Transcription of Chapter 3 Cartesian Tensors - DAMTP
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Chapter 3 Cartesian Tensors - DAMTP
Chapter 3 Cartesian Suffix Notation and the Summation ConventionWe will consider vectors in 3D, though the notation we shall introduce applies (mostly)just as well tondimensions. For a general vectorx= (x1, x2, x3)we shall refer toxi, theithcomponent ofx. The indeximay take any of the values1, 2 or 3, and we refer to the vectorxi to mean the vector whose components are(x1, x2, x3) . However, we cannot writex=xi, since the LHS is a vector and the RHS ascalar. Instead, we can write [x]i=xi, and similarly [x+y]i=xi+ that the expressionyi=xiimplies thaty=x; the statement in suffix notationis implicitly true for all three possible values ofi(one at a time!).Einstein introduced a convention whereby if a particular suffix ( ,i) appears twicein a single term of an expression then it is implicitly summed. For example, in traditionalnotationx . y=x1y1+x2y2+x3y3=3 i=1xiyi;using summation convention we simply writex.
Cartesian Tensors 3.1 Suffix Notation and the Summation Convention We will consider vectors in 3D, though the notation we shall introduce applies (mostly) just as well to n dimensions. For a general vector x = (x 1,x 2,x 3) we shall refer to x i, the ith component of x. The index i may take any of the values 1, 2 or 3, and we refer to “the ...
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