EN0175-05

EN0175-05 - EN0175 09 19 06 Today we introduce the...

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Unformatted text preview: EN0175 09 / 19 / 06 Today we introduce the application of index notations in vector and tensor algebra. Index notation : e.g. Kronecker delta: . In matrix form: ⎩ ⎨ ⎧ ≠ = = j i j i ij , , 1 δ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 1 1 1 Dot product of vectors: ( ) ( ) ( ) i i ij j i j i j i j j i i b a b a e e b a e b e a b a = = ⋅ = ⋅ = ⋅ δ v v v v v v Cross product of vectors: , b a c v v v × = The magnitude of cross product is: θ sin ab c = v The direction of cross product is: a v b v c v θ For base vectors 1 e v , 2 e v , : 3 e v 1 1 = × e e v v , 3 2 1 e e e v v v = × , 2 3 1 e e e v v v − = × 3 1 2 e e e v v v − = × , 2 2 = × e e v v , 1 3 2 e e e v v v = × 2 1 3 e e e v v v = × , 1 2 3 e e e v v v − = × , 3 3 = × e e v v Or we can write the nine equations in a concise form as: k ijk j i e e e v v v ε = × where (called permutation symbol) ⎪ ⎩ ⎪ ⎨ ⎧ = − = = otherwise , 321 , 132 , 213 ijk , 1 312 , 231 , 123 ijk , 1 ijk ε 1 EN0175 09 / 19 / 06 1 2 3 Vector operations are useful in representing areas and volumes in a solid body....
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EN0175-05 - EN0175 09 19 06 Today we introduce the...

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