TENSOR ANALYSIS FOR ENGINEERS

By P. HILID

CONTENTS INTRODUCTION STRUCTURES AND SUPERSTRUCTURES WHAT IS A SCALAR? WHAT IS A VECTOR? WHAT IS A TENSOR? EXAMPLES OF TENSORS IN MATERIALS MECHANICS THE STRESS TENSOR THE DEFORMATION GRADIENT TENSOR VECTOR AND TENSOR NOTATION — PHILOSOPHY TERMINOLOGY FROM FUNCTIONAL ANALYSIS MATRIX ANALYSIS (AND SOME MATRIX CALCULUS) DEFINITION OF A MATRIX COMPONENT MATRICES ASSOCIATED WITH VECTORS AND TENSORS (NOTATION EXPLANATION) THE MATRIX PRODUCT SPECIAL CASE: A MATRIX TIMES AN ARRAY SPECIAL CASE: INNER PRODUCT OF TWO ARRAYS SPECIAL CASE: OUTER PRODUCT OF TWO ARRAYS EXAMPLE THE KRONECKER DELTA THE IDENTITY MATRIX DERIVATIVES OF VECTOR AND MATRIX EXPRESSIONS DERIVATIVE OF AN ARRAY WITH RESPECT TO ITSELF DERIVATIVE OF A MATRIX WITH RESPECT TO ITSELF THE TRANSPOSE OF A MATRIX DERIVATIVE OF THE TRANSPOSE THE INNER PRODUCT OF TWO COLUMN MATRICES DERIVATIVES OF THE INNER PRODUCT THE OUTER PRODUCT OF TWO COLUMN MATRICES THE TRACE OF A SQUARE MATRIX DERIVATIVE OF THE TRACE THE MATRIX INNER PRODUCT DERIVATIVE OF THE MATRIX INNER PRODUCT MAGNITUDES AND POSITIVITY PROPERTY OF THE INNER PRODUCT DERIVATIVE OF THE MAGNITUDE NORMS WEIGHTED OR “ENERGY” NORMS DERIVATIVE OF THE ENERGY NORM THE 3D PERMUTATION SYMBOL THE ?-? (E-DELTA) IDENTITY THE ?-? (E-DELTA) IDENTITY WITH MULTIPLE SUMMED INDICES DETERMINANT OF A SQUARE MATRIX MORE ABOUT COFACTORS COFACTOR-INVERSE RELATIONSHIP DERIVATIVE OF THE COFACTOR DERIVATIVE OF A DETERMINANT (IMPORTANT) RATES OF DETERMINANTS DERIVATIVES OF DETERMINANTS WITH RESPECT TO VECTORS PRINCIPAL SUB-MATRICES AND PRINCIPAL MINORS MATRIX INVARIANTS ALTERNATIVE INVARIANT SETS POSITIVE DEFINITE THE COFACTOR-DETERMINANT CONNECTION INVERSE EIGENVALUES AND EIGENVECTORS SIMILARITY TRANSFORMATIONS FINDING EIGENVECTORS BY USING THE ADJUGATE EIGENPROJECTORS FINDING EIGENPROJECTORS WITHOUT FINDING EIGENVECTORS VECTOR/TENSOR NOTATION “ORDINARY” ENGINEERING VECTORS ENGINEERING “LABORATORY” BASE VECTORS OTHER CHOICES FOR THE BASE VECTORS BASIS EXPANSION OF A VECTOR SUMMATION CONVENTION — DETAILS DON’T FORGET WHAT REPEATED INDICES REALLY MEAN FURTHER SPECIAL-SITUATION SUMMATION RULES INDICIAL NOTATION IN DERIVATIVES BEWARE: AVOID IMPLICIT SUMS AS INDEPENDENT VARIABLES SUSPENDING THE SUMMATION CONVENTION COMBINING INDICIAL EQUATIONS SUMMING THE KRONECKER DELTA ITSELF OUR (UNCONVENTIONAL) “UNDER-TILDE” NOTATION TENSOR INVARIANT OPERATIONS SIMPLE VECTOR OPERATIONS AND PROPERTIES DOT PRODUCT BETWEEN TWO VECTORS DOT PRODUCT BETWEEN ORTHONORMAL BASE VECTORS A “QUOTIENT” RULE (DECIDING IF A VECTOR IS