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Acknowledgments 5 Preface 17 1. Introduction 19 STRUCTURES and SUPERSTRUCTURES 20 What is a scalar? What is a vector? 23 What is a tensor? 24 Examples of tensors in materials mechanics 27 The stress tensor 27 The deformation gradient tensor 29 Vector and Tensor notation - philosophy 30 2. Terminology from functional analysis 32 3. Matrix Analysis (and some matrix calculus) 39 Definition of a matrix 39 Component matrices associated with vectors and tensors (notation explanation) 40 The matrix product 40 SPECIAL CASE: a matrix times an array 40 SPECIAL CASE: inner product of two arrays 41 SPECIAL CASE: outer product of two arrays 41 EXAMPLE: 41 The Kronecker delta 43 The identity matrix 43 Derivatives of vector and matrix expressions 44 Derivative of an array with respect to itself 45 Derivative of a matrix with respect to itself 46 The transpose of a matrix 47 Derivative of the transpose: 47 The inner product of two column matrices 47 Derivatives of the inner product: 48 The outer product of two column matrices. 49 The trace of a square matrix 49 Derivative of the trace 49 The matrix inner product 50 Derivative of the matrix inner product 50 Magnitudes and positivity property of the inner product 51 Derivative of the magnitude 52 Norms 52 Weighted or “energy” norms 53 Derivative of the energy norm 53 The 3D permutation symbol 54 The e-d (E-delta) identity 54 The e-d (E-delta) identity with multiple summed indices 56 Determinant of a square matrix 57 More about cofactors 60 Cofactor-inverse relationship 61 Derivative of the cofactor 62 Derivative of a determinant (IMPORTANT) 62 Rates of determinants 63 Derivatives of determinants with respect to vectors 64 Principal sub-matrices and principal minors 64 Matrix invariants 64 Alternative invariant sets 65 Positive definite 65 The cofactor-determinant connection 66 Inverse 67 Eigenvalues and eigenvectors 67 Similarity transformations 69 Finding eigenvectors by using the adjugate 70 Eigenprojectors 71 Finding eigenprojectors without finding eigenvectors. 72 4. Vector/tensor notation 73 “Ordinary” engineering vectors 73 Engineering “laboratory” base vectors 73 Other choices for the base vectors 73 Basis expansion of a vector 74 Summation convention - details 75 Don’t forget what repeated indices really mean 76 Further special-situation summation rules 77 Indicial notation in derivatives 78 BEWARE: avoid implicit sums as independent variables 78 Reading index STRUCTURE, not index SYMBOLS 79 Aesthetic (courteous) indexing 80 Suspending the summation convention 80 Combining indicial equations 81 Index-changing properties of the Kronecker delta 82 Summing the Kronecker delta itself 87 Our (unconventional) “under-tilde” notation 87 Tensor invariant operations 87 5. Simple vector operations and properties 89 Dot product between two vectors 89 GOAL: Define, cite properties, show application to find angle between two vectors, show application to decide if a vector is zero. 89 Dot product between orthonormal base vectors 90 A “quotient” rule (deciding if a vector is zero) 90 GOAL: Explain that you can’t define division by vectors, but there is an extended viewpoint that is similar. 90 Deciding if one vector equals another vector 91 Finding the i-th component of a vector 91 GOAL: Show that the ith component of a vector can be found by dotting that vector by the ith base vector. 91 Even and odd vector functions 92 GOAL: Define, show function decomposition into even plus odd parts 92 Homogeneous functions 92 GOAL: Define, show identities 92 Vector orientation and sense 93 GOAL: Clarify terminology 93 Simple scalar components 93 GOAL: Find the scalar component of one vector in the direction of another. 93 Cross product 94 GOAL: Define, show identities, show how to find the area of the parallelogram formed by two vectors 94 Cross product between orthonormal base vectors 94 GOAL: Cite important special-case of the cross product between base vectors. 94 Triple scalar product 96 GOAL: Define, cite properties, show application to deciding if three vectors are linearly independent. 96 Triple scalar product between orthonormal RIGHT-HANDED base vectors 97 GOAL: Cite the triple scalar product between right-handed base vectors and argue against redefining the permutation symbol for other types of basis triads (instead, advocate using different values for the permutation tensor components). 