m = Tjk T jk = g j T g k (1 50)

18 1.3 Gradient, covariant and contravariant derivatives 17 Similarly: T jk = g j T g k Tj k = g j T g k T j k = gj T g k T jk = i j T i k (1 51) A fourth order tensor C can be expanded in any tensor base with tetrads made up of any combination of linear independent vectors of the Cartesian, the covariant or the contravariant vector bases as follows: C = C jklm g j g k g l g m = C klm j = C jkl mg j g k g l g m = C lm jk g j g k g l g m = C j lm k g j g k g l g m = C jk m l g j g k g l g m g j g k g l g m = C k m j l g j g k g l g m = C kl j mg j g k g l g m = C j m kl g j g k g l g m = C j l k m g jg k g l g m = C jk lm g jg k g l g m = C j klm g jg k g l g m = Cj k lm gj g k g l g m = Cjk l m gj g k g l g m = Cjkl m g j g k g l g m = C jklm g j g k g l g m = C jklm i j i k i l i m (1 52) Formally, the relations between the various mixed tensor components can be derived by raising and lowering indices by means of the covariant components g jk and the contravariant componentsg jk of the identity tensorg, cf. (1-40). As an example: C jk lm = g lrg ms C jkrs (1 53) 1.3 Gradient, covariant and contravariant derivatives Consider a scalar function a = a(x l ) = a(θ l ) of the Cartesian or curvilinear coordinates. The gradient of a, denoted as a, is a vector with the following Cartesian and contravariant representations: a = a x j i j = a θ j gj (1 54) Let θ k denote the covariant components of the incremental curvilinear coordinate vector θ = θ k g k. The corresponding increment a of the scalarais determined by: a = a θ = a θ j gj θ k g k = a θ j θj (1 55) The gradient of a vector functionv = v(x l ) = v(θ l ), denoted as v, is a second order tensor, which in complete analogy to (1-55) associates to any incremental curvilinear coordinate vector

19 18 Chapter 1 Tensor Calculus θ = θ k g k a corresponding increment v = v θ = v θ j θ j of the vector v. v has the following representations: v = v x i k k = ( vj i j ) i x k k = vj x i ji k k ( v θ k gk = (vj g j ) v g k j = θ k θ g k j +v j g j θ k v θ k gk = (v ( j g j ) g k vj g j = θ k θ k gj +v j θ k ) g k ) g k (1 56) At the derivation of the Cartesian representation it has been used that the base vector i j is constant as a function of x l. In contrast, the curvilinear base vectors depend on the curvilinear coordinates, which accounts for the second term within the parentheses in (1-56). Clearly, g j θ k and gj are vectors, which may hence be decomposed in the covariant and contravariant vector θ k bases as follows: g j θ k = g j θ k = { l j k { j l k g l g l (1 57) { l g signify the covariant components of j, and { j θ k l k is the contravariant components { l j k is denoted the Christoffel symbol. This is related to the co- and contravariant j k of gj. θ k components of the identity tensor as: { l j k = 1 ( gjm 2 glm + g km g ) jk θ k θ j θ m (1 58) (1-57) and (1-58) have been proved in Box 1.1. From (1-22) and (1-58) follows that the Christoffel symbols fulfill the symmetry condition: { { l l = j k k j From (1-8) follows that: (1 59) g j θ k = 2 x θ j θ k = 2 x θ k θ j = g k θ j (1 60) Alternatively, the symmetry property (1-59) may be proved by insertion of (1-57) in (1-60).

20 1.3 Gradient, covariant and contravariant derivatives 19 Box 1.1: Proof of (1-57) and (1-58) Consider the first relation (1-57) as a definition of the Christoffel symbol, and prove that this implies the second relation (1-57). From (1-15) and the first relation (1-57) follows: θ kδj m = ( gm g j) = g m θ k g j θ g k m = { j g l g m l k { l m k θ k gj +g m gj θ = 0 k { { g l g j l = δ j j l = m k m k { j = δm l l k = ( { ) g j j θ + g l g k m = 0 (1 61) l k Since (1-61) is valid for any of the N linear independent covariant base vectors g m, the term within the bracket must be equal to0. This proves the validity of the second relation (1-57). From (1-22) and the first relation (1-57) follows: g jm = (g j g m ) θ k θ k From (1-62) follows: g jm = g j θ k g m + g m θ k g j = { { n n g nm + g nj (1 62) j k m k θ + g km g jk k θ j θ = m { { { { { { n n n n n n g nm + g nj + g nm + g nk g nk g nj = j k m k k j m j j m k m 2 { n g nm j k (1 63) where the symmetry property (1-59) has been used. Next, (1-58) follows from (1-63) upon pre-multiplication on both sides withg lm and use of (1-23).

