L03_Eigenvalues___Eigenvectors

L03_Eigenvalues___Eigenvectors - CE507 Lecture 3 More Index...

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CE507 Lecture 3 More Index Notations ; Eigenvalues & Eigenvectors I. More Index Notation Index Notation for Differentiation Given: ) , , ( ) ~ ( 3 2 1 x x x f x f = is a scalar field (scalar function) of space. Definition: 1 1 , f f x = Thus = = = k j i ijk j i ij i i x x x f f x x f f x f f 3 2 , , , Therefore 222 2 11 22 33 123 ,,,, , ii fff f f xxx ∂∂∂ =++= + + = the Laplacian of f Vector Field: i i e x f x f x f x f x f ˆ ) ~ ( )) ~ ( ), ~ ( ), ~ ( ( ) ~ ( ~ 3 2 1 = = Gradient of a Scalar Field: i i e f f f f x f x f x f f grad f ˆ , ) , , , , , ( ) , , ( . ~ 3 2 1 3 2 1 = = = = Note: k k f e f , ˆ . ~ = Del Operator:
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123 ˆ (, ,) , kk e xx x ∂∂ ∇= = ± Divergence of a Vector Field: k k f x f x f x f f f f x x x f , ) , , , , , ( * ) , , ( ~ ~ 3 3 2 2 1 1 3 2 1 3 2 1 = + + = = Curl of a Vector Field: 3 2 1 1 2 2 1 3 3 1 1 3 2 2 3 3 2 1 3 2 1 3 2 1 ˆ ˆ ˆ ˆ ˆ ˆ ~ ~ ~ e x f x f e x f x f e x f x f f f f x x x e e e f Curl f + + = = = × , ? ( ???) jj i ijk k ijk k i fe f e x εε == Divergence Theorem: Given a closed surface S enclosing a volume “V”. At each infinitesimal area “ds” on the surface, let “ n ˆ ” be its outward normal, then for a given vector field,” f ~ ” : ˆ VS f dV f n dS ∇⋅ = ∫∫∫ ∫∫ ±± ± w x f ~ is assumed to be continuous and differentiable. Physically, f ~ can be considered as a current density (volume of flow / time) of a hypothetical fluid. L.H.S is then the net Rate at which the “fluid” passes out of the volume V.The R.H.S is the rate at which
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L03_Eigenvalues___Eigenvectors - CE507 Lecture 3 More Index...

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