Appendix E (h) and (i)

Rewrite of Appendix E, rewrite of (h), addition of new section (i) : 10.3.12 Installed this on 10.19.12 at 8:40 PM. (h) Expansions of tensors on unit tangent base vectors We start with the general Picture A (and later specialize to orthogonal coordinates), In section (b) above it was established that one can expand a tensor A on the tangent base vectors en as A = Σijk... A' ijk... (eiejek...) A' ijk... = contravariant components of A in x'-space A' ijk... = Rii'Rjj'Rkk'...... A i'j'k'... . Since en = h'n n, this same expansion for tensor A can be written A = Σijk... h'ih'jh'k......A' ijk... (ijk...) = Σijk... [A()]ijk... (ijk...) where the unit-vector expansion coefficients are given by [A()]ijk... = h'ih'jh'k......A' ijk... = h'ih'jh'k...... Rii'Rjj'Rkk'...... A i'j'k'... = (h'i Rii')( h'j Rjj')( h'k Rkk') ..... A i'j'k'... Coefficient notation. In a curvilinear coordinates application of these expansions, the expansion coefficients are usually written in the following manner, [A()]ijk... = Ax'x'x' ...... where the x'n are the names of the coordinates. For example, for a rank-4 tensor in spherical coordinates with coordinates x'1 = r, x'2 = θ and x'3 = φ one might write [A()]2213 = Aθθrφ . Since [A()]ijk... is not a tensor (with respect to F), there is no particular reason to put the indices "up" and for that reason they are usually written down, as in Aθθrφ . Matrices M and N. It is convenient now to define Mab ≡ h'a Rab so that then [A()]ijk... = Mii'Mjj'Mkk'...... A i'j'k'... . Defining Nab to be the inverse of Mab, one has Nab ≡ h'b-1Sab = h'b-1Rba so A ijk... = Nii'Njj'Nkk'...... [A()]i'j'k'... . To verify that this N is the correct inverse or M, MakNkc = (h'a Rak)( h'c-1Rck) = (h'a/ h'c) Rak Rck = (h'a/ h'c)δac = δac making use of the orthogonality rule of Section 7 (r), Rak Rck = δac . M and N can be written in terms of the tangent base vectors as follows: Mni ≡ h'n Rni = h'n(en)i Nin = h'n-1Rni = h'n-1(en)i = (n)i which says that Nin = { 1 , 2, .... } -- the columns of Nin are the unit tangent base vectors. Rank-1 tensors. For a vector, the above coefficient relation is written [A()]i = Mij Aj or A() = M A and A = N A() where A has these two familiar expansions, A = Anun = [A()]n n [A()]n = Ax' for example [A()]1 = Ar . Rank-2 tensors. Here the coefficient relation is [A()]ij = Mii'Mjj'A i'j' = Mii' A i'j' Mjj' which can be written [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm Mni = h'n(en)i . Defining bn = h'nen , then [bn]i = h'n(en)i = Mni = Bni of section (g) . Meanwhile, from Section 6 (b) we know that wnk = bn bk = h'nh'k(en ek) , and then wnk = (w-1)nk . In any event, whatever wnk is, it is the object which can lower the n or m indices on both sides of the above equation. Lowering just the m index and then reversing the j tilt gives [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm = Mni A ij (MT)jm and we replicate the section (g) result with B = M : A() = M A MT // [A() = M A MT ]SN,dt . Matrix H and x"-space. In the discussion above one has x-space with basis vectors un and x'-space with basis vectors en (the tangent base vectors). It is useful then to define x"-space as the space whose basis vectors are the n unit vectors, which are generally not orthogonal. The relation Mab ≡ h'a Rab given above can be written in down-tilt form as M = HR where Hij ≡ diag(h'1, h'2.....). It then follow that N = M-1 = R-1H-1 = SH-1 Finally, note that x = xiui = x'iei = x'i(h'ii) = x"ii => x"i = h'i x'i or x" = H x' which shows that the transformation from x'-space to x"-space is linear with matrix H. We can now show all three spaces in the same picture as follows: This picture shows that the transformation directly from x-space to x"-space is FM(x) = H F(x) . Here F(x) is a (generally) non-linear transformation assumed to connect x'-space to x-space. This is then concatenated with linear transformation H to get non-linear