21 20 Chapter 1 Tensor Calculus Insertion of (1-57) in (1-56) provides the following representations of v: v = v j ;k g jg k = v j;k g j g k (1 64) where: { v j ;k = vj j θ + k k l v j;k = v { j l θ k j k v l v l (1 65) Hence, v j ;k specifies the mixed co-and contravariant components, and v j;k specifies the contravariant components of v. By the use of (1-57) the partial derivative of the vector functionv(θ l ) may be written as: v θ k = (v j g j ) θ k (v j g j ) θ k = vj θ kg j + g j θ kvj = v j ;k g j = v j θ kgj + gj θ kv j = v j;k g j (1 66) Hence, alternatively v j ;k and v j;k may be interpreted as the covariant components and the contravariant components of the vector v. For this reason v j θ k ;k is referred to as the covariant derivative, and v j;k as the contravariate derivative of the components v j and v j, respective. In the applied notation these derivatives will always be indicated by a semicolon. Further, by the use of (1-57) the following results for the derivatives of the dyads entering the covariant, mixed and contravariant tensor bases become: θ l(g jg k ) = g j θ l g k +g j g k θ l = θ l(g jg k ) = g j θ l gk +g j g k θ l = θ l(gj g k ) = gj θ l g k +g j g k θ l = θ l(gj g k ) = gj θ l gk +g j gk θ l = { m j l { m j l { j m l { j m l { m g m g k + k l { g m g k k { g m m g k + { g m g k k m l g j g m g j g m m l g j g m k l g j g m (1 67) Finally, the covariant and contravariant components of the differential incrementdv of the vector v due to the differential incremental vector dθ = dθ k g k of the curvilinear coordinates becomes:

22 1.3 Gradient, covariant and contravariant derivatives 21 dv = v dθ = v j ;k g jg k dθ l g l = v j;k g j g k dθ l g l dv = v j ;k dθk g j = v j;k dθ k g j (1 68) The gradient of a second order tensor function T = T(θ l ), denoted as T, is a third order tensor with the following representations: T = T θ l gl = (T jk g j g k ) θ l g l = (T j k g jg k ) θ l g l = (T k j g j g k ) θ l g l = (T jk g j g k ) θ l g l = ( T jk g θ l j g k +T jk g j θ g l k +T jk g k g j θ l ( T j k θ l g j g k +T j k ( T k j g j θ l gk +T j k g g k j θ l g j θ g l k +Tj k g j g k θ l g j g θ l k +Tj k ( Tjk g j g k g j +T θ l jk θ l gk +T jk g j gk θ l ) g l ( { { ) T jk +T mk j +T jm k g θ l j g k g l = T jk ;l m l m l g jg k g l ( T j { { ) k +T m j θ l k T j m m g j g k g l = T j k;l m l k l g jg k g l T = ( T k { { ) j T k m θ l m +Tj m k g j g k g l = Tj k ;l g j g k g l j l m l ( Tjk θ l T km where: { T jk ;l = Tjk +T mk j θ l m l { T j k;l = Tj k +T m j θ l k m l Tj k ;l = T j k { T k m θ l m T jk;l = T jk θ l T km j l { m j l { m j l { +T jm k m l { T j m m k l { +Tj m k m l T jm { m k l (1-57) has been used in the last statements of (1-69). { ) m T jm k l ) ) ) g l g l g l g j g k g l = T jk;l g j g k g l (1 69) (1 70)

