Mathematical Preliminaries Introductory Course on Multiphysics - PowerPoint PPT Presentation

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Mathematical Preliminaries Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI ´ NSKI bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Outline 1 Vectors, tensors, and index notation Generalization of the concept of vector Summation convention and index notation Kronecker delta and permutation symbol Tensors and their representations Multiplication of vectors and tensors Vertical-bar convention and Nabla operator

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Outline 1 Vectors, tensors, and index notation Generalization of the concept of vector Summation convention and index notation Kronecker delta and permutation symbol Tensors and their representations Multiplication of vectors and tensors Vertical-bar convention and Nabla operator Integral theorems 2 General idea Stokes’ theorem Gauss-Ostrogradsky theorem

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Outline 1 Vectors, tensors, and index notation Generalization of the concept of vector Summation convention and index notation Kronecker delta and permutation symbol Tensors and their representations Multiplication of vectors and tensors Vertical-bar convention and Nabla operator Integral theorems 2 General idea Stokes’ theorem Gauss-Ostrogradsky theorem 3 Time-harmonic approach Types of dynamic problems Complex-valued notation A practical example

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Outline 1 Vectors, tensors, and index notation Generalization of the concept of vector Summation convention and index notation Kronecker delta and permutation symbol Tensors and their representations Multiplication of vectors and tensors Vertical-bar convention and Nabla operator Integral theorems 2 General idea Stokes’ theorem Gauss-Ostrogradsky theorem 3 Time-harmonic approach Types of dynamic problems Complex-valued notation A practical example

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Generalization of the concept of vector A vector is a quantity that possesses both a magnitude and a direction and obeys certain laws (of vector algebra ): • the vector addition and the commutative and associative laws, • the associative and distributive laws for the multiplication with scalars. The vectors are suited to describe physical phenomena , since they are independent of any system of reference . The concept of a vector that is independent of any coordinate system can be generalised to higher-order quantities, which are called tensors . Consequently, vectors and scalars can be treated as lower-rank tensors.

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Generalization of the concept of vector The vectors are suited to describe physical phenomena , since they are independent of any system of reference . The concept of a vector that is independent of any coordinate system can be generalised to higher-order quantities, which are called tensors . Consequently, vectors and scalars can be treated as lower-rank tensors. Scalars have a magnitude but no direction. They are tensors of order 0. Example: the mass density. Vectors are characterised by their magnitude and direction. They are tensors of order 1. Example: the velocity vector. Tensors of second order are quantities which multiplied by a vector give as the result another vector. Example: the stress tensor. Higher-order tensors are often encountered in constitutive relations between second-order tensor quantities. Example: the fourth-order elasticity tensor.

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Summation convention and index notation Einstein’s summation convention A summation is carried out over repeated indices in an expression and the summation symbol is skipped. Example 3 � a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 a i b i ≡ i = 1 3 � A ii = A 11 + A 22 + A 33 A ii ≡ i = 1 3 � A ij b j ≡ A ij b j = A i 1 b 1 + A i 2 b 2 + A i 3 b 3 ( i = 1 , 2 , 3 ) [3 expressions] j = 1 3 3 � � T ij S ij = T 11 S 11 + T 12 S 12 + T 13 S 13 T ij S ij ≡ i = 1 j = 1 + T 21 S 21 + T 22 S 22 + T 23 S 23 + T 31 S 31 + T 32 S 32 + T 33 S 33

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Summation convention and index notation Einstein’s summation convention A summation is carried out over repeated indices in an expression and the summation symbol is skipped. The principles of index notation: An index cannot appear more than twice in one term! � � � If necessary, the standard summation symbol must be used. A repeated index is called a bound or dummy index . Example A ii , C ijkl S kl , A ij b i c j ← Correct ← Wrong! A ij b j c j � ← Correct A ij b j c j j A term with an index repeated more than two times is correct if: � the summation sign is used: a i b i c i = a 1 b 1 c 1 + a 2 b 2 c 2 + a 3 b 3 c 3 , or i the dummy index is underlined: a i b i c i = a 1 b 1 c 1 or a 2 b 2 c 2 or a 3 b 3 c 3 .

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Summation convention and index notation Einstein’s summation convention A summation is carried out over repeated indices in an expression and the summation symbol is skipped. The principles of index notation: An index cannot appear more than twice in one term! � � � If necessary, the standard summation symbol must be used. A repeated index is called a bound or dummy index . If an index appears once, it is called a free index . The number of free indices determines the order of a tensor. Example A ii , a i b i , T ij S ij ← scalars (no free indices) ← A ij b j a vector (one free index: i ) ← C ijkl S kl a second-order tensor (two free indices: i , j )

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Summation convention and index notation Einstein’s summation convention A summation is carried out over repeated indices in an expression and the summation symbol is skipped. The principles of index notation: An index cannot appear more than twice in one term! � � � If necessary, the standard summation symbol must be used. A repeated index is called a bound or dummy index . If an index appears once, it is called a free index . The number of free indices determines the order of a tensor. The denomination of dummy index (in a term) is arbitrary, since it vanishes after summation, namely: a i b i ≡ a j b j ≡ a k b k , etc. Example a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 = a j b j A ii ≡ A jj , T ij S ij ≡ T kl S kl , T ij + C ijkl S kl ≡ T ij + C ijmn S mn

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Kronecker delta and permutation symbol Definition (Kronecker delta) � for i = j 1 δ ij = 0 for i � = j

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Kronecker delta and permutation symbol Definition (Kronecker delta) � for i = j 1 δ ij = for i � = j 0 The Kronecker delta can be used to substitute one index by another, for example: a i δ ij = a 1 δ 1 j + a 2 δ 2 j + a 3 δ 3 j = a j , i.e., here i → j . When Cartesian coordinates are used (with orthonormal base vectors e 1 , e 2 , e 3 ) the Kronecker delta δ ij is the (matrix) representation of the unity tensor I = e 1 ⊗ e 1 + e 2 ⊗ e 2 + e 3 ⊗ e 3 = δ ij e i ⊗ e j . A • I = A ij δ ij = A ii which is the trace of the matrix (tensor) A .

Vectors, tensors, and index notation Integral theorems Time-harmonic approach Vectors, tensors, and index notation Kronecker delta and permutation symbol Definition (Kronecker delta) � for i = j 1 δ ij = for i � = j 0 The Kronecker delta can be used to substitute one index by another, for example: a i δ ij = a 1 δ 1 j + a 2 δ 2 j + a 3 δ 3 j = a j , i.e., here i → j . When Cartesian coordinates are used (with orthonormal base vectors e 1 , e 2 , e 3 ) the Kronecker delta δ ij is the (matrix) representation of the unity tensor I = e 1 ⊗ e 1 + e 2 ⊗ e 2 + e 3 ⊗ e 3 = δ ij e i ⊗ e j . A • I = A ij δ ij = A ii which is the trace of the matrix (tensor) A . Definition (Permutation symbol)   1 for even permutations: 123, 231, 312   ǫ ijk = − 1 for odd permutations: 132, 321, 213   0 if an index is repeated 