Outline Outline 4 Basic Rules 4 Basic Rules 4 Vectors and Tensors 4 Vectors and Tensors 4 Tensor Operation 4 Tensor Operation 4 Isotropic Tensors 4 Isotropic Tensors G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II (Cartesian Tensor) (Cartesian Tensor) x = * Q x y* y Basic Rules Basic Rules i ij j Change Change x* h A free index appears only once in each h A free index appears only once in each of Frame of Frame x = * Q x θ term of a tensor equation. The equation term of a tensor equation. The equation x j ij i then holds for all possible values of that then holds for all possible values of that index. index. = ± det Q 1 ij h Summation is implied on an index, which h Summation is implied on an index, which appears twice. appears twice. = δ = δ Q ij Q Q ij Q h No index can appear more than twice in h No index can appear more than twice in ik jk kj ik any term. any term. G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II 1
= + = + * * * * r x i y j x i y j Transformation in Transformation in θ θ ⎡ ⎤ cos sin [ ] = Q Two Dimension Two Dimension ⎢ ⎥ = θ + θ * i i cos j sin − θ θ sin cos ⎣ ⎦ y* y r * = − θ + θ j i sin j cos x* ⎡ ⎤ [ ] 1 0 δ = Kronecker Kronecker = θ + θ x * ⎢ ⎥ x cos y sin j θ ij Delta Delta 0 1 j* i* ⎣ ⎦ x i = − θ + θ y * x sin y cos G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II T * = v = * Q v T Vector Vector Scalar Scalar i ij j Second Second t = * Q Q t = ⋅ * v Q v Order Tensor Order Tensor ij ik jl kl Vector Vector Vector Third Order Second Order Second Order Third Order λ = λ = ⋅ ⋅ * * T Q Q Q τ Q t Q Tensor ijk im jn kl mnl Tensor Tensor Tensor G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II 2
∂ ϕ ε Alternating ∇ ϕ = = ϕ Alternating ( ) i , i ∂ x ijk Symbol Gradient Symbol Gradient i ∂ v ∇ = j = ( v ) v ij j , i ∂ x ε = i 1 , for i , j , k even permutatio n ijk Divergence Divergence ∇ v ⋅ = v ε = − 1 , for i , j , k odd permutatio n i , i ijk ε = 0 , when two indices are equal ∂ τ ijk ∇ τ ⋅ = ij = τ ( ) j ij , i ∂ x i G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II ∂ U All Scalars Rank Zero: Rank Zero: All Scalars All Scalars ∇ × = ε = ε ( U ) k U Curl Curl i ijk ∂ ijk k , j x j None Rank One: Rank One: None None = ε det A A A A Determinant Determinant ijk 1 i 2 j 3 k αδ Rank Two: Rank Two: ij ε ε = δ δ − δ δ Identity Identity ijk imn jm kn jn km αε ∂ ϕ 2 ∇ ϕ = = ϕ 2 Rank Three: Laplacian Laplacian Rank Three: ijk ∂ ∂ , ii x x i i G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II 3
● Basic Rules Basic Rules ● Basic Rules ● Rank Four: Rank Four: ● Vectors and Tensors Vectors and Tensors ● Vectors and Tensors ● αδ δ + β δ δ + δ δ ( ) ● Tensor Operation Tensor Operation ● Tensor Operation ● ij kl ik jl il jk + γ δ δ − δ δ ( ) ● Isotropic Tensors Isotropic Tensors ● Isotropic Tensors ● ik jl il jk G. Ahmadi G. Ahmadi ME 637-Particle-II ME 637-Particle-II G. Ahmadi ME 637-Particle-II 4
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