23 22 Chapter 1 Tensor Calculus Then, the partial derivative of the tensor functiont(θ m ) may be described as any of the following second order tensor representations, cf. (1-66): T θ l = T jk ;l g jg k = T j k;l g jg k = T jk;l g j g k (1 71) Partial differentiation of both sides of (1-44) provides: v θ = g v v+g k θk θ = g v v+ k θk θ k g θ v = 0 k Sincevis arbitrary, (1-72) implies that: g θ k = 0 (1 72) (1 73) In turn this means that g = g θ l g l = 0. Then, from (1-69) follows that g jk ;l = g jk;l = 0, so the components of the identity tensor vanish under covariant and contravariant differentiation. 1.4 Riemann-Christoffel tensor The gradient v of a vectorv with components in Cartesian and curvilinear tensor bases have been indicated by (1-56). The gradient of this second order tensor is given as, cf. (1-69): ( ) v x l x i k k i l = ( ) ( v j i j ) i x l x k k i l = 2 v j x k x i ji l k i l ( v) = θ l ( v θ k gk ) g l = θ l ( ) v θ l θ k gk g l = θ l ( ( v j θ + k ( ( vj θ k { ) j v )g m j g k g l = v j ;kl k m g jg k g l { ) m v m )g j g k g l = v j;kl g j g k g l j k (1 74) where the following tensor components have been introduced: { { { v j ;kl = (v j ;k ) ;l = 2 v j j v m j v m m v j θ k θ + + l k m θ l l m θ k k l θ m { { n j v m + { { { j v m j n + v m (1 75) k l n m θ l k m l n k m { { { v j;kl = (v j;k ) ;l = 2 v j m θ k θ vm m vm m l j k θ l j l θ vj k k l θ + m { { n m v m { { { m n m v k l n j θ l m + v m (1 76) j k j l n k

24 1.4 Riemann-Christoffel tensor 23 From (1-75) and (1-76) follow that the indicesk andl can be interchanged in the first five terms on the right hand sides without changing the value of this part of the expressions, whereas this is not the case for the last two terms. This implies that the sequence in which the covariant differentiations is performed is significant, i.e. in general v j;kl v j;lk. In order to investigate this further consider the quantity: ( { { { { n m n m v j;kl v j;kl = + { m { ) m v j l n k j k n l θ k j l θ l m = j k R m jkl v m where: R m jkl = { { { { n m n m j l n k j k n l + { m θ k j l { m θ l j k (1 77) (1 78) R m jkl signifies the mixed components of the so-called Riemann-Christoffel tensor R, i.e. R = R m jkl g mg j g k g l. The components of R, and hence the right hand side of (1-77), are not vanishing due to the curvature of the Riemann space. Obviously, R m jkl = R m jlk (1 79) Further, the following socalled Bianchi s first identity applies: R m jkl +R m klj +R m ljk = 0 (1 80) (1-80) follows by insertion of (1-78) and use of the symmetry property (1-59) of the Christoffel symbol. In an Euclidean N-dimensional space, i.e. a space spanned by a constant Cartesian vector basis, the Cartesian components ofrfollow from (1-74): R jklm v m = 2 v j x k x 2 v j l x l x = 0 k R jklm = 0 (1 81) Hence, it can be concluded thatr = 0 in an Euclidean space. In turn this means that the curvilinear components (1-78) must also vanish in this space. A space, where everywhere R = 0 is called flat. Reversely, a non-vanishing curvature tensor indicates a curved space. In a flat space R m jkl = 0, with the implication thatv j;kl = v j;lk. The three-dimensional Euclidian space is flat, and any plane in this space forms a flat two-dimensional subspace. In contrast, a curved surface in the Euclidian space is not a flat subspace. An example of a curved four-dimensional space is the time-space manifold used at the formulation of the general theory of relativity, where the indices correspondingly range over j = 1, 2, 3, 4.

