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index notation and summation

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

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Chapter 4 Index Notation and the Summation Convention Syllabus covered: 3. Index notation and the Summation Convention; summation over repeated indices; Kronecker delta and ε ijk ; formula for ε ijk ε 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 ij 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 ij 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
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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 ij b j . Indices that appear twice and are summed over are called dummy indices. Indices that appear once (more exactly, once in every term in an equation or expression) are called free. Free indices tell us how many equations have been compressed into one. For example, a i b j c k = d ijk stands for 27 equations such as a 2 b 1 c 3 = d 213 .
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