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Unformatted text preview: Appendix A: Vector calculus We shall revise some vectors operations that you should have already met before this course. These may be presented slightly differently to the way you have previously seen them. A.1 Suffix notation and summation convention Suppose that we have two vectors u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ). Then the dot product is defined to be u v = 3 summationdisplay i =1 u i v i or, more simply, write u v = u i v i (drop the summation symbol on the understanding that repeated suffices imply summa- tion. Defn A.1.2 : The Kronecker delta is defined by ij = braceleftbigg 1 , i = j , i negationslash = j bracerightbigg So in summation convention, ij a j = a i since ij a j 3 summationdisplay j =1 ij a j = a i Examples : 1. ii = 3 2. ij u i v j = u j v j u v . Defn A.1.3 : The antisymmetric symbol ijk is defined by 123 = 1 ijk is zero if there are any repeated suffices. E.g. 113 = 0. Interchanging any two suffices reverses the sign: e.g. ijk = jik = kji Above implies invariant under cyclic rotation of suffices: ijk = jki = kij With this definition all 27 permutations are defined. There are only 6 non-zero compo-With this definition all 27 permutations are defined....
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