25 24 Chapter 1 Tensor Calculus Because of the relations (1-79), (1-80) only 1 12 N2 (N 2 1) of the tensor componentsr m jkl are independent and non-trivial, (Spain, 1965). Hence, for a two-dimensional Riemann space merely one independent component exists, which can be chosen as R In the three-dimensional case six independent and non-trivial components exist, which may be chosen as R 1 212, R 1 213, R 1 223, R1 313, R1 323 and R Example 1.1: Covariant base vectors, identity tensor, Christoffel symbols and Riemann- Christoffel tensor in spherical coordinates By the use of (1-10) the first equation (1-57) becomes: (c m j i { m) θ k = cm j l θ k i m = c m l i m j k { c m j l θ k = c m l j k (1 82) The Cartesian components c m j of the covariant base vector g j is stored in the column matrix g j = [c m j ]. Then, (1-82) may be written in the following matrix form: g j θ k = { { { { l g j k l = g j k 1 + g j k 2 + g j k 3 (1 83) The spherical coordinate system defined by (1-1) is considered. In this case the column matrices attain the form, cf. (1-8): g 1 = θ 3 cosθ 1 cosθ 2 θ 3 sinθ 1 sinθ 2 sinθ 1 cosθ 2 θ 3 cosθ 1 sinθ 2, g 2 = θ 3 sinθ 1 cosθ 2, g 3 = sinθ 1 sinθ 2 (1 84) θ 3 sinθ 1 0 cosθ 1 The covariant components of the identity tensor is given by (1-22) as g jk = g j g k = g Tg, where the last j k statement is obtained by evaluating the scalar product in Cartesian coordinates. The covariant and contravariant components of the identity tensor are conveniently stored in matrices. Using (1-84) these becomes: (θ 3 ) [g jk ] = 0 (θ 3 ) 2 sin 2 θ , 1 [ g jk ] (θ 3 ) = 1 0 (θ 3 ) 2 sin 2 0 θ (1 85) The result for the contravariant components follows from (1-23). Next, (1-83) and (1-84) will be used to determine the Christoffel. The following results are obtained:

26 1.4 Riemann-Christoffel tensor 25 g 1 θ 1 = θ 3 sinθ 1 cosθ 2 θ 3 sinθ 1 sinθ 2 = 0 g 1 +0 g 2 θ 3 g 3 θ 3 cosθ 1 θ g 3 cosθ 1 sinθ 2 1 θ 2 = θ 3 cosθ 1 cosθ 2 = 0 g tanθ 1 g +0 g cosθ g 1 cosθ 2 1 θ 3 = cosθ 1 sinθ 2 = 1 θ 3 g +0 g +0 g sinθ 1 θ g 3 cosθ 1 sinθ 2 2 θ 1 = θ 3 cosθ 1 cosθ 2 = 0 g tanθ 1 g +0 g g 2 θ 2 = θ 3 sinθ 1 cosθ 2 θ 3 sinθ 1 sinθ 2 = 1 2 sin(2θ1 ) g 1 +0 g 2 θ 3 sin 2 θ 1 g 3 0 sinθ g 1 sinθ 2 2 θ 3 = sinθ 1 cosθ 2 = 0 g θ 3 g +0 g cosθ g 1 cosθ 2 3 θ 1 = cosθ 1 sinθ 2 = 1 θ 3 g +0 g +0 g sinθ 1 sinθ g 1 sinθ 2 3 θ 2 = sinθ 1 cosθ 2 = 0 g θ 3 g +0 g g 3 θ 3 = 0 0 = 0 g 1 +0 g 2 +0 g 3 0 { { { { { { { { { { { { { { { { { { { { { { { { { { { = 0 = 0 = θ 3 = 0 1 = tanθ 1 = 0 = 1 = 0 = 0 θ 3 = 0 1 = tanθ 1 = 0 1 = 2 sin(2θ1 ) = 0 = θ 3 sin 2 θ 1 = 0 1 = = 0 θ 3 = 1 = 0 = 0 θ 3 = 0 1 = = 0 = 0 = 0 = 0 θ 3 (1 86)