ZERO) DECIDING IF ONE VECTOR EQUALS ANOTHER VECTOR FINDING THE I-TH COMPONENT OF A VECTOR EVEN AND ODD VECTOR FUNCTIONS HOMOGENEOUS FUNCTIONS VECTOR ORIENTATION AND SENSE SIMPLE SCALAR COMPONENTS CROSS PRODUCT CROSS PRODUCT BETWEEN ORTHONORMAL BASE VECTORS TRIPLE SCALAR PRODUCT TRIPLE SCALAR PRODUCT BETWEEN ORTHONORMAL RIGHT-HANDED BASE VECTORS PROJECTIONS ORTHOGONAL (PERPENDICULAR) LINEAR PROJECTIONS RANK-1 ORTHOGONAL PROJECTIONS RANK-2 ORTHOGONAL PROJECTIONS BASIS INTERPRETATION OF ORTHOGONAL PROJECTIONS RANK-2 OBLIQUE LINEAR PROJECTION RANK-1 OBLIQUE LINEAR PROJECTION DEGENERATE (TRIVIAL) RANK-0 LINEAR PROJECTION DEGENERATE (TRIVIAL) RANK-3 PROJECTION IN 3D SPACE COMPLEMENTARY PROJECTORS NORMALIZED VERSIONS OF THE PROJECTORS EXPRESSING A VECTOR AS A LINEAR COMBINATION OF THREE ARBITRARY (NOT NECESSARILY ORTHONORMAL) VECTORS GENERALIZED PROJECTIONS LINEAR PROJECTIONS NONLINEAR PROJECTIONS THE VECTOR “SIGNUM” FUNCTION GRAVITATIONAL (DISTORTED LIGHT RAY) PROJECTIONS SELF-ADJOINT PROJECTIONS GRAM-SCHMIDT ORTHOGONALIZATION SPECIAL CASE: ORTHOGONALIZATION OF TWO VECTORS THE PROJECTION THEOREM TENSORS ANALOGY BETWEEN TENSORS AND OTHER (MORE FAMILIAR) CONCEPTS LINEAR OPERATORS (TRANSFORMATIONS) DYADS AND DYADIC MULTIPLICATION SIMPLER “NO-SYMBOL” DYADIC NOTATION THE MATRIX ASSOCIATED WITH A DYAD THE SUM OF DYADS A SUM OF TWO OR THREE DYADS IS NOT (GENERALLY) REDUCIBLE SCALAR MULTIPLICATION OF A DYAD THE SUM OF FOUR OR MORE DYADS IS REDUCIBLE! (NOT A SUPERSET) THE DYAD DEFINITION OF A SECOND-ORDER TENSOR EXPANSION OF A SECOND-ORDER TENSOR IN TERMS OF BASIS DYADS TRIADS AND HIGHER-ORDER TENSORS OUR VM N TENSOR “CLASS” NOTATION COMMENT TENSOR OPERATIONS DOTTING A TENSOR FROM THE RIGHT BY A VECTOR THE TRANSPOSE OF A TENSOR DOTTING A TENSOR FROM THE LEFT BY A VECTOR DOTTING A TENSOR BY VECTORS FROM BOTH SIDES EXTRACTING A PARTICULAR TENSOR COMPONENT DOTTING A TENSOR INTO A TENSOR (TENSOR COMPOSITION) TENSOR ANALYSIS PRIMITIVES THREE KINDS OF VECTOR AND TENSOR NOTATION REPRESENTATION THEOREM FOR LINEAR FORMS REPRESENTATION THEOREM FOR VECTOR-TO-SCALAR LINEAR FUNCTIONS ADVANCED REPRESENTATION THEOREM (TO BE READ ONCE YOU LEARN ABOUT HIGHER-ORDER TENSORS AND THE VM N CLASS NOTATION) FINDING THE TENSOR ASSOCIATED WITH A LINEAR FUNCTION METHOD #1 METHOD #2 METHOD #3 THE IDENTITY TENSOR TENSOR ASSOCIATED WITH COMPOSITION OF TWO LINEAR TRANSFORMATIONS THE POWER OF HEURISTICALLY CONSISTENT NOTATION THE INVERSE OF A TENSOR THE COFACTOR TENSOR AXIAL TENSORS (TENSOR ASSOCIATED WITH A CROSS-PRODUCT) GLIDE PLANE EXPRESSIONS AXIAL VECTORS COFACTOR TENSOR