97 6. Projections 98 Orthogonal (perpendicular) linear projections 98 GOAL: Show how to find the part of a vector in the direction of another vector (and clarify distinction between this and finding the scalar component in the direction of a vector). 98 Rank-1 orthogonal projections 100 GOAL: Explain that finding the part of a vector in the direction of another vector is a projection operation, and explain why this projection has “rank 1”. 100 Rank-2 orthogonal projections 101 GOAL: Show how to find the orthogonal (nearest) projection of a vector onto the plane perpendicular to another vector. 101 Basis interpretation of orthogonal projections 101 GOAL: Emphasize that the rank-1 and rank-2 projections effectively extract “pieces” of the starting vector that break it down into smaller parts relative to a locally aligned basis. 101 Rank-2 oblique linear projection 102 GOAL: Explain oblique (not nearest point) projections onto a plane. 102 Rank-1 oblique linear projection 103 GOAL: Explain oblique projections onto a second vector. 103 Degenerate (trivial) Rank-0 linear projection 103 GOAL: Explain that the zero operator is a degenerate projector. 103 Degenerate (trivial) Rank-3 projection in 3D space 104 GOAL: Explain that the identity operator is also a projection. 104 Complementary projectors 104 GOAL: define, begin introducing concepts needed for the projection theorem 104 Normalized versions of the projectors 104 GOAL: Generalize the “aligned” basis description of a projector to show that the structure is very similar for oblique projections except that the “aligned” basis is non-orthonormal. 104 Expressing a vector as a linear combination of three arbitrary (not necessarily orthonormal) vectors. 106 GOAL: outline most straightforward process, refine notation for the process to be more consistent with curvilinear notation. 106 Generalized projections 108 GOAL: Reiterate the mathematician’s definition of the term “projection” (idempotent) 108 Linear projections 108 GOAL: Define very special (linear) class of projectors, explain that oblique projectors are also linear. 108 Nonlinear projections 108 GOAL: To clarify linear projections, give examples of some nonlinear projections. 108 The vector “signum” function 108 Gravitational (distorted light ray) projections 109 Self-adjoint projections 109 GOAL: Show that orthogonal (nearest point) projections are self-adjoint, whereas oblique projections are not. Set stage for later showing that orthogonal projection tensors are symmetric, whereas oblique projection tensors are not. 109 Gram-Schmidt orthogonalization 110 GOAL: show how to convert a set of vectors into a minimal orthonormal basis that will span the same space as the original set of vectors. 110 Special case: orthogonalization of two vectors 111 The projection theorem 111 7. Tensors 113 Analogy between tensors and other (more familiar) concepts 114 GOAL: Explain how vectors share several axiomatic properties in common with smooth scalar-valued functions. Then demonstrate that tensors are quite similar to smooth functions of two variables. 114 Linear operators (transformations) 117 GOAL: Set stage for the “linear transformation” definition of a tensor by showing how a matrix arises naturally to characterize ... 117 Dyads and dyadic multiplication 121 GOAL: Define, cite properties 121 Simpler “no-symbol” dyadic notation 122 GOAL: Advocate in favor of not using the dyadic multiplication symbol. 122 The matrix associated with a dyad 122 GOAL: Show that a dyad has an associated matrix that is equivalent to the outer product of the vectors. Set stage for making connection between dyads and tensors - both have associated matrices. Dyads are special kinds of tensors. 122 The sum of dyads 123 GOAL: define, cite properties 123 A sum of two or three dyads is NOT (generally) reducible 124 GOAL: Show that the sum of two or three dyads cannot always be rearranged to become just a single dyad - the sum of dyads is its... 124 Scalar multiplication of a dyad 124 GOAL: Define this operation, cite properties, emphasize that scalar multiplication can act on any of the individual vectors forming a dyad. 124 The sum of four or more dyads is reducible! (not a superset) 125 GOAL: Show that the sum of more than three dyads (in 3D) can always be reduced to the sum of three or fewer dyads. 125 The dyad definition of a second-order tensor 125 GOAL: Use result of previous section to define a “tensor” to be any dyad or sum of dyads. 