27 26 Chapter 1 Tensor Calculus The non-trivial components of the Riemann-Christoffel tensor follow from (1-77) and (1-86): { { { { n 1 n 1 R = 2 2 n n sin(2θ1 ) θ 3 sin 2 θ 1 { { { n 1 n R = 2 3 n { { + 1 θ θ θ tanθ sin(2θ1 ) 0 0 cos(2θ 1 ) 0 = 0 { 1 n 3 { + 1 θ θ θ θ 3 1 tanθ = 0 { { n 1 R = 2 3 n θ 3 1 R = { n { { n n 3 θ 3 { { { + 1 θ θ sin(2θ1 ) sin(2θ1 ) θ θ3 sin 2 θ = 0 { { { { { 1 n n n 3 θ θ 3 = θ 3 1 θ 3 1 θ (θ 3 ) 2 = 0 { { { { n 1 n 1 R = 3 3 n n sin(2θ1 ) { { n 2 R = 3 3 n 2 0 { { + 1 θ θ θ = 0 { { n n 3 { { + 2 θ θ tanθ θ θ 3 1 θ (θ 3 ) 2 = 0 = = = = = (1 87) As expected all components of the Riemann-Christoffel tensor vanish as a consequence of the flatness of the threedimensional Euclidean space. 1.5 Geodesics In Euclidean 3-dimensional space the shortest distance between two points A and B is a straight line. On a surface embedded in the Euclidian 3-dimensional space, where both principal curvatures everywhere are either positive or negative, the curve with the shortest distance between the points is indicated by an inflexible string stretched between the points. Especially, on a sphere the curve makes up a part of a great circle. In this section this principle is carried over to a general Riemann space in terms of a socalled geodesics, which is defined as the curve with the minimum length connecting two points in the Riemann space with the length measured by the fundamental tensor of the space. A geodesics joining the point A and B must fulfill the following variational principle:

28 1.5 Geodesics 27 δ B A ds = 0 (1 88) where ds is a differential length element along an arbitrary curve connecting the points A and B given as: ds 2 = dx dx = dx j dx k g j g k = g jk dx j dx k (1 89) dx j indicates the covariant components of the differential increment dx of the position vectorx tangential to the arc length increment. Further, (1-22) has been used in the last statement. The geodesics turns out to be given by the following non-linear differential equation: { d 2 x j j dx k dx l + ds 2 k l ds ds = 0 (1 90) where x j indicates the coordinates in a given referential curvilinear coordinate system in the N-dimensional Riemann space of a running point along the geodesic. Eq. (1-90) is solved with the initial valuex j A and the unit tangential vectortj A = dxj A specified at a given point pointaon ds the geodesics. A proof of (1-90) is given in Box 1.2. Consider a point A on a differential surface embedded in the three-dimensional Euclidian s- pace. At an arbitrary pointaon the surface two linear independent tangential vectorst 1 and t 2 may be defined, which span the tangent plane ata. The tangent plane is a two-dimensional flat space Euclidian subspace, in contrast to the underlying differential surface. As a consequence the Riemann tensor vanish in the tangential subspace. In the following this observation is generalized from a two-dimensional to an arbitrary N-dimensional curved Riemann space. Consider the N linear independent unit tangential vectors t j, j = 1,...,N, indicating the direction of the geodesics drawn out from an arbitrary point Ain the Riemann space. The vectors may be organized to form a local arc length vector basis known as the Riemann vector basis, which span a tangential manifold to the Riemann space at A. The related fundamental tensorg has the contravariant components, cf. (1-22): g jk = t j t k (1 91) Let x j denote the covariant components in the local Riemann vector basis of a position vector along a geodesic curve. The differential equation of the geodesic in Riemann coordinates reads: { d 2 x j j dx k dx l + ds 2 k l ds ds = 0 (1 92) where { j k l indicates the Christoffel symbol evaluated by the co- and contravariant components of the fundamental tensorg.

