TENSOR ALGEBRA AND TENSOR ANALYSIS

By MICHAEL HARRIS

CONTENTS VECTORS AND TENSORS IN A FINITE-DIMENSIONAL SPACE NOTION OF THE VECTOR SPACE BASIS AND DIMENSION OF THE VECTOR SPACE COMPONENTS OF A VECTOR, SUMMATION CONVENTION SCALAR PRODUCT, EUCLIDEAN SPACE, ORTHONORMAL BASIS DUAL BASES SECOND-ORDER TENSOR AS A LINEAR MAPPING TENSOR PRODUCT, REPRESENTATION OF A TENSOR WITH RESPECT TO A BASIS CHANGE OF THE BASIS, TRANSFORMATION RULES SPECIAL OPERATIONS WITH SECOND-ORDER TENSORS 1 10 SCALAR PRODUCT OF SECOND-ORDER TENSORS 1 11 DECOMPOSITIONS OF SECOND-ORDER TENSORS 1 12 TENSORS OF HIGHER ORDERS EXERCISES VECTOR AND TENSOR ANALYSIS IN EUCLIDEAN SPACE VECTOR- AND TENSOR-VALUED FUNCTIONS, DI?ERENTIAL CALCULUS COORDINATES IN EUCLIDEAN SPACE, TANGENT VECTORS COORDINATE TRANSFORMATION CO-, CONTRA- AND MIXED VARIANT COMPONENTS GRADIENT, COVARIANT AND CONTRAVARIANT DERIVATIVES CHRISTO?EL SYMBOLS, REPRESENTATION OF THE COVARIANT DERIVATIVE APPLICATIONS IN THREE-DIMENSIONAL SPACE: DIVERGENCE AND CURL CURVES AND SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE CURVES IN THREE-DIMENSIONAL EUCLIDEAN SPACE SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE APPLICATION TO SHELL THEORY EIGENVALUE PROBLEM AND SPECTRAL DECOMPOSITION OF SECOND-ORDER TENSORS COMPLEXI?CATION EIGENVALUE PROBLEM, EIGENVALUES AND EIGENVECTORS CHARACTERISTIC POLYNOMIAL SPECTRAL DECOMPOSITION AND EIGENPROJECTIONS SPECTRAL DECOMPOSITION OF SYMMETRIC SECOND-ORDER TENSORS SPECTRAL DECOMPOSITION OF ORTHOGONAL AND SKEW-SYMMETRIC SECOND-ORDER TENSORS CAYLEY-HAMILTON THEOREM FOURTH-ORDER TENSORS FOURTH-ORDER TENSORS AS A LINEAR MAPPING TENSOR PRODUCTS, REPRESENTATION OF FOURTH-ORDER TENSORS WITH RESPECT TO A BASIS SPECIAL OPERATIONS WITH FOURTH-ORDER TENSORS SUPER-SYMMETRIC FOURTH-ORDER TENSORS SPECIAL FOURTH-ORDER TENSORS ANALYSIS OF TENSOR FUNCTIONS SCALAR-VALUED ISOTROPIC TENSOR FUNCTIONS SCALAR-VALUED ANISOTROPIC TENSOR FUNCTIONS DERIVATIVES OF SCALAR-VALUED TENSOR FUNCTIONS TENSOR-VALUED ISOTROPIC AND ANISOTROPIC TENSOR FUNCTIONS DERIVATIVES OF TENSOR-VALUED TENSOR FUNCTIONS GENERALIZED RIVLIN’S IDENTITIES ANALYTIC TENSOR FUNCTIONS INTRODUCTION CLOSED-FORM REPRESENTATION FOR ANALYTIC TENSOR FUNCTIONS AND THEIR DERIVATIVES SPECIAL CASE: DIAGONALIZABLE TENSOR FUNCTIONS SPECIAL CASE: THREE-DIMENSIONAL SPACE RECURRENT CALCULATION OF TENSOR POWER SERIES AND THEIR DERIVATIVES APPLICATIONS TO CONTINUUM MECHANICS POLAR DECOMPOSITION OF THE DEFORMATION GRADIENT THE DERIVATIVE OF THE STRETCH AND ROTATION TENSOR WITH RESPECT TO THE DEFORMATION GRADIENT TIME RATE OF GENERALIZED STRAINS STRESS CONJUGATE TO A GENERALIZED STRAIN FINITE PLASTICITY BASED ON THE ADDITIVE DECOMPOSITION OF GENERALIZED STRAINS SOLUTIONS REFERENCES