125 Expansion of a second-order tensor in terms of basis dyads 126 GOAL: show that expanding an ordinary engineering tensor as a linear combination of the NINE possible ways to form basis dyads is similar in spirit to expanding an ordinary engineering vector as a linear combination of the laboratory orthonormal basis. 126 Triads and higher-order tensors 128 GOAL: Introduce higher-order tensors 128 Our Vmn tensor “class” notation 129 GOAL: Define the meaning of Vmn and show that any tensor of class Vmn will have mn components. 129 Comment 132 8. Tensor operations 133 Dotting a tensor from the right by a vector 133 The transpose of a tensor 133 Dotting a tensor from the left by a vector 134 Dotting a tensor by vectors from both sides 135 Extracting a particular tensor component 135 Dotting a tensor into a tensor (tensor composition) 135 9. Tensor analysis primitives 137 Three kinds of vector and tensor notation 137 REPRESENTATION THEOREM for linear forms 140 GOAL: Explain that there exists a unique tensor that characterizes each function that linearly transforms vectors to vectors. 140 Representation theorem for vector-to-scalar linear functions 141 Advanced Representation Theorem (to be read once you learn about higher-order tensors and the Vmn class notation) 142 Finding the tensor associated with a linear function 143 Method #1 143 Method #2 143 Method #3 144 EXAMPLE 144 The identity tensor 144 Tensor associated with composition of two linear transformations 145 The power of heuristically consistent notation 146 The inverse of a tensor 147 The COFACTOR tensor 147 Axial tensors (tensor associated with a cross-product) 149 Glide plane expressions 151 Axial vectors 151 Cofactor tensor associated with a vector 152 Cramer’s rule for the inverse 152 Inverse of a rank-1 modification (Sherman-Morrison formula) 153 Derivative of a determinant 153 Exploiting operator invariance with “preferred” bases 154 GOAL: 154 10. Projectors in tensor notation 156 Nonlinear projections do not have a tensor representation 156 Linear orthogonal projectors expressed in terms of dyads 157 Just one esoteric application of projectors 159 GOAL: Give an illustration of a physical problem whose governing equations are improved through the use of projectors. 159 IMPORTANT: Finding a projection to a desired target space 159 Properties of complementary projection tensors 161 Self-adjoint (orthogonal) projectors 161 Non-self-adjoint (oblique) projectors 162 GOAL: Call out some differences between orthogonal and oblique projectors 162 Generalized complementary projectors 163 11. More Tensor primitives 165 Tensor properties 165 GOAL: List the properties that a person should look for in a tensor because numerous useful theorems exist for tensors with certain properties. 165 Orthogonal (unitary) tensors 166 Tensor associated with the cross product 169 Cross-products in left-handed and general bases 170 Physical application of axial vectors 172 Symmetric and skew-symmetric tensors 173 Positive definite tensors 174 Faster way to check for positive definiteness 174 Positive semi-definite 175 Negative definite and negative semi-definite tensors 175 Isotropic and deviatoric tensors 176 12. Tensor operations 177 Second-order tensor inner product 177 A NON-recommended scalar-valued product 178 GOAL: Explain why a commonly used alternative scalar-valued product should be avoided. 178 Fourth-order tensor inner product 179 Fourth-order Sherman-Morrison formula 180 Higher-order tensor inner product 181 Self-defining notation 181 The magnitude of a tensor or a vector 183 Useful inner product identities 183 Distinction between an Nth-order tensor and an Nth-rank tensor 184 Fourth-order oblique tensor projections 185 Leafing and palming operations 185 GOAL: Introduce a simple, but obscure, higher-order tensor operation 185 Symmetric Leafing 187 13. Coordinate/basis transformations 188 Change of basis (and coordinate transformations) 188 EXAMPLE 191 Definition of a vector and a tensor 193 Basis coupling tensor 194 14. Tensor (and Tensor function) invariance 195 What’s the difference between a matrix and a tensor? 