29 28 Chapter 1 Tensor Calculus Consider a geodesic curve at the pointadefined by the covariant componentst j in the Riemann vector basis with origin ata, and letp be a neighbouring point placed on the geodesics defined by the unit tangent t j a small arc length s from A. Then, the covariant coordinates of the point P is approximately given as: x j st j (1 93) (1-93) holds asymptotically as s 0. Insertion of (1-93) into (1-92) provides in the limit: { j t j t k = 0 k l (1 94) The unit tangential vector with the curvilinear components t j in (1-94) has been arbitrarily selected. Hence, this equation must be fulfilled for all N unit tangential vectors related to the geodesics drawn out of point A. This can only hold, if the Christoffel symbol vanishes at the origin of the Riemann coordinate system at the running pointa, i.e.: { j = 0 k l (1 95) As consequence of (1-95), the covariant derivatives with respect tox j becomes equal to partial derivatives, cf. (1-65). Further, from (1-62) follows that: g jm x k = 0 Finally, in a Riemannian coordinate system we have, cf. (1-78): (1 96) { R m jkl;n = 2 m x n x k j l Bianchi s second identity reads: { 2 m (1 97) x n x l j k R m jkl;n + Rm jln;k + Rm jnk;l = 0 (1 98) (1-98) is proved in a Riemannian coordinate system by the use of (1-97). Then, being valid in one coordinate system it is also valid in any curvilinear coordinate system.

30 1.5 Geodesics 29 Box 1.2: Proof of (1-90) v B u A Fig. 1-3: Family of curves connecting pointsaand B. Consider a family of curves connecting two pointaand B specified by the parametrization: x j = x j (u,v) (1 99) The parameterv characterizes a certain curve in the family, and u [a A,u B ] is a parameter defining a certain point on the curve specified by v. Especially,umay be chosen as the length parametersalong the curve. Then the lengthlof the curve defined by v is given as: L = ub u A w 1 2 du (1 100) wherew is given as, cf. (1-89): w = g jk t j t k (1 101) and t j signifies the quantity: t j = dxj du (1 102) Hence, w is a function of the independent variables x j and t j. δl denotes the variation of the length of the curve defined by the parameters (u,v) due to a variationδv ofv for fixed u. Then (1-88) attains the form: ub ub 1 ( ) δl = δ w 1 w ub 2 2 du = u A u A v δvdu = w 1 2 x j u A x j v + w 1 2 t j δvdu = 0 t j v (1 103) From (1-102) follows: t j v = d x j du v (1 104)

31 30 Chapter 1 Tensor Calculus Insertion of (1-104) in (1-103) and followed by integration by part provides: [ ] w 1 2 x j ub ( ) ub w 1 2 δl = d w 1 2 δx j du = 0 (1 105) t j v x j du t j u A + u A whereδx j = xj δv has been introduced. v x j (u A ) and x j (u B ) are common for all curve, and hence independent of v. = 0, so the boundary terms in (1-105) vanish. x j (u A ) = xj (u B ) v v Then, In the integrand δx j can be varied independently for any u ]u A,u B [. This leads to the following Euler condition necessary for stationarity: w 1 2 d w 1 2 = 0 (1 106) x j du t j (1-106) may be rewritten on the form: d w w = 1 du t j x j 2w dw du w t j (1 107) Especially, let u be chosen as the arc-length s along the geodesic, so that: u = s, t j = dxj ds, w = g jk t j t k 1 dw du = 0 (1 108) Now, t j indicates the covariant components of the unit tangential vector along the geodesic. Insertion of (1-108) in (1-107) provides: d w w = 0 ds t j x j 2 d ( gjk t k) g kl ds x j tk t l = d 2 x k g jk + g jk dx l ds 2 x l ds dx k ds 1 g kl dx k dx l 2 x j ds ds = 0 (1 109) where it has been used that g jk = g jk (x l ), so dg jk = g jk. Further, due to the symmetry property (1-22) of the components of the metric tensor it follows by interchanging ds ds the name of the dummy indices l and k that g ( jk dx l dx k = 1 gjk + g ) jl dx k dx l. Then, x l ds ds 2 x l x k ds ds (1-109) may be written: g jk d 2 x k ds x l dx l ( gjk + g jl x l x g ) kl dx k dx l k x j ds ds = 0 (1 110) Finally, Eq. (1-90) follows by pre-multiplication and contraction on both sides of (1-110) withg mj, and use of (1-43) and (1-58).

