TOP SURFACESINTERFACES EXP
TOP SURFACESINTERFACES EXP PHYS 591
Virginia Commonwealth University
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This 11 page Class Notes was uploaded by Dr. Nia O'Kon on Wednesday October 28, 2015. The Class Notes belongs to PHYS 591 at Virginia Commonwealth University taught by Robert Gowdy in Fall. Since its upload, it has received 13 views. For similar materials see /class/230666/phys-591-virginia-commonwealth-university in Physics 2 at Virginia Commonwealth University.
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1 Linear Functions 11 of Real Numbers The key concept behind differential geometry is the idea of homogeneous lineare ity For a realevalued function f with one real argument homogeneous linearity over the rents is de ned by the requirements HM WW facy ffy for any real numbers A at y To save space I will often abbreviate homogeneous linearity to just linearity 12 of Complex Numbers Homogeneous linearity over the complex numbers is de ned in the same way as for real numbers except that the function f and the numbers A at y are allowed to be complex numbers There is however a new possibility because each complex number at a 1 has a complex conjugate partner 30 a 7 2391 For a complexevalued function g of one complex argument homogeneous antielinearity is de ned by the requirements 90 V900 gacy 99y 1 3 of Vectors As you will see from the rst homework assignment linear functions of one real or complex number are pretty trivial What about functions whose arguments or values are not single real or complex numbers To de ne linearity for such functions it must be possible to add their arguments or values together and to multiply them by real numbers in a consistent way Such arguments and values belong to vector spaces A vector space over the real numbers or over the complex numbers is de ned to be a set V with an addition operation and a scalar multiplication operation with the usual associative and commutative properties and a unique zero element The addition operation takes two elements at and y of V and forms a new object at y that also belongs to V Similarly the scalar multiplie cation operation 96 takes a real or complex number a and an element at of V and forms an object a 96 at that also belongs to V In the usual terse notation of mathematics we would describe these operations as mappings VgtltVgtV ERXVHV The usual associative and commutative properties for real or complex numbers a b and vectors at y z are omitting the 96 as is the usual convention abx axbx 101 11135 xyz xyz 12 yac aacy axay and the zero element 0 is a member of V such that x0x 200 for any real or complex number a and vector at A realevalued linear function de ned on a vector space V is called a linear form over V The set of linear forms over V is itself a vector space7 called the space V dual to V 14 lsomorphisms The de nition of a linear function works perfectly well if the function assigns vectors in a vector space rather than just real numbers A vectorevalued linear function on a vector space is best thought of as a mapping f V A U of a vector space V to a vector space U If the mapping is oneetoeone7 it is called an isomorphism lsomorphisms give us a convenient way to compare different vector spaces If the set of possible values f V is actually equal to U7 then we say that U and V are isomorphic In such a case7 the spaces U and V are interchangeable copies of each other and we can use the function f and its inverse f 1 to go back and forth between them 15 Examples of Vector Spaces 151 Real ntuples The set ER of real numbers is obviously a vector space with and 96 just the usual operations of addition and multiplication The set 9 2 ERXER of number pairs such as 07 d is a vector space with the de nitions a 96 07d aqad 11 07d acbd The zero vector in this case is the pair 00 The set C of complex numbers is also an example of a vector space Similarly7 the set DR of realenumber netuples with the corresponding de nie tions of addition and multiplication by a scalar is also a vector space An even larger but still manageable vector space is the set 9 of reale number in nite sequences such as 120121 12 which can be multiplied by a scalar and added to one another in the obvious way This space is an example of a vector space with a countable basis 1 5 2 thction spaces An example of a vector space that is much larger than 9 is the set 9 of realevalued functions of real numbers Functions f and 9 can be added to give a function f 9 de ned by fgxfxgx for all real numbers at and a function f can be multiplied by a real number a to give a function af de ned in the obvious way af av af 36 The space 9 is extremely large 7 larger than we ever need in physics In classical physics we pick out subspaces of 9 that consists of functions with nite derivatives up to some order In quantum theory the state of a particle that is moving in one dimension can be described by a complexevalued wave function 11 The set of all such functions is the vector space 0 That space is also much larger than we need in physics so we pick out a subspace L2 6 of squareeintegrable functions to represent physical states 153 Directional Derivatives as Vectors Because differential operators can be added and multiplied by constants they can be regarded as vectors Consider the set Tam of directional derivatives of functions on 1R2 at the point 00 The operators and act on a function f at y to produce its partial derive atives and A linear combination of these operators such 39 xy0 3y xy0 as v 2 acts on a function f to produce 7 3f 3f vf72ax 08iy 1 11113 These operators are directional derivatives The vector space of directional derivatives at the point 00 is called the tangent space at 00 There is a separate tangent space for each point in 1R2 Moving the evaluation point to 0 12 would produce the tangent space To 0 1 2 A few notational conventions for operators such as the directional derivative 1 should be recalled First of all when parentheses are omitted they are always assumed to be nested from right to left as in the expression uvgwfuv9wf where uvw are directional derivatives and fg are functions Thus each op erator acts on everything to the right of it Second one can use singleeterm parentheses to turn off this righteaction as in the expression u w wf where 1 acts on g to produce the function 119 and does nothing else Third a relation between operators always assumes that the operators are acting on arbitrary functions even if those functions are not explicitly shown as in the relation 7 1 2 v 7 e112 e21 which really means vf e111 e2v2 f 1U1f 2v2f for any function f In this last example notice that the parenthesis does not turn off the right action of the operators e1 e2 because it includes more than one term An example that uses both multiple and singleeterm parentheses would e e111 e2v2 f 1U1f52v2f 61111 fv1 1f 62112 fv2 2f where the product rule is applied to show e1 e2 acting on each factor separately 1 54 Differential forms The space TO TD of linear forms over the tangent space Tam is the set of reale valued linear functions of directional derivatives One way to produce such a function is to let each vector 1 act on a particular function f The resulting linear form is called df and is de ned by df U Ufl00 Some alternative notations for the result of df acting on the vector v are dfvdfvdfvvdf One also sometimes sees the notation df v df7vgt Corresponding to the coordinate functions at and y there are linear forms dac and dy These objects are called di erential forms and Will be discussed at length later 2 Basis Vectors 21 Linear Independence and Dimensionality A set 1211221137 vn of vectors is said to be linearly independent if the equa tion 11121 12112 13113 anvn 0 implies that all of the coef cients ai must be zero The maximum number of linearly independent vectors in a space is called the dimensionality of the space 22 The Basis Representation A set of n linearly independent vectors 6