195 Example of a “scalar rule” that satisfies tensor invariance 197 Example of a “scalar rule” that violates tensor invariance 198 Example of a 3x3 matrix that does not correspond to a tensor 199 The inertia TENSOR 201 15. Scalar invariants and spectral analysis 203 Invariants of vectors or tensors 203 Primitive invariants 203 Trace invariants 205 Characteristic invariants 205 Direct notation definitions of the characteristic invariants 207 The cofactor in the triple scalar product 207 Invariants of a sum of two tensors 208 CASE: invariants of the sum of a tensor plus a dyad 208 The Cayley-Hamilton theorem: 210 CASE: Expressing the inverse in terms of powers and invariants 210 CASE: Expressing the cofactor in terms of powers and invariants 210 Eigenvalue problems 210 Algebraic and geometric multiplicity of eigenvalues 211 Diagonalizable tensors (the spectral theorem) 213 Eigenprojectors 213 16. Geometrical entities 216 Equation of a plane 216 Equation of a line 217 Equation of a sphere 218 Equation of an ellipsoid 218 Example 219 Equation of a cylinder with an ellipse-cross-section 220 Equation of a right circular cylinder 220 Equation of a general quadric (including hyperboloid) 220 Generalization of the quadratic formula and “completing the square” 221 17. Polar decomposition 223 Singular value decomposition 223 Special case: 223 The polar decomposition theorem: 224 Polar decomposition is a nonlinear projection 227 The *FAST* way to do a polar decomposition in 2D 227 A fast and accurate numerical 3D polar decomposition 228 Dilation-Distortion (volumetric-isochoric) decomposition 229 GOAL: Describe another (less common) multiplicative decomposition that breaks a tensor transformation into two parts, one part that captures size changes and the other part characterizing shape changes. 229 Thermomechanics application 230 18. Material symmetry 233 What is isotropy? 233 GOAL: Describe two competing definitions of isotropy and the relative merits of each. 233 Important consequence 235 Isotropic second-order tensors in 3D space 236 Isotropic second-order tensors in 2D space 237 GOAL: Demonstrate that the proper-isotropic space is two dimensional for tensors of class V22, in stark contrast to the result for V23. The strict-isotropic space is one-dimensional. 237 Isotropic fourth-order tensors 240 Finding the isotropic part of a fourth-order tensor 241 GOAL: Reiterate the concept of projections by showing an advanced, higher-order, application. 241 A scalar measure of “percent anisotropy” 242 GOAL: 242 Transverse isotropy 242 19. Abstract vector/tensor algebra 245 Structures 245 GOAL: What are structures, and why use them? Motivate writing down operations in purposely abstract, counter-intuitive, notation to ensure that you don’t introduce your own bias into the analysis. 245 Definition of an abstract vector 248 What does this mathematician’s definition of a vector have to do with the definition used in applied mechanics? 250 Inner product spaces 251 Alternative inner product structures 251 Some examples of inner product spaces 252 Continuous functions are vectors! 253 Tensors are vectors! 254 Vector subspaces 255 Example: 256 Example: commuting space 256 Subspaces and the projection theorem 258 Abstract contraction and swap (exchange) operators 258 The contraction tensor 262 The swap tensor 262 20. Vector and Tensor Visualization 265 Mohr’s circle for 2D tensors 266 GOAL: Describe how to generate Mohr’s circle for a 2x2 matrix that is not necessarily symmetric. 266 21. Vector/tensor differential calculus 269 Stilted definitions of grad, div, and curl 269 Gradients in curvilinear coordinates 270 When do you NOT have to worry about curvilinear formulas? 272 Spatial gradients of higher-order tensors 274 Product rule for gradient operations 275 Identities involving the “nabla” 277 Compound differential operator notation (and unfortunate pampering) 279 Right and left gradient operations (we love them both!) 280 Casual (non-rigorous) tensor calculus 283 SIDEBAR: “total” and “partial” derivative notation 284 The “nabla” or “del” gradient operator 287 Okay, if the above relation does not hold, does anything LIKE IT hold? 289 Directed derivative 291 EXAMPLE 292 Derivatives in reduced dimension spaces 293 A more physically significant example 297 Series expansion of a nonlinear vector function 298 Exact differentials of one variable 300 Exact differentials of two variables 301 The same result in a different notation 302 Exact differentials in three dimensions 302 Coupled inexact differentials 303 22. Vector/tensor Integral calculus 304 Gauss theorems 304 Stokes theorem 304 Divergence theorem 304 Integration by parts 304 Leibniz theorem 306 LONG EXAMPLE: conservation of mass 309 Generalized integral formulas for discontinuous integrands 313 23. Closing remarks 314 24. Solved problems 315 REFERENCES 317 INDEX This index is a work in progress. Please notify the author of any critical omissions or errors. 