32 1.6 Exercises Exercises 1.1 Given the following vectorsa, b, c and d. Prove the following vector identities (a.) a (b c) = (a c)b (a b)c (b.) (a b) (c d) = (a c)(b d) (a d)(b c) (c.) (a b) c = (b c) a = (c a) b 1.2 Prove the e δ relation (1-19). 1.3 Given the vectorsa, b, c and d with the Cartesian components [āj ] 1 = 2, [ bj ] 3 j = 2, [ c ] 1 = 2, [ dj ] 3 = Calculate the dyadic scalar productsab cd and ab cd. 1.4 Prove the relations (1-37) and (1-40). 1.5 Given the Cartesian components C rstu of the fourth order tensor C. Calculate the mixed curvilinear componentsc k m j l. 1.6 Prove the last three relations in (1-67). 1.7 Prove (1-73) by applying (1-69) to the mixed representation g = δ k j g j g k of the identity tensor. 1.8 Determine the geodesics on a cylindrical surface in the three dimensional Euclidean space with arbitrary directrix. 1.9 Prove (1-97).

33 32 Chapter 1 Tensor Calculus

34 CHAPTER 2 Differential Theory of Surfaces 2.1 Differential geometry of surfaces, first fundamental form a) b) θ 2 x 2 b 2 θ 2 b 1 p ω s 3 n P a 1 r Ω i 2 s 1 i3 x 3 a 2 s 2 s a 1 θ 1 a 2 θ 1 i 1 x 1 Fig. 2 1 a) Parameter space. b) Surface space. The following section concerns surfaces in the three dimensional Euclidean space. Let the spherical θ 3 coordinate be fixed at the value θ 3 = r. Then, the mapping (1-1) of the spherical coordinates onto the Cartesian coordinates takes the form: x 1 rsinθ 1 cosθ 2 x 2 = rsinθ 1 sinθ 2 (2 1) rcosθ 1 x 3 (x 1,x 2,x 3 ) denotes the Cartesian components of the position vector x to a given point of the surface. Obviously, (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 = r 2. Hence, with the zenith angle varied in the intervalθ 1 [0,π], and the azimuth angle varied in the intervalθ 2 ]0,2π], (2-1) represents the parametric description of a sphere with the radius r and the center at (x 1,x 2,x 3 ) = (0,0,0). In what follows it is assumed that the parametric description of all considered surfaces is defined 33

35 34 Chapter 2 Differential Theory of Surfaces by a constant value of the curvilinear coordinate θ 3 = c in the mapping (1-2). Then, a given surface Ω is given as, cf. (1-5): x j = f j (θ 1,θ 2,c) = f j (θ α ) (2 2) In the last statement of (2-2) the explicit dependence on the constantcis ignored, as will also be the case in the following. Let the mapping (2-2) be defined within a domainω in the parameter space. For each point p ω determined by the parameters (θ 1,θ 2 ), a given point P is defined on Ω with Cartesian coordinates given by (2-2). Assume that a curve through p is specified by the parametric description ( θ 1 (t),θ 2 (t) ), where t is the free parameter. Then, this curve is mapped onto a curve s = s(t) through P on Ω as shown on Fig Especially, if the curvilinear coordinateθ 2 is fixed, whereas θ 1 is varied independently a curves 1 through P is defined on Ω. Similarly, a curves 2 on the surface through P is obtained, if θ 1 is fixed and θ 2 is varied. The positive direction of s 1 and s 2 are defined, so positive increments of θ α correspond to positive increments of s α. Then, these curves define a local two-dimensional arch length coordinate system(s 1,s 2 ) throughp. Assume that ω is a rectangular domain [a 1,a 2 ] [b 1,b 2 ] with sides parallel to the θ α axes as shown on Fig. 2-1a. As an example this is the case for the mapping (2-1), where ω = [0,π] ]0,2π]. In such cases the surface Ω will be bounded by the arc length coordinate curves given by the parameter descriptions x j = f j (a 1,θ 2 ), x j = f j (a 2,θ 2 ), x j = f j (θ 1,b 1 ) and x j = f j (θ 1,b 2 ), see Fig. 2-1b. n P a 2 a 1 ϕ s 2 s 1 a 1 a 2 A Fig. 2 2 Covariant and contravariant base vectors and surface normal unit vector. Similar to (1-8) a covariant vector base (a 1,a 2 ) may be defined at each point of the surface via the relation: a α = x θ α, α = 1,2 (2 3) Obviously, a α are tangential to the arch length curves at P, see Fig. 2-1b. Then, a α may be interpreted as a local two-dimensional covariant vector bases, which spans the tangent plane at the pointp, see Fig. 2-2.

