Consider the coordinate system illustrated in Figure 1 Instead of using the typical axis labels and we use and or 1 The corresponding unit basis vectors are then and or 1 The basis vectors and have the following properties 1 1 0 2 Figu ID: 22811 Download Pdf

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Consider the coordinate system illustrated in Figure 1 Instead of using the typical axis labels and we use and or 1 The corresponding unit basis vectors are then and or 1 The basis vectors and have the following properties 1 1 0 2 Figu

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Index Notation for Vector Calculus by Ilan Ben-Yaacov and Francesc Roig Copyright 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 , and we use , and , or = 1 The corresponding unit basis vectors are then , and , or = 1 The basis vectors , and have the following properties: = = = 1 (1) = = = 0 (2) Figure 1: Reference coordinate system.

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2 Index Notation We now introduce the

Kronecker delta symbol ij ij has the following prop- erties: ij i,j = 1 (3) Using Eqn 3, Eqns 1 and 2 may be written in index notation as follows: ij i,j = 1 (4) In standard vector notation, a vector may be written in component form as A = (5) Using index notation, we can express the vector as A = =1 (6) Notice that in the expression within the summation, the index 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 can be expressed as A = =1 (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 = etc. (8) Copyright 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 , which can be expressed using index notation as ~a (9) Note that we use different indices

( and ) for the two vectors to indicate that the index for is completely independent of that used for ~a . We will ﬁrst write out the scalar product ~a in long-hand form, and then express it more compactly using some of the properties of index notation. ~a =1 =1 =1 =1 [( )] =1 =1 ( )] (commutative property) =1 =1 ij ) (from Eqn 3) Summing over all values of and , we get ~a 11 12 13 21 22 23 31 32 33 11 22 33 =1 Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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4 Index Notation Doing this in a more compact notation gives us ~a = ( ij Notice that when we have an

expression containing ij , we simply get rid of the ij and set everywhere in the expression. Example 1: Kronecker delta reduction Reduce ij jk ki ij jk ki ik ki (remove ij set everywhere) δii (remove ik set everywhere) =1 ii =1 1 = 1 + 1 + 1 = 3 Here we can see that ii = 3 (Einstein convention implied) (10) Note also that ij jk ik (11) Example 2: ~r and in index notation (a) Express ~r using index notation. ~r Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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Index Notation 5 (b) Express using index notation. ~r ~r ~r ~r ~r (c) Express ~a using index notation. ~a ~a ~r

~r The Cross Product in Index Notation Consider again the coordinate system in Figure 1. Using the conventional right- hand rule for cross products, we have = = = 0 = = = (12) To write the expressions in Eqn 12 using index notation, we must introduce the symbol ijk , which is commonly known as the Levi-Civita tensor, the alternating unit tensor, or the permutation symbol (in this text it will be referred to as the permutation symbol). ijk has the following properties: ijk = 1 if ( ijk ) is an even (cyclic) permutation of (123), i.e. 123 231 312 = 1 ijk if ( ijk ) is an odd (noncyclic)

permutation of (123), i.e. 213 321 132 ijk = 0 if two or more subscripts are the same, i.e. 111 112 313 = 0 etc. Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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6 Index Notation Hence, we may rewrite the expressions in Eqn 12 as follows: 123 213 231 321 312 132 (13) Now, we may write a single generalized expression for all the terms in Eqn 13: ijk (14) Here ijk =1 ijk is a dummy index). That is, this works because 12 =1 12 121 122 123 = The same is true for all of the other expressions in Eqn 13. Note that iik = 0 , since iik for all values of ijk is also given by the

following formula. ijk )( )( i,j,k = 1 (15) This is a remarkable formula that works for ijk if you do not want to calculate the parity of the permutation ijk . Also note the following property of ijk ijk jik kji i.e. switching any two subscripts reverses the sign of the permutation symbol (or in other words ijk is anti-symmetric ). Also, ijk kij jki i.e. cyclic permutations of the subscripts do not change the sign of ijk . These Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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Index Notation 7 properties also follow from the formula in Eqn 15. Now, let’s consider the cross

product of two vectors ~a and , where ~a Then ~a = ( ) = ijk Thus we write for the cross product: ~a ijk (16) All indices in Eqn 16 are dummy indices (and are therefore summed over) since they are repeated. We can always relabel dummy indices, so Eqn 16 may be written equivalently as ~a pqr Returning to Eqn 16, the th component of ~a is ~a ijk where now only and are dummy indices. Note that the cross product may also be written in determinant form as follows: ~a (17) The follwoing is a very important identity involoving the product of two per- mutation symbols. ijk lmn il im in jl jm jn kl km

kn (18) Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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8 Index Notation The proof of this identity is as follows: If any two of the indices i,j,k or l,m,n are the same, then clearly the left- hand side of Eqn 18 must be zero. This condition would also result in two of the rows or two of the columns in the determinant being the same, so therefore the right-hand side must also equal zero. If i,j,k and l,m,n both equal (1,2,3), then both sides of Eqn 18 are equal to one. The left-hand side will be , and the right-hand side will be the determinant of the identity matrix. If

any two of the indices i,j,k or l,m,n are interchanged, the corresponding permutation symbol on the left-hand side will change signs, thus reversing the sign of the left-hand side. On the right-hand side, an interchange of two indices results in an interchange of two rows or two columns in the determinant, thus reversing its sign. Therefore, all possible combinations of indices result in the two sides of Eqn 18 being equal. Now consider the special case of Eqn 18 where In this case, the repeated index implies a summation over all values of . The product of the two permutation symbols is now

ijk lmk il im ik jl jm jk kl km kk (note kk = 3) = 3 il jm im jl im jk kl ik jm kl ik jl km il jk km = 3 il jm im jl im jl il jm im jl il jm (from Eqn 11) (19) Or ﬁnally ijk lmk il jm im jl (20) Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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Index Notation 9 Eqn 20 is an extremely useful property in vector algebra and vector calculus applications. It can also be expressed compactly in determinant form as ijk lmk il im jl jm (21) The cyclic property of the permutation symbol allows us to write also ijk klm il jm im jl To recap: ij and ~a ijk and ~a ijk These

relationships, along with Eqn 20, allow us to prove any vector identity. Example 3: Vector identity proof Show for the double cross product: ~a ~c ) = ( ~a ~c ~a ~c Start with the left-hand side (LHS): ~a ~c ) = ( jkl jkl ( jkl ilh jkl hil = ( jh ki ji kh jh ki ji kh = ( ) ( ) ( ~a ~c ~a ~c Copyright 2006 by Ilan Ben-Yaacov and Francesc Roig

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10 Index Notation Example 4: The scalar triple product Show that ~a ~c ) = ~c ~a ) = ~c ~a ~a ~c ) = ( jkm jkm ( jkm im jki or ~a ~c ) = ijk From our permutation rules, it follows that ~a ~c ) = ijk kij ~c ~a jki ~c ~a Copyright 2006 by

Ilan Ben-Yaacov and Francesc Roig

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