319 Acknowledgments......Page 5 Preface......Page 17 1. Introduction......Page 19 STRUCTURES and SUPERSTRUCTURES......Page 20 What is a scalar? What is a vector?......Page 23 What is a tensor?......Page 24 The stress tensor......Page 27 The deformation gradient tensor......Page 29 Vector and Tensor notation - philosophy......Page 30 2. Terminology from functional analysis......Page 32 Definition of a matrix......Page 39 SPECIAL CASE: a matrix times an array......Page 40 EXAMPLE:......Page 41 The identity matrix......Page 43 Derivatives of vector and matrix expressions......Page 44 Derivative of an array with respect to itself......Page 45 Derivative of a matrix with respect to itself......Page 46 The inner product of two column matrices......Page 47 Derivatives of the inner product:......Page 48 Derivative of the trace......Page 49 Derivative of the matrix inner product......Page 50 Magnitudes and positivity property of the inner product......Page 51 Norms......Page 52 Derivative of the energy norm......Page 53 The e-d (E-delta) identity......Page 54 The e-d (E-delta) identity with multiple summed indices......Page 56 Determinant of a square matrix......Page 57 More about cofactors......Page 60 Cofactor-inverse relationship......Page 61 Derivative of a determinant (IMPORTANT)......Page 62 Rates of determinants......Page 63 Matrix invariants......Page 64 Positive definite......Page 65 The cofactor-determinant connection......Page 66 Eigenvalues and eigenvectors......Page 67 Similarity transformations......Page 69 Finding eigenvectors by using the adjugate......Page 70 Eigenprojectors......Page 71 Finding eigenprojectors without finding eigenvectors.......Page 72 Other choices for the base vectors......Page 73 Basis expansion of a vector......Page 74 Summation convention - details......Page 75 Don’t forget what repeated indices really mean......Page 76 Further special-situation summation rules......Page 77 BEWARE: avoid implicit sums as independent variables......Page 78 Reading index STRUCTURE, not index SYMBOLS......Page 79 Suspending the summation convention......Page 80 Combining indicial equations......Page 81 Index-changing properties of the Kronecker delta......Page 82 Tensor invariant operations......Page 87 GOAL: Define, cite properties, show application to find angle between two vectors, show application to decide if a vector is zero.......Page 89 GOAL: Explain that you can’t define division by vectors, but there is an extended viewpoint that is similar.......Page 90 GOAL: Show that the ith component of a vector can be found by dotting that vector by the ith base vector.......Page 91 GOAL: Define, show identities......Page 92 GOAL: Find the scalar component of one vector in the direction of another.......Page 93 GOAL: Cite important special-case of the cross product between base vectors.......Page 94 GOAL: Define, cite properties, show application to deciding if three vectors are linearly independent.......Page 96 GOAL: Cite the triple scalar product between right-handed base vectors and argue against redefining the permutation symbol for other types of basis triads (instead, advocate using different values for the permutation tensor components).......Page 97 GOAL: Show how to find the part of a vector in the direction of another vector (and clarify distinction between this and finding the scalar component in the direction of a vector).......Page 98 GOAL: Explain that finding the part of a vector in the direction of another vector is a projection operation, and explain why this projection has “rank 1”.......Page 100 GOAL: Emphasize that the rank-1 and rank-2 projections effectively extract “pieces” of the starting vector that break it down into smaller parts relative to a locally aligned basis.......Page 101 GOAL: Explain oblique (not nearest point) projections onto a plane.......Page 102 GOAL: Explain that the zero operator is a degenerate projector.......Page 103 GOAL: Generalize the “aligned” basis description of a projector to show that the structure is very similar for oblique projections except that the “aligned” basis is non-orthonormal.......Page 104 GOAL: outline most straightforward process, refine notation for the process to be more consistent with curvilinear notation.......Page 106 The vector “signum” function......Page 108 GOAL: Show that