Vectors and Index Notation - UCA

1 Vectors and Index NotationStephen R. AddisonJanuary 12, 20041 Basic vector Unit VectorsWe will denote a unit vector with a superscript caret, thus adenotes a unit vector . a | a|=1If~xis a vector in thex-direction x=~x|~x|is a unit vector . We will usei,j,and k, or x, y,and z, ore1,e2and e3and a variety of variations without further Addition and SubtractionAddition and subtraction are depicted Scalar Products~a.~b=|~a||~b|cos 1~a.~b=~b.~asince cos =cos( ) vector Products|~a ~b|=|~a||~b|sin ~a ~b= ~b ~a,Why?~c=~a ~b,~cis perpendicular to~aand~ Geometric InterpretationQPqArea of a triangle=12base perpendicular height=12|~Q||~P|sin SoA=12|~Q ~P|is the area of a triangle and, accordingly,|~Q ~P|=the area of a Area as a vectorAreas can also be expressed as vector quantities, for the parallelogram considered above, we couldhave written~A=~Q ~P.

2 If nis a unit vector normal to a plane of area A, then~A=|~A| n=A n,where A is the numerical value of the Direction of the resultant of a vector productWe have options, in simple cases we often use the right-hand screw rule:If~c=~a ~b, the direction of~cis the direction in which a right-handed screw would advance inmoving from~ato~ we can use the right hand rule, as seen in the right handfor direction ofvector product I prefer this version of the right-hand rule - it doesn t require the contortions of the version typicallyfound in beginning Components and unit vectorsWe can write Vectors in component form, for example:~a=axi+ayj+azk,~b=bxi+byj+bzk,an d~c=cxi+cyj+ order to calculate in terms of components, we need to be familiar with the scalar and vectorproducts of unit a right-handed coordinate system with axes labeledx,y,andz, as shown in the ,j,andkare the unit Vectors , we the scalar products, andi i=j j=k k=0andi j=k,i k= j,j k=ifor the vector these results we can compute~a ~b=(axi+ayj+azk) (bxi+byj+bzk)=axbx(i i)+axby(i j)+axbz(i k)+aybx(j i)+ayby(j j)+aybz(j k)+azbx(k i)+azby(k j)+azbz(k k)Simplifying,~a ~b=(axby aybx)(i j) k+(axbz azbx)(i k) j+(aybz azby)(j k) i,and finally we have~a ~b=(aybz azby)i+(azbx axbz)j+(axby aybx) a mnemonic, this is often written in the form of a determinant.

3 While the mnemonic is useful,the vector product is not a determinant. (All terms in a determinant must be numbers.)~a ~b= ijkaxayazbxbybz Your book goes on to triple products of various types, at this point, I am going to introduce indexnotation - a far better way of doing vector Index NotationYou will usually find that Index Notation for Vectors is far more useful than the Notation that youhave used before. Index Notation has the dual advantages of being more concise and more trans-parent. Proofs are shorter and simpler. It becomes easier to visualize what the different terms inequations Index Notation and the Einstein summation conventionWe begin with a change of Notation , instead of writing~A=Axi+Ayj+Azkwe write~A=A1e1+A2e2+A3e3=3 i= simplify this further by introducing the Einstein summation convention: if an Index appearstwice in a term, then it is understood that the indices are to be summed from 1 to 3.

4 Thus we write~A=AieiIn practice, it is even simpler because we omit the basis vectorseiand just write the Cartesian com-ponentsAi. Recall a basis of a vector space is a linearly independent subset that spans (generates)the whole space. The unit vectorsi,j,andkare a basis of we will often denote~AasAiwith the understanding that the Index can assume the values 1, 2,or 3 for the scalar components(A1,A2,A3); we ll refer to the vectorAieven though to get~Awe need to Examplesai=bi a1=b1,a2=b2,a3=b3aibi=a1b1+a2b2+ Summation convention and dummy indicesConsider~a.~b=axbx+ayby+azbzthis indicates that we can write a scalar product as~a.~b=aibi5In the termaibi, an Index likeithat is summed over is called a dummy Index (or more cavalierlyas a dummy variable).