36 2.1 Differential geometry of surfaces, first fundamental form 35 At the pointp the unit normal vectornto the surface and the tangent plane is defined as: n = n(θ 1,θ 2 ) = a 1 a 2 a 1 a 2 = 1 A a 1 a 2 (2 4) A denotes the area of the parallelogram spanned bya 1 and a 2, and given as: A = a 1 a 2 = a 1 a 2 sinϕ (2 5) The related contravariant base vectors follows from (1-16). In the present casev = A 1 = A. Then: a 1 = 1 A a 2 n = 1 A a 2 2 (a 1 a 2 ) = 1 ( ) a A a 1 (a 1 a 2 )a 2 a 2 = 1 A n a 1 = 1 A (a 2 1 a 2 ) a 1 = 1 ( ) (2 6) a A a 2 (a 1 a 2 )a 1 The last statements in (2-6) follow from the vector identitya (b c) = (a c)b (b a)c, cf. Exercise 1.1. It is easily shown that the covariant the contravariant base vectors in the tangent plane as given by (2-6) fulfill the orthonormality relation: a α a β = δ β α (2 7) whereδα β indicates the Kronecker s delta in two dimensions. A surface vector function v = v(θ 1,θ 2 ) is a vector field, which everywhere (i.e. for any parameters (θ 1,θ 2 )) is tangential to the surface. Then, v may be represented by the following Cartesian, covariant and contravariant representations, cf. (1-20): v = v α i α = v α a α = v α a α (2 8) i α = i α indicates a Cartesian vector base in the tangential plane, and v j, v α and v α denote the Cartesian, the covariant and the contravariant components ofv. The Cartesian and the covariant components are related as, cf. (1-25): v α = c α β vβ, c α β = iα a β v α = d α β vβ, d α β = i β a α (2 9) where the orthogonality conditionsi α i β = δα β and a α a β = δα β have been used. Similarly, the covariant and contravariant components are related by the following relations: v α = a αβ v β (2 10) v α = a αβ v β where:

37 36 Chapter 2 Differential Theory of Surfaces a αβ = a α a β a αβ = a α a β (2 11) Both a α and a α are surface vectors. Then, these can be expanded in two-dimensional contravariant and covariant vector bases as follows, cf. (1-24): a α = a αβ a β (2 12) a α = a αβ a β (2-12) is proved by scalar multiplication with a γ and a γ on both sides of the equation and use of (2-7) and (2-11). Scalar multiplication on both sides of the first equation (2-12) witha γ and use of (2-7) provides: a αβ a βγ = δ γ α (2 13) In analogy to (1-27) a surface second order tensor is defined as a second order tensor, which everywhere maps a surface vectorvonto another surface vectoruby means of a scalar product. A second order surface tensor T admits the following Cartesian, covariant, contravariant and mixed representations, cf. (1-35): T = T jk i j i k = T αβ a α a β = T αβ a α a β = T α β a αa β = T β α aα a β (2 14) T αβ, T αβ, T α β and T α β denotes the covariant, the contravariant and the mixed covariant and contravariant components of the surface tensor. These are related as, cf. (1-39), (1-40): T αβ = a αγ a βδ T γδ, T α β = aαγ T γβ T αβ = a αγ a βδ T γδ, Tα β = a αγ T γβ (2 15) T αβ = a αγ T γ β, T αβ = a αγ T β γ The relations in (2-15) follows by insertion of the expansions (2-12) in (2-14). The surface identity tensorais defined as a second order tensor with the covariant components a αβ, the contravariant components a αβ, and the mixed components δα β, corresponding to the following representations, cf. (1-42): a = a αβ a α a β = a αβ a α a β = δ β αa α a β (2 16) The tensoramaps any surface vector onto itself, i.e.: a v = v (2 17) (2-17) is proved in the same way as (1-44), using the representation (2-16).