orthogonal (nearest point) projections are self-adjoint, whereas oblique projections are not. Set stage for later showing that orthogonal projection tensors are symmetric, whereas oblique projection tensors are not.......Page 109 GOAL: show how to convert a set of vectors into a minimal orthonormal basis that will span the same space as the original set of vectors.......Page 110 The projection theorem......Page 111 7. Tensors......Page 113 GOAL: Explain how vectors share several axiomatic properties in common with smooth scalar-valued functions. Then demonstrate that tensors are quite similar to smooth functions of two variables.......Page 114 GOAL: Set stage for the “linear transformation” definition of a tensor by showing how a matrix arises naturally to characterize .........Page 117 GOAL: Define, cite properties......Page 121 GOAL: Show that a dyad has an associated matrix that is equivalent to the outer product of the vectors. Set stage for making connection between dyads and tensors - both have associated matrices. Dyads are special kinds of tensors.......Page 122 GOAL: define, cite properties......Page 123 GOAL: Define this operation, cite properties, emphasize that scalar multiplication can act on any of the individual vectors forming a dyad.......Page 124 GOAL: Use result of previous section to define a “tensor” to be any dyad or sum of dyads.......Page 125 GOAL: show that expanding an ordinary engineering tensor as a linear combination of the NINE possible ways to form basis dyads is similar in spirit to expanding an ordinary engineering vector as a linear combination of the laboratory orthonormal basis.......Page 126 GOAL: Introduce higher-order tensors......Page 128 GOAL: Define the meaning of Vmn and show that any tensor of class Vmn will have mn components.......Page 129 Comment......Page 132 The transpose of a tensor......Page 133 Dotting a tensor from the left by a vector......Page 134 Dotting a tensor into a tensor (tensor composition)......Page 135 Three kinds of vector and tensor notation......Page 137 GOAL: Explain that there exists a unique tensor that characterizes each function that linearly transforms vectors to vectors.......Page 140 Representation theorem for vector-to-scalar linear functions......Page 141 Advanced Representation Theorem (to be read once you learn about higher-order tensors and the Vmn class notation)......Page 142 Method #2......Page 143 The identity tensor......Page 144 Tensor associated with composition of two linear transformations......Page 145 The power of heuristically consistent notation......Page 146 The COFACTOR tensor......Page 147 Axial tensors (tensor associated with a cross-product)......Page 149 Axial vectors......Page 151 Cramer’s rule for the inverse......Page 152 Derivative of a determinant......Page 153 GOAL:......Page 154 Nonlinear projections do not have a tensor representation......Page 156 Linear orthogonal projectors expressed in terms of dyads......Page 157 IMPORTANT: Finding a projection to a desired target space......Page 159 Self-adjoint (orthogonal) projectors......Page 161 GOAL: Call out some differences between orthogonal and oblique projectors......Page 162 Generalized complementary projectors......Page 163 GOAL: List the properties that a person should look for in a tensor because numerous useful theorems exist for tensors with certain properties.......Page 165 Orthogonal (unitary) tensors......Page 166 Tensor associated with the cross product......Page 169 Cross-products in left-handed and general bases......Page 170 Physical application of axial vectors......Page 172 Symmetric and skew-symmetric tensors......Page 173 Faster way to check for positive definiteness......Page 174 Negative definite and negative semi-definite tensors......Page 175 Isotropic and deviatoric tensors......Page 176 Second-order tensor inner product......Page 177 GOAL: Explain why a commonly used alternative scalar-valued product should be avoided.......Page 178 Fourth-order tensor inner product......Page 179 Fourth-order Sherman-Morrison formula......Page 180 Self-defining notation......Page 181 Useful inner product identities......Page 183 Distinction between an Nth-order tensor and an Nth-rank tensor......Page 184 GOAL: Introduce a simple, but obscure, higher-order tensor operation......Page 185 Symmetric Leafing......Page 187 Change of basis (and coordinate transformations)......Page 188 EXAMPLE......Page 191 Definition of a vector and a