5 The Index used is irrelevant - just as the integration variable is irrelevant inan integral (though in this case the term dummy variable is entirely appropriate).This meansaibi=ajbj=ambm=a1b1+a2b2+ term cannot contain an Index more than twice, if a compound calculation would lead to such asituation, the dummy Index should be changed. An Index that appears only once in a term is calledafreeorfloating exampleaibicj=(a1b1+a2b2+a3b3)cjIn an equation, all terms must contain the same free indices, in particular you should note thataibicj6=a1bickIf we haveS=AiBiand we wantSCi, as noted above we change the subscripts onAandBbecausethey are the dummy indices. Do not change the free indices because you risk changing the ,SCi= The Kronecker delta or the substitution operatorThe Kronecker delta, i j=1 ifi=j,=0i f i6= 11= 22= 33and 12= 23= 13=0We will sometimes find it convenient this result in an array i j= 11 12 13 21 22 23 31 32 33 = 1 0 00 1 00 0 1.

6 Why is the Kronecker delta slso known as the substitution operator? We can figure this out bymaking a jiai= i jai, letjtake on the values 1, 2, and 3. Then we have:j=1 : 11a1+ 12a2+ 13a3=a16j=2 : 21a1+ 22a2+ 23a3=a2j=3 : 31a1+ 32a2+ 33a3= this we can see that i jai= applying Kronecker delta allows us to drop a repeated Index and changes one Index Further ExamplesArBsCt st=ArBsCs=ArBtCtSo we see that if two indices are repeated, only one is dropped. We should note the followingobvious results: ii=1+1+1=3and i j jk= ik4 The permutation symbol or the Levi-Civita tensorThe numbers 1, 2, 3 are in cyclic order if they occur in the order 1,2,3 on a counterclockwise pathstarting from Permutations1 2 3 2 3 1 3 1 2 7 Non-Cyclic PermutationsWhen the path is clockwise the permutations are non-cyclic.

7 3 2 1 1 3 22 1 3 Cyclic permutations are sometimes called even, non-cyclic permuations are sometimes called idea can be used in the evaluation of vector products. The idea is introduced through thepermutation symbol i jk. i jk=+1 if ijk is a cyclic permutation of 1,2,3 i jk= 1 if ijk is a non-cyclic permuation of 1,2,3 i jk=0 otherwise, an Index is we find 123= 231= 312=+1,and 132= 213= 321= 1,while 122= 133= 112= should also note the following properties: i jk= jki= ki jbut when we swap indices i jk= jikand i jk= ik vector Product in Index notationRecall~a ~b= ijkaxayazbxbybz Now considerci= i jkajbkThis is a vector characterized by a single free Index i.

8 The indices j and k are dummy indices andare summed out. We get the three values ofciby lettingi=1,2,3 independently. This is usefulbut the method is made more powerful by the methods of the next The identity i jk irs= jr ks js rkThis identity can be used to generate all the identities of vector analysis, it has four free prove it by exhaustion, we would need to show that all 81 cases that the s have the repeated Index first, and that in the s, the free indices are take in thisorder:1. both second2. both third3. one second, one third4. the other second, the other thirdLet s put this to use by proving what would be a tough identity using ordinary vector ll prove the bac-cab that~A (~B ~C)=(~A.)

9 ~C)~B (~A.~B)~CTo prove this, let~A (~B ~C)=~A ~D=~Ewe the convert to Index Notation as follows: Writing~B ~C= i jkBjCk=Di9then~A ~D= rsiAsDi= rsiAs i jkBjCk= terms, we haveEr= rsi i jkAsBjCk= irs i jkAsBjCk,and using the identityEr=( r j sk rk s j)AsBjCk,thenEr= r j skAsBjCk rk s jAsBjCkthen using the substitution properties of the knronecker deltas, this becomesEr=AkBrCk AjBjCr=Br(AkCk) Cr(AjBj)=~B(~A.~C) ~C(~A.~B)