appendix_A

appendix_A - Appendix A Vector calculus We shall revise...

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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 δ ij = braceleftbigg 1 , i = j , i negationslash = j bracerightbigg So in summation convention, δ ij a j = a i since δ ij a j ≡ 3 summationdisplay j =1 δ ij a j = a i Examples : 1. δ ii = 3 2. δ ij u i v j = u j v j ≡ u · v . Defn A.1.3 : The antisymmetric symbol ǫ ijk is defined by • ǫ 123 = 1 • ǫ ijk is zero if there are any repeated suffices. E.g. ǫ 113 = 0. • Interchanging any two suffices reverses the sign: e.g. ǫ ijk = − ǫ jik = − ǫ kji • Above implies invariant under cyclic rotation of suffices: ǫ ijk = ǫ jki = ǫ 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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appendix_A - Appendix A Vector calculus We shall revise...

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