ASSOCIATED WITH A VECTOR CRAMER’S RULE FOR THE INVERSE INVERSE OF A RANK-1 MODIFICATION (SHERMAN-MORRISON FORMULA) DERIVATIVE OF A DETERMINANT EXPLOITING OPERATOR INVARIANCE WITH “PREFERRED” BASES PROJECTORS IN TENSOR NOTATION NONLINEAR PROJECTIONS DO NOT HAVE A TENSOR REPRESENTATION LINEAR ORTHOGONAL PROJECTORS EXPRESSED IN TERMS OF DYADS JUST ONE ESOTERIC APPLICATION OF PROJECTORS IMPORTANT: FINDING A PROJECTION TO A DESIRED TARGET SPACE PROPERTIES OF COMPLEMENTARY PROJECTION TENSORS SELF-ADJOINT (ORTHOGONAL) PROJECTORS NON-SELF-ADJOINT (OBLIQUE) PROJECTORS GENERALIZED COMPLEMENTARY PROJECTORS MORE TENSOR PRIMITIVES TENSOR PROPERTIES ORTHOGONAL (UNITARY) TENSORS TENSOR ASSOCIATED WITH THE CROSS PRODUCT CROSS-PRODUCTS IN LEFT-HANDED AND GENERAL BASES PHYSICAL APPLICATION OF AXIAL VECTORS SYMMETRIC AND SKEW-SYMMETRIC TENSORS POSITIVE DEFINITE TENSORS FASTER WAY TO CHECK FOR POSITIVE DEFINITENESS POSITIVE SEMI-DEFINITE NEGATIVE DEFINITE AND NEGATIVE SEMI-DEFINITE TENSORS ISOTROPIC AND DEVIATORIC TENSORS SECOND-ORDER TENSOR INNER PRODUCT A NON-RECOMMENDED SCALAR-VALUED PRODUCT FOURTH-ORDER TENSOR INNER PRODUCT FOURTH-ORDER SHERMAN-MORRISON FORMULA HIGHER-ORDER TENSOR INNER PRODUCT SELF-DEFINING NOTATION THE MAGNITUDE OF A TENSOR OR A VECTOR USEFUL INNER PRODUCT IDENTITIES DISTINCTION BETWEEN AN NTH-ORDER TENSOR AND AN NTH-RANK TENSOR FOURTH-ORDER OBLIQUE TENSOR PROJECTIONS LEAFING AND PALMING OPERATIONS SYMMETRIC LEAFING COORDINATE/BASIS TRANSFORMATIONS CHANGE OF BASIS (AND COORDINATE TRANSFORMATIONS) DEFINITION OF A VECTOR AND A TENSOR BASIS COUPLING TENSOR TENSOR (AND TENSOR FUNCTION) INVARIANCE WHAT’S THE DIFFERENCE BETWEEN A MATRIX AND A TENSOR? EXAMPLE OF A “SCALAR RULE” THAT SATISFIES TENSOR INVARIANCE EXAMPLE OF A “SCALAR RULE” THAT VIOLATES TENSOR INVARIANCE EXAMPLE OF A 3X3 MATRIX THAT DOES NOT CORRESPOND TO A TENSOR THE INERTIA TENSOR SCALAR INVARIANTS AND SPECTRAL ANALYSIS INVARIANTS OF VECTORS OR TENSORS PRIMITIVE INVARIANTS TRACE INVARIANTS CHARACTERISTIC INVARIANTS DIRECT NOTATION DEFINITIONS OF THE CHARACTERISTIC INVARIANTS THE COFACTOR IN THE TRIPLE SCALAR PRODUCT INVARIANTS OF A SUM OF TWO TENSORS CASE: INVARIANTS OF THE SUM OF A TENSOR PLUS A DYAD THE CAYLEY-HAMILTON THEOREM CASE: EXPRESSING THE INVERSE IN TERMS OF POWERS AND INVARIANTS CASE: EXPRESSING THE COFACTOR IN TERMS OF POWERS AND INVARIANTS EIGENVALUE PROBLEMS ALGEBRAIC AND GEOMETRIC MULTIPLICITY OF EIGENVALUES DIAGONALIZABLE TENSORS (THE SPECTRAL THEOREM) GEOMETRICAL ENTITIES EQUATION OF A PLANE EQUATION OF A LINE EQUATION OF A SPHERE EQUATION OF AN ELLIPSOID