tensor......Page 193 Basis coupling tensor......Page 194 What’s the difference between a matrix and a tensor?......Page 195 Example of a “scalar rule” that satisfies tensor invariance......Page 197 Example of a “scalar rule” that violates tensor invariance......Page 198 Example of a 3x3 matrix that does not correspond to a tensor......Page 199 The inertia TENSOR......Page 201 Primitive invariants......Page 203 Characteristic invariants......Page 205 The cofactor in the triple scalar product......Page 207 CASE: invariants of the sum of a tensor plus a dyad......Page 208 Eigenvalue problems......Page 210 Algebraic and geometric multiplicity of eigenvalues......Page 211 Eigenprojectors......Page 213 Equation of a plane......Page 216 Equation of a line......Page 217 Equation of an ellipsoid......Page 218 Example......Page 219 Equation of a general quadric (including hyperboloid)......Page 220 Generalization of the quadratic formula and “completing the square”......Page 221 Special case:......Page 223 The polar decomposition theorem:......Page 224 The *FAST* way to do a polar decomposition in 2D......Page 227 A fast and accurate numerical 3D polar decomposition......Page 228 GOAL: Describe another (less common) multiplicative decomposition that breaks a tensor transformation into two parts, one part that captures size changes and the other part characterizing shape changes.......Page 229 Thermomechanics application......Page 230 GOAL: Describe two competing definitions of isotropy and the relative merits of each.......Page 233 Important consequence......Page 235 Isotropic second-order tensors in 3D space......Page 236 GOAL: Demonstrate that the proper-isotropic space is two dimensional for tensors of class V22, in stark contrast to the result for V23. The strict-isotropic space is one-dimensional.......Page 237 Isotropic fourth-order tensors......Page 240 GOAL: Reiterate the concept of projections by showing an advanced, higher-order, application.......Page 241 Transverse isotropy......Page 242 GOAL: What are structures, and why use them? Motivate writing down operations in purposely abstract, counter-intuitive, notation to ensure that you don’t introduce your own bias into the analysis.......Page 245 Definition of an abstract vector......Page 248 What does this mathematician’s definition of a vector have to do with the definition used in applied mechanics?......Page 250 Alternative inner product structures......Page 251 Some examples of inner product spaces......Page 252 Continuous functions are vectors!......Page 253 Tensors are vectors!......Page 254 Vector subspaces......Page 255 Example: commuting space......Page 256 Abstract contraction and swap (exchange) operators......Page 258 The swap tensor......Page 262 20. Vector and Tensor Visualization......Page 265 GOAL: Describe how to generate Mohr’s circle for a 2x2 matrix that is not necessarily symmetric.......Page 266 Stilted definitions of grad, div, and curl......Page 269 Gradients in curvilinear coordinates......Page 270 When do you NOT have to worry about curvilinear formulas?......Page 272 Spatial gradients of higher-order tensors......Page 274 Product rule for gradient operations......Page 275 Identities involving the “nabla”......Page 277 Compound differential operator notation (and unfortunate pampering)......Page 279 Right and left gradient operations (we love them both!)......Page 280 Casual (non-rigorous) tensor calculus......Page 283 SIDEBAR: “total” and “partial” derivative notation......Page 284 The “nabla” or “del” gradient operator......Page 287 Okay, if the above relation does not hold, does anything LIKE IT hold?......Page 289 Directed derivative......Page 291 EXAMPLE......Page 292 Derivatives in reduced dimension spaces......Page 293 A more physically significant example......Page 297 Series expansion of a nonlinear vector function......Page 298 Exact differentials of one variable......Page 300 Exact differentials of two variables......Page 301 Exact differentials in three dimensions......Page 302 Coupled inexact differentials......Page 303 Integration by parts......Page 304 Leibniz theorem......Page 306 LONG EXAMPLE: conservation of mass......Page 309 Generalized integral formulas for discontinuous integrands......Page 313 23. Closing remarks......Page 314 24. Solved problems......Page 315 REFERENCES......Page 317 INDEX This index is a work in progress. Please notify the author of any critical omissions or errors.......Page 319