transformation FM(x). We can now write A() as A" and restate equations above as A = Σijk... A" ijk... (ijk...) // rank-n tensor expanded on ijk... A" ijk... = Mii'Mjj'Mkk'...... A i'j'k'... // rank-n tensor transformation A ijk... = Nii'Njj'Nkk'...... A" i'j'k'... // inverse of the above A" i = Mij Aj // rank-1 tensor A" ij = Mii'Mjj'A i'j' // rank-2 tensor A" = M A MT // rank-2 tensor, matrix notation The word "tensor" suddenly has a new meaning in the above equations. The equations indicate objects being tensors with respect to this non-linear transformation FM(x) whose linearized-at-a-point matrix is RM = M = HR, where R is the linearized-at-a-point matrix version of F(x), and H is the diagonal matrix of scale factors h'i which are associated with g'ij in x'-space. The Aijk... are contravariant components of tensor A in x-space, while A"ijk... are the corresponding contravariant components of A in x"-space, all with respect to FM(x) and its matrix RM = M where, for example, dx" = M dx. At this point the spaces are completely general, and none of R, RH = H, RM = M is a rotation matrix. In the next section, we shall specialize the above picture so that x-space is Cartesian with g = 1, and x'-space is the space of a set of orthogonal curvilinear coordinates x'. In this scenario, F(x) is non-linear and so then is FM(x) = H F(x) . Since the i now form a frame of orthonormal vectors, and since ui also form such a frame, one will not be surprised to find that M is now a rotation which relates these two frame sets. Orthogonal curvilinear coordinates application We now switch to picture B (g=1) and assume that the x'i are orthogonal coordinates, and our three-frame picture above then becomes In this situation, x-space is Cartesian with gab = gab = δab and g'ab = h'a2δab and g'ab = h'a-2δab. Since the metric tensors are diagonal, one could write H = , but we continue to use H. In what follows, the plan is simply to exercise both the developmental and standard notations with regard to the M and N matrices. To this end, we first collect the following facts from Section 7 (o), Rab = Rab' gb'b = Rab Rab = g'aa'Ra'b' gb'b = g'aa'Ra'b = h'a2 Rab = h'a2 Rab Rab= g'aa'Ra'b = h'a2 Rab or Rab = Rab Rab = Rab = h'a2 Rab Rab = h'a-2 Rab . Also from Section 7 (11), g'ab = Raa'Rbb'ga'b' = Raa'Rba' = RacRbc g'ab = Sa'a Sb'b ga'b' = Raa' Rbb'ga'b' = Raa' Rba' = RacRbc or g'ab = RacRbc and g'ab = RacRbc . In this scenario, regardless of what R and S are, M is a "rotation" (verified below), where we include in this term possible axis reflections. What we really mean is that in developmental notation M is a real-orthogonal matrix, MMT = 1. Since N=M-1, N is then also a rotation. To prove that M is a rotation in developmental notation, the standard notation equation Mab ≡ h'a Rab can be reverse-translated to Mab = h'aRab . Then (now g' = RRT from Section 5 (l) ) (MMT)ac = MabMcb = h'aRab h'cRcb = h'ah'c RabRTbc = h'ah'c(RRT)ac = h'ah'cg'ac = h'ah'c [ h'a–2 δa,c] = δa,c => MMT = 1 . Proving the same thing directly in standard notation requires showing that Mab Mcb = δa,c ( see the end of Section 7 (i) ) Mab Mcb = h'a Rab h'c Rcb = h'a h'c Rab Rcb = h'a h'c Rab (h'c-2 Rcb) = (h'a/h'c) (Rab Rcb) = (h'a/h'c)δac = δac = δa,c where use was again made of the orthogonality rule of Section 7 (r), Rab Rcb = δac. Relation beween M and N. Looking at Mab Mcb = δa,c and knowing that Mab (M-1)bc = δac = δa,c one concludes that (M-1)bc = Mcb . But (M-1)bc = Nbc so Nbc = Mcb // reminder: this does not say that N = MT in standard notation which can be verified from the above expressions for N and M. Interpretation of N and M. Since en = S un ( Section 3 (a) with e'n = un) and since en = h'nn , it follows that n = h'n-1 S un or (n)a = h'n-1 Sab (un)b = h'n-1 Sab δnb = h'n-1 San = Nan = Nabδbn = Nab(un)b or n = N un and (n)a = Nan . // => un = M n Since the un are the Cartesian unit vectors, it seems intuitively