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unit2_math_aside - 16.21 Techniques of Structural Analysis...

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16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #2 - Mathematical aside: Vectors, indicial notation and summation convention Ra´ul Radovitzky February 6, 2005 Indicial notation In 16.21 we’ll work in a an euclidean three-dimensional space R 3 . Free index: A subscript index () i will be denoted a free index if it is not repeated in the same additive term where the index appears. Free means that the index represents all the values in its range. Latin indices will range from 1 to, ( i, j, k, ... = 1 , 2 , 3), greek indices will range from 1 to 2, ( α, β, γ, ... = 1 , 2). Examples: 1. a i 1 implies a 11 , a 21 , a 31 . (one free index) 2. x α y β implies x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 (two free indices). 3. a ij implies a 11 , a 12 , a 13 , a 21 , a 22 , a 23 , a 31 , a 32 , a 33 (two free indices implies 9 values). 4. ∂σ ij + b i = 0 ∂x j 1
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has a free index ( i ), therefore it represents three equations: ∂σ 1 j + b 1 = 0 ∂x j ∂σ 2 j + b 2 = 0 ∂x j ∂σ 3 j + b 3 = 0 ∂x j Summation convention: When a repeated index is found in an expression (inside an additive
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