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index-notation - Index Notation for Vector Calculus by Ilan...

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Index Notation for Vector Calculus by Ilan Ben-Yaacov and Francesc Roig Copyright c 2006 Index notation, also commonly known as subscript notation or tensor notation, is an extremely useful tool for performing vector algebra. Consider the coordinate system illustrated in Figure 1. Instead of using the typical axis labels x , y , and z , we use x 1 , x 2 , and x 3 , or x i i = 1 , 2 , 3 The corresponding unit basis vectors are then ˆ e 1 , ˆ e 2 , and ˆ e 3 , or ˆ e i i = 1 , 2 , 3 The basis vectors ˆ e 1 , ˆ e 2 , and ˆ e 3 have the following properties: ˆ e 1 · ˆ e 1 = ˆ e 2 · ˆ e 2 = ˆ e 3 · ˆ e 3 = 1 (1) ˆ e 1 · ˆ e 2 = ˆ e 1 · ˆ e 3 = ˆ e 2 · ˆ e 3 = 0 (2) x 1 x 2 x 3 a 1 a 2 a 3 a e 1 e 2 e 3 Figure 1: Reference coordinate system. 1
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2 Index Notation We now introduce the Kronecker delta symbol δ ij . δ ij has the following prop- erties: δ ij = 0 i = j 1 i = j i, j = 1 , 2 , 3 (3) Using Eqn 3, Eqns 1 and 2 may be written in index notation as follows: ˆ e i · ˆ e j = δ ij i, j = 1 , 2 , 3 (4) In standard vector notation, a vector A may be written in component form as A = A x ˆ i + A y ˆ j + A z ˆ k (5) Using index notation, we can express the vector A as A = A 1 ˆ e 1 + A 2 ˆ e 2 + A 3 ˆ e 3 = 3 i =1 A i ˆ e i (6) Notice that in the expression within the summation, the index i is repeated . Re- peated indices are always contained within summations, or phrased differently a repeated index implies a summation. Therefore, the summation symbol is typi- cally dropped, so that A can be expressed as A = A i ˆ e i 3 i =1 A i ˆ e i (7) This repeated index notation is known as Einstein’s convention. Any repeated index is called a dummy index . Since a repeated index implies a summation over all possible values of the index, one can always relabel a dummy index, i.e. A = A i ˆ e i = A j ˆ e j = A k ˆ e k etc. A 1 ˆ e 1 + A 2 ˆ e 2 + A 3 ˆ e 3 (8) Copyright c 2006 by Ilan Ben-Yaacov and Francesc Roig
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Index Notation 3 The Scalar Product in Index Notation We now show how to express scalar products (also known as inner products or dot products) using index notation. Consider the vectors a and b , which can be expressed using index notation as a = a 1 ˆ e 1 + a 2 ˆ e 2 + a 3 ˆ e 3 = a i ˆ e i b = b 1 ˆ e 1 + b 2 ˆ e 2 + b 3 ˆ e 3 = b j ˆ e j (9) Note that we use different indices ( i and j ) for the two vectors to indicate that the index for b is completely independent of that used for a . We will first write out the scalar product a · b in long-hand form, and then express it more compactly using some of the properties of index notation.
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