obvious that the transformation that moves this frame of orthonormal unit vectors {un} into the orthonormal frame {n} must be a "rotation". Above it was shown that Nan = Mna , therefore Mna = Nan = (n)a The rotation matrix Nan = (n)a has the orthogonal basis vectors n as its columns, while the rotation matrix Mna = (n)a has the orthogonal basis vectors n as its rows. From this point of view, it seems pretty reasonable that MN = 1. It might be noted that, in our situation with Cartesian x-space and orthogonal coordinates, n = n : n = en/|en| |en|2 = en en = g'nn = h'n-2 so n = en h'n = h'n g'nn en = h'nh'n-2 en = h'n-1 en = n . The relation un = M n can be written un = M(x) n(x) to emphasize that the rotation M(x) = RM(x) is really a different rotation at every point x, since the n(x) vary with x. This is very different from a global rotation which is the same at all points. For a global rotation R, F = R = linear and ∂iuj is a tensor. For RM being a rotation which varies from point to point, FM is non-linear just as F defining the curvilinear coordinates is non-linear, and ∂iuj fails to be a tensor under either F or FM. One implication of the above picture relates to tensor equations being covariant, as discussed in Section 7 (u). If one has a tensor field equation in x-space, Qadc(x) = Hab(x)Tbc(x) Bd(x) , in which all the objects transform as tensors with respect to the underlying x" = FM(x) (and its linear approximation M(x) as in dx" = M(x) dx), then the equation is covariant and takes the same form in x"-space, Q"adc(x") = H"ab(x")T"bc(x") B"d(x") . An x-space observer has axes un (Frame S) while an x"-space observer has axes n(x) (Frame S"), and these two sets of observation axes are related by un = M(x) n(x) where M(x) is a rotation. If the first equation describes something at location x in the realm of Newtonian mechanics, we expect the equation to have the same form in both Frame S and Frame S" which are related by this local rotation M(x). In other words, rotations are an invariance of Newtonian mechanics, and this means equations are covariant with respect to rotations. The above example, which might apply to fluid dynamics, has this covariance at each point x in the fluid, and it happens that the rotation is a different rotation at different points x, but it is always a rotation. Example: Polar Coordinates. In polar coordinates now with ordering r,θ = 1,2 one has S11 = (∂x/∂r) = cosθ x = rcosθ S12 = (∂x/∂θ) = -rsinθ y = rsinθ S21 = (∂y/∂r) = sinθ S22 = (∂y/∂θ) = rcosθ Sij = Rij = R = S-1 [g' = RRT]DN = = → g'ab = so hr = 1 and hθ = r . The N and M matrices may be computed as follows: Mab ≡ h'a Rab = = = Rz(-θ) Nab ≡ Sab h'b-1 = = = Rz(θ) Therefore, the relation between a rank-2 tensor's n-expanded form components and the Cartesian form components is given by the expression stated above for rank-2 tensors, A() = M A MT or = . // Lai p 316 Problem 5.71 Airy functions. In isotropic elastic stress analysis for states of plane stress and plane strain, the Cartesian stress tensor Tij has a simple form in which the upper left four components can be represented as derivatives of a potential-like function called an Airy function φ, so that T11 = ∂22φ, T22 = ∂12φ, and T12 = T21 = – ∂1∂2φ. In this case, the above equation becomes = (*) where ∂1 = cosθ ∂r - (sinθ/r)∂θ ∂1 = ∂/∂x1 ∂2 = sinθ ∂r + (cosθ/r)∂θ ∂2 = ∂/∂x2 Using Maple's dchange function, one can have Maple compute from (*) to be // Lai p 264 (5.27.3) This then is a real-world example of using a rank-2 tensor in curvilinear coordinates expanded on the unit tangent base vectors. The mentioned plane of strain or stress has Cartesian coordinates x1,x2 which are converted to polar coordinates r,θ. The third Cartesian coordinate x3 is more or less ignored. The relation between the stress tensor Tij and the infinitesimal strain tensor Eij for an isotropic material is stated in Cartesian coordinate x-space as ( a