EXAMPLE EQUATION OF A CYLINDER WITH AN ELLIPSE-CROSS-SECTION EQUATION OF A RIGHT CIRCULAR CYLINDER EQUATION OF A GENERAL QUADRIC (INCLUDING HYPERBOLOID) GENERALIZATION OF THE QUADRATIC FORMULA AND “COMPLETING THE SQUARE” POLAR DECOMPOSITION SINGULAR VALUE DECOMPOSITION SPECIAL CASE THE POLAR DECOMPOSITION THEOREM POLAR DECOMPOSITION IS A NONLINEAR PROJECTION THE *FAST* WAY TO DO A POLAR DECOMPOSITION IN 2D A FAST AND ACCURATE NUMERICAL 3D POLAR DECOMPOSITION DILATION-DISTORTION (VOLUMETRIC-ISOCHORIC) DECOMPOSITION THERMOMECHANICS APPLICATION MATERIAL SYMMETRY WHAT IS ISOTROPY? IMPORTANT CONSEQUENCE ISOTROPIC SECOND-ORDER TENSORS IN 3D SPACE ISOTROPIC SECOND-ORDER TENSORS IN 2D SPACE ISOTROPIC FOURTH-ORDER TENSORS FINDING THE ISOTROPIC PART OF A FOURTH-ORDER TENSOR A SCALAR MEASURE OF “PERCENT ANISOTROPY” TRANSVERSE ISOTROPY ABSTRACT VECTOR/TENSOR ALGEBRA STRUCTURES DEFINITION OF AN ABSTRACT VECTOR WHAT DOES THIS MATHEMATICIAN’S DEFINITION OF A VECTOR HAVE TO DO WITH THE DEFINITION USED IN APPLIED MECHANICS? INNER PRODUCT SPACES ALTERNATIVE INNER PRODUCT STRUCTURES SOME EXAMPLES OF INNER PRODUCT SPACES CONTINUOUS FUNCTIONS ARE VECTORS! TENSORS ARE VECTORS! VECTOR SUBSPACES EXAMPLE: COMMUTING SPACE SUBSPACES AND THE PROJECTION THEOREM ABSTRACT CONTRACTION AND SWAP (EXCHANGE) OPERATORS THE CONTRACTION TENSOR THE SWAP TENSOR VECTOR AND TENSOR VISUALIZATION MOHR’S CIRCLE FOR 2D TENSORS VECTOR/TENSOR DIFFERENTIAL CALCULUS STILTED DEFINITIONS OF GRAD, DIV, AND CURL GRADIENTS IN CURVILINEAR COORDINATES WHEN DO YOU NOT HAVE TO WORRY ABOUT CURVILINEAR FORMULAS? SPATIAL GRADIENTS OF HIGHER-ORDER TENSORS PRODUCT RULE FOR GRADIENT OPERATIONS IDENTITIES INVOLVING THE “NABLA” COMPOUND DIFFERENTIAL OPERATOR NOTATION (AND UNFORTUNATE PAMPERING) RIGHT AND LEFT GRADIENT OPERATIONS (WE LOVE THEM BOTH!) CASUAL (NON-RIGOROUS) TENSOR CALCULUS SIDEBAR: “TOTAL” AND “PARTIAL” DERIVATIVE NOTATION THE “NABLA” OR “DEL” GRADIENT OPERATOR OKAY, IF THE ABOVE RELATION DOES NOT HOLD, DOES ANYTHING LIKE IT HOLD? DIRECTED DERIVATIVE DERIVATIVES IN REDUCED DIMENSION SPACES A MORE PHYSICALLY SIGNIFICANT EXAMPLE SERIES EXPANSION OF A NONLINEAR VECTOR FUNCTION EXACT DIFFERENTIALS OF ONE VARIABLE EXACT DIFFERENTIALS OF TWO VARIABLES THE SAME RESULT IN A DIFFERENT NOTATION EXACT DIFFERENTIALS IN THREE DIMENSIONS COUPLED INEXACT DIFFERENTIALS VECTOR/TENSOR INTEGRAL CALCULUS GAUSS THEOREMS STOKES THEOREM DIVERGENCE THEOREM INTEGRATION BY PARTS LEIBNIZ THEOREM LONG EXAMPLE: CONSERVATION OF MASS GENERALIZED INTEGRAL FORMULAS FOR DISCONTINUOUS INTEGRANDS CLOSING REMARKS