index notation and summation

index notation and summation - Chapter 4 Index Notation and...

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Unformatted text preview: Chapter 4 Index Notation and the Summation Convention Syllabus covered: 3. Index notation and the Summation Convention; summation over repeated indices; Kronecker delta and i jk ; formula for i jk klm . We now introduce a very useful notation. In particular it makes proving identities such as those in Chapter 2 much simpler. There are many other uses: an extended version of it is used in Relativity, and it can be widely used in linear algebra and its applications (e.g. input-output models in economics). Index notation In this notation we abbreviate a vector a = ( a 1 , a 2 , a 3 ) a 1 i + a 2 j + a 3 k to a i . The special vector r will be written as x i , so x 1 = x , x 2 = y , x 3 = z in the usual notation. The name of the index (here, i ) is irrelevant: a i and a j mean exactly the same thing. However, it becomes relevant when we write an equation such as a i = b i (which implies 3 equations in fact). This means a 1 = b 1 , a 2 = b 2 and a 3 = b 3 . The same equations could be written a j = b j . But they cannot be written a i = b j , because we would have no idea whether, for example, a 1 was equal to b 1 , b 2 or b 3 . Really, the notation means the i th component of a , or the i-th entry in a . The same idea works equally well in any number of dimensions, or even for infinite sequences ( a 1 , a 2 , . . . , a n , . . . ) . However, we shall assume that indices always run from 1 to 3. We can have objects with more than one index 1 . For example, we can use C i j as a notation for a 3 3 (or more generally n n ) matrix C . The equation a = Cb could then be written a i = 3 j = 1 C i j b j . If we want to mix the vector and index ways of writing things we must write ( a ) i = a i and so on. Einstein summation convention This further compression of notation allows us to drop summation signs. It is that if an index is repeated, 1 In general these will be objects called Cartesian tensors (since we stick to Cartesian coordinates here) or more generally just tensors. In this course we avoid a proper definition of tensors and discussions of their general properties. 43 the repetition signals that one should sum over its allowed values (1 to 3 in our case). Thus a i b i 3 i = 1 a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 = a . b . (4.1) Again, the name of the repeated index is irrelevant: a j b j means exactly the same thing. Using this convention, we can write the matrix product above, Cb , more briefly still as C i j b j ....
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index notation and summation - Chapter 4 Index Notation and...

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