form of Hooke's Law generalizing F = -kx), Tij = λ tr(E)δij + 2μEij or the same thing Tij = λ tr(E)δij + 2μEij where λ and μ are Lamé's constants. With respect to transformation FM, this is a "true tensor equation" (tr(E) = Ekk is scalar under rotations), so according to Section 7 (u) it is "covariant" and in x"-space may be written T"ij = λ tr(E")δij + 2μE"ij or [T()]ij = λ [T()]kk δij + 2μ[E()]ij . For example, using the notation convention described above, Trr = λ [ Trr+ Tθθ] + 2μ Err Trθ = 2μ Erθ . Notice that δ"ij = δij under transformation FM , since δ"ij = MiaMjbδab = MiaMja = δij, whereas under transformation F one has δ'ij = RiaRjbδab = RiaRja = g'ij. By way of contrast, the Cartesian-coordinates equation Eij = (∂iuj + ∂jui)/2, which relates strain tensor Eij to the vector displacement u of a continuum particle, is not a "true tensor equation", so Erθ ≠ (∂ruθ + ∂θur)/2. In fact, this relation is E = [(u)T + (u)] / 2 and (u) for polar coordinates is computed in Appendix G and one ends up with Erθ = (∂ruθ + (1/r) ∂θur - uθ/r) / 2 . (i) Tensor expansions in a mixed basis Recall the rank-n tensor expansion from section (b) above, A = Σijk... αijk... (bibjbk...) αijk = A (bibjbk) where αijk... are the coefficients of the expansion of A on the direct product basis shown. A might be a tensor, or it might be a tensor-like object. To make explicit the fact that the coefficients depend on the choice of basis, one might write (one b for each index, number of b's is the rank of the tensor) , αijk... = [ A(b,b,b...)]ijk... . The fact that the indices ijk... are "up" indicates that the b label stands for the bi basis and not bi. So here is an example showing the expansion of a rank-3 tensor, A = Σijk [ A(b,b,b)]ijk (bibjbk) [ A(b,b,b)]ijk = A (bibjbk) . Earlier we used the simpler notation [A(b)]ijk for the above coefficient, but now we want to show all the basis elements because now we want to consider a "mixed basis expansion" such as A = Σijk [ A(b,e,u)]ijk (biejuk) [ A(b,e,u)]ijk = A (biejuk) . This is a completely viable expansion since the b, e and u basis vectors are each a complete set within their part of the direct-product space. To verify the validity of this expansion, consider : [ A(b,e,u)]ijk = {A} (biejuk) = { Σi'j'k' [ A(b,e,u)]i'j'k' (bi'ej'uk')} (biejuk) = { Σi'j'k' [ A(b,e,u)]i'j'k' (bi' bi) (ej' ej) (uk' uk) = { Σi'j'k' [ A(b,e,u)]i'j'k' δi'iδj'jδk'k = [ A(b,e,u)]ijk . Alternatively, one could relate this coefficient to the coefficients expanded on uiujuk, [ A(b,e,u)]ijk = A (biejuk) = Aabc (bi)a (ei)a (ui)a as was shown near the start of section (b). In the case of a rank-2 tensor, one has the option of using the other notations discussed above, A = Σij [ A(b,u)]ij (biuj) = Σij [ A(b,u)]ij (biuj) direct product dyadic = Σij [ A(b,u)]ij (biujT) = Σij [ A(b,u)]ij |bi><uj| matrix bra-ket where [ A(b,u)]ij = A (biuj) = Aab (bi)a (uj)b = (bi)T A (ujj) = < bi| A | uj > . One can of course use unit versions of the en basis vectors, n, and then one might write for example A = Σijk [ A(,,u)]ijk (ijuk) where the hats are replicated into the superscript tensor label. Example of a mixed-basis expansion of a tensor The identity tensor can be expanded this way, since ui uj = (ui)Tuj = δij = <ui| uj> , 1 = Σj uj uj = Σj uj(uj)T = Σj ujuj = Σj | uj>< uj| . direct product matrix dyadic bra-ket The ui are related to the ei according to uj = Rij ei . Proof: (uj)a = Rij (ei)a => δja = Rij Ria which is an orthogonality rule of Section 7 (r). It follows that 1 = Σij Rij ei uj = Σij Rij ei(uj)T = Σij Rij eiuj = Σij Rij | ei><uj| direct product matrix dyadic bra-ket where Rij = (ei)T 1 uj = ei uj = eiuj = <ei | uj> // as in Section 7 (s) matrix dot dyadic bra-ket This then is a mixed-basis expansion of the identity tensor. Another form would be 1 = Σij Rij ei uj = Σij [1(e,u)]ij ei uj following the notation discussed above, leading to this rather obscure way of writing Rij , Rij = [1(e,u)]ij . (j) What is a tensor? We are now in a better position to examine some possible answers to this question. (1) A tensor is an operator like A which lives inside a direct product Hilbert Space. We can associate with this tensor A a large variety of up-indexed objects such as [A(b,e,u)]ijk in the example above. If one has at hand K different bases of interest, then for a tensor of rank n there would be Kn possible up-indexed objects. In the case of rank 2, these K2 different indexed objects are matrices. Given the un and en bases used throughout this document, there are two special up-indexed objects of rank 3: [A(u,u,u)]ijk and [A(e,e,e)]ijk which we abbreviate as [A(u)]ijk and [A(e)]ijk or as [A]ijk and [A']ijk . The indexed object [A]ijk is a set of N3 contravariant components of a rank-3 tensor in x-space, and [A']ijk is a set of N3 contravariant components of the same rank-3 tensor in x'-space, where these two spaces are linked by a transformation x' = F(x) which has a linearized form dx' = R dx at a point x. The two sets of contravariant components are related by [A']ijk = Rii'Rjj'Rkk' [A]i'j'k' . These sets of components "transform as a rank-3 tensor with respect to the underlying transformation x' = F(x). As outlined in Section 7, the all-up indexed tensor of rank n is just one of a family of 2n tensors where the indices take all possibly up and down positions, and each such tensor has a corresponding transformation rule, such as [A']ijk = Rii'Rjj'Rkk' [A]i'j'k' for A = Σijk [ A']ijk (eiejek) . For this definition of "tensor" as an abstract operator A, there are many possible set pairs of components (indexed objects where the values of all indices are set in all possible ways) which do NOT transform as a rank-3 tensor with respect to F as just described. The component sets that do transform as tensors are those for which the components are coefficients of an expansion of A on a direct product basis where the individual basis vectors are selected from ei or ei, or are selected from ui or ui. If some other generic basis vector bi appears, then the component set does not transform as a tensor under F. Definition (1) is basically the definition of "tensor" used in this document. (2) Another definition of tensor might be: a tensor is any of the indexed objects mentioned in (1) above. The set of components like [A(b,e,u)]ijk is called a "tensor" because it is a possible coefficient set that can be obtained by expanding the "tensor operator" A on suitable basis vectors. Just as a matrix is sometimes written without its indices, so this tensor object might be represented just as A(b,e,u). The components of this particular indexed object do not transform as either end of the transformation rule stated above, so this tensor is a tensor, but does not transform as a tensor. (3) A third possible definition: a tensor is any indexed object each of whose indices ranges from 1 to N where N is the dimension of one's space of interest. By this definition, any NxN matrix Aij would be a tensor of rank 2. It would be very unlikely that a random matrix like this would be part of the transformation rule [A']ij = Rii'Rjj' [A]i'j' so this matrix Aij is then a tensor, but it probably doesn't transform as a tensor. In fields of physics involving relativity, the first definition is normally used, and an indexed object is called a tensor only if it transforms the way a tensor should transform with respect to a transformation of interest. For example, the affine connection Γcab is never called a tensor. In most other areas of physics definition (3) seems more common, where any matrix is a rank-2 tensor, also known as a second-order tensor. The whole subject of second order tensors is then identified with linear algebra where the operators are matrices. It may turn out that a particular matrix is in fact a tensor by definition (1) with respect to rotations. This is the case for basic matrices involved in equations which must be covariant. In continuum mechanics, which generally uses definition (3), there are many tensors which do not transform as tensors under rotations or other transformations. In that field, when a tensor in fact transforms as a tensor, it is called an objective tensor (sometimes an indifferent tensor). Equations which are covariant in the sense of Section 7 (u) are called "frame indifferent". See Appendix J for examples. ****************************************************************************** Here is the section that got replaced (h) Expansions of tensors on unit tangent base vectors We start with the general Picture A (and later specialize to orthogonal coordinates), In section (b) above it was established that one can expand a tensor A on the tangent base vectors en as A = Σijk... A' ijk... (eiejek...) A' ijk... = contravariant components of A in x'-space A' ijk... = Rii'Rjj'Rkk'...... A i'j'k'... Since en = h'n n, this same expansion for tensor A can be written A = Σijk... h'ih'jh'k......A' ijk... (ijk...) = Σijk... [A()]ijk... (ijk...) where the unit-vector expansion coefficients are given by [A()]ijk... = h'ih'jh'k......A' ijk... = h'ih'jh'k...... Rii'Rjj'Rkk'...... A i'j'k'... = (h'i Rii')( h'j Rjj')( h'k Rkk') ..... A i'j'k'... Coefficient notation. In a curvilinear coordinates application of these expansions, the expansion coefficients are usually written in the following manner, [A()]ijk... = Ax'x'x' ...... where the x'n are the names of the coordinates. For example, for a rank-4 tensor in spherical coordinates with coordinates x'1 = r, x'2 = θ and x'3 = φ one might write [A()]2213 = Aθθrφ . Since [A()]ijk... is not a tensor (with respect to F), there is no particular reason to put the indices "up" and for that reason they are usually written down, as in Aθθrφ . Matrices M and N. It is convenient now to define Mab ≡ h'a Rab so that then [A()]ijk... = Mii'Mjj'Mkk'...... A i'j'k'... . Defining Nab to be the inverse of Mab, one has Nab ≡ h'b-1Sab = h'b-1Rba A ijk... = Nii'Njj'Nkk'...... [A()]i'j'k'... . To verify that this N is the correct inverse or M, MakNkc = (h'a Rak)( h'c-1Rck) = (h'a/ h'c) Rak Rck = (h'a/ h'c)δac = δac making use of the orthogonality rule of Section 7 (r), Rak Rck = δac . M and N can be written in terms of the tangent base vectors as follows: Mni ≡ h'n Rni = h'n(en)i Nin = h'n-1Rni = h'n-1(en)i = (n)i which says that Nin = { 1 , 2, .... } -- the columns of Nin are the unit tangent base vectors. Rank-1 tensors. For a vector, the above coefficient relation is written [A()]i = Mij Aj or A() = M A and A = N A() where A has these two familiar expansions, A = Anun = [A()]n n [A()]n = Ax' for example [A()]1 = Ar . Rank-2 tensors. Here the coefficient relation is [A()]ij = Mii'Mjj'A i'j' = Mii' A i'j' Mjj' which can be written [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm Mni = h'n(en)i . Defining bn = h'nen , then [bn]i = h'n(en)i = Mni = Bni of section (g) . Meanwhile, from Section 6 (b) we know that wnk = bn bk = h'nh'k(en ek) , and then wnk = (w-1)nk . In any event, whatever wnk is, it is the object which can lower the n or m indices on both sides of the above equation. Lowering just the m index and then reversing the j tilt gives [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm = Mni A ij (MT)jm and we replicate the section (g) result with B = M : A() = M A MT // [A() = M A MT ]SN,dt . Orthogonal curvilinear coordinates application We now switch to picture B (g=1) and assume that the x'i are orthogonal coordinates, In this situation, x-space is Cartesian with gab = gab = δab and g'ab = h'a2δab and g'ab = h'a-2δab. From Section 7 (o) one then has, Rab = Rab' gb'b = Rab Rab = g'aa'Ra'b' gb'b = g'aa'Ra'b = h'a2 Rab = h'a2 Rab Rab= g'aa'Ra'b = h'a2 Rab or Rab = Rab Rab = Rab = h'a2 Rab Rab = h'a-2 Rab . Also from Section 7 (11), g'ab = Raa'Rbb'ga'b' = Raa'Rba' = RacRbc g'ab = Sa'a Sb'b ga'b' = Raa' Rbb'ga'b' = Raa' Rba' = RacRbc or g'ab = RacRbc and g'ab = RacRbc . In this scenario, regardless of what R and S are, M is a "rotation" (shown below), where we include in this term possible axis reflections. What we really mean is that in developmental notation M is a real-orthogonal matrix, MMT = 1. Since N=M-1, N is then also a rotation. To prove that M is a rotation in developmental notation, the standard notation equation Mab ≡ h'a Rab can be reverse-translated to Mab = h'aRab . Then (now g' = RRT from Section 5 (l) ) (MMT)ac = MabMcb = h'aRab h'cRcb = h'ah'c RabRTbc = h'ah'c(RRT)ac = h'ah'cg'ac = h'ah'c [ h'a–2 δa,c] = δa,c => MMT = 1 . Proving the same thing directly in standard notation requires showing that Mab Mcb = δa,c ( see the end of Section 7 (i) ) Mab Mcb = h'a Rab h'c Rcb = h'a h'c Rab Rcb = h'a h'c Rab (h'c-2 Rcb) = (ha'/h'c) (Rab Rcb) = (ha'/h'c)δac = δac = δa,c where use was again made of the orthogonality rule of Section 7 (r), Rab Rcb = δac. Relation beween M and N. Looking at Mab Mcb = δa,c and knowing that Mab (M-1)bc = δac = δa,c one concludes that (M-1)bc = Mcb . But (M-1)bc = Nbc so Nbc = Mcb // reminder: this does not say that N = MT in standard notation which can be verified from the above expressions for N and M. Interpretation of N and M. Since en = S un ( Section 3 (a) with e'n = un) and since en = h'nn , it follows that n = h'n-1 S un or (n)a = h'n-1 Sab (un)b = h'n-1 Sab δnb = h'n-1 San = Nan = Nabδbn = Nab(un)b or n = N un and (n)a = Nan . Since the un are the Cartesian unit vectors, it seems intuitively obvious that the transformation that moves this frame of orthonormal unit vectors {un} into the orthonormal frame {n} must be a "rotation". Above it was shown that Nan = Mna , therefore Mna = Nan = (n)a The rotation matrix Nan = (n)a has the orthogonal basis vectors n as its columns, while the rotation matrix Mna = (n)a has the orthogonal basis vectors n as its rows. It might be noted that, in our situation with Cartesian x-space and orthogonal coordinates, n = n : n = en/|en| |en|2 = en en = g'nn = h'n-2 so n = en h'n = h'n g'nn en = h'nh'n-2 en = h'n-1 en = n . Paradigm Shift. In the curvilinear coordinates "application", x-space is Cartesian and x'-space has coordinates x'i which are the curvilinear coordinates. The connection between x-space and x'-space is some underlying transformation F. The inverse transformation x = F-1(x') defines the curvilinear coordinates, as for example in the case x = rcosθ and y = rsinθ. Transformation F is non-linear, and associated with it is the linearized-at-a point transformation matrix R, and R is in general not a rotation matrix. Now, if we are dealing with orthogonal curvilinear coordinates, and when we expand tensors onto the unit tangent base vectors, we end up with (as shown above) A = Σijk... [A()]ijk... (ijk...) [A()]ijk... = Mii'Mjj'Mkk'...... Ai'j'k'... where M is a rotation matrix. At this point, one can make a paradigm shift and regard M as the R-matrix (call it RM) of a different underlying transformation (call it FM). We can think of this new transformation as connecting the same original Cartesian x-space to a new x"-space. Since the connection has RM = M = a rotation, g" = 1 as well. Then the above expansion and transformation can be written as A = Σijk... A" ijk... (ijk...) n = N un un = M n A" ijk... = Mii'Mjj'Mkk'...... Ai'j'k'... A" ijk... = [A()]ijk... for example: x"i = Mii'xi' or x" = Mx . // wrong, true for dx's. With respect to this new underlying transformation FM having R matrix RM = M, the components A" ijk... are the contravariant components of the tensor A in x"-space, while the Aijk... continue to be the contravariant components of the same tensor A in x-space. Here is a picture: The tensor components are still A" ijk.. = [A()]ijk.. (e.g., A" 2213 = [A()]2213 = Aθθrφ ). At this point, one can ignore the curvilinear transformation on the right above, and concentrate only on the transformation on the left. This transformation is a rotation which does n = M-1un, so the usual Cartesian basis unit vectors un in x-space are back-rotated into a set of Cartesian unit vectors n in x"-space. The right side transformation is lurking in this notation, however, since the n are unit vector versions of the tangent base vectors en of the transformation on the right. One implication of the above picture relates to tensor equations being covariant, as discussed in Section 7 (u). If one has a tensor equation in x-space Qadc = HabTbc Bd in which all the objects are true tensor with respect to the underlyling FM, then the equation is covariant and takes the same form in x"-space Q"adc = H"abT"bc B"d An example will be given below. The relation un = M n can be written un = M(x) n(x) to emphasize that the rotation M = RM is really a different rotation at every point x, since the n(x) vary with x. This is very different from a global rotation which is the same at all points. For a global rotation R, F = R = linear and ∂iuj is a tensor. For RM being a rotation which varies from point to point, FM is non-linear just as F defining the curvilinear coordinates is non-linear, and ∂iuj fails to be a tensor under either F or FM. Example: Polar Coordinates. In polar coordinates now with ordering r,θ = 1,2 one has S11 = (∂x/∂r) = cosθ x = rcosθ S12 = (∂x/∂θ) = -rsinθ y = rsinθ S21 = (∂y/∂r) = sinθ S22 = (∂y/∂θ) = rcosθ Sij = Rij = R = S-1 [g' = RRT]DN = = → g'ab = so hr = 1 and hθ = r The N and M matrices may be computed as follows: Mab ≡ h'a Rab = = = Rz(-θ) Nab ≡ Sab h'b-1 = = = Rz(θ) Therefore, the relation between a rank-2 tensor's n-expanded form components and the Cartesian form components is given by the expression stated above for rank-2 tensors, A() = M A MT or = . // Lai p 316 Problem 5.71 Airy functions. In isotropic elastic stress analysis for states of plane stress and plane strain, the Cartesian stress tensor Tij has a simple form in which the upper left four components can be represented as derivatives of a potential-like function called an Airy function φ, so that T11 = ∂22φ, T22 = ∂12φ, and T12 = T21 = – ∂1∂2φ. In this case, the above equation becomes = (*) where ∂1 = cosθ ∂r - (sinθ/r)∂θ ∂1 = ∂/∂x1 ∂2 = sinθ ∂r + (cosθ/r)∂θ ∂2 = ∂/∂x2 Using Maple's dchange function, one can have Maple compute from (*) to be // Lai p 264 (5.27.3) This then is a real-world example of using a rank-2 tensor in curvilinear coordinates expanded on the unit tangent base vectors. The mentioned plane of strain or stress has Cartesian coordinates x1,x2 which are converted to polar coordinates r,θ. The third Cartesian coordinate x3 is more or less ignored. The relation between the stress tensor Tij and the infinitesimal strain tensor Eij for an isotropic material is stated in Cartesian coordinate x-space as ( a form of Hooke's Law generalizing F = -kx), Tij = λ tr(E)δij + 2μEij or the same thing Tij = λ tr(E)δij + 2μEij where λ and μ are Lamé's constants. With respect to transformation FM, this is a "true tensor equation" (tr(E) = Ekk is scalar under rotations), so according to Section 7 (u) it is "covariant" and in x"-space may be written T"ij = λ tr(E")δij + 2μE"ij or [T()]ij = λ [T()]kk δij + 2μ[E()]ij . For example, using the notation convention described above, Trr = λ [ Trr+ Tθθ] + 2μ Err Trθ = 2μ Erθ . Notice that δ"ij = δij under transformation FM , since δ"ij = MiaMjbδab = MiaMja = δij, whereas under transformation F one has δ'ij = RiaRjbδab = RiaRja = g'ij. By way of contrast, the Cartesian-coordinates equation Eij = (∂iuj + ∂jui)/2 which relates strain tensor Eij to the vector displacement u of a continuum particle is not a "true tensor equation", so Erθ ≠ (∂ruθ + ∂θur)/2. In fact, this relation is E = [(u)T + (u)] / 2 and (u) for polar coordinates is computed in Appendix G and one ends up with Erθ = (∂ruθ + (1/r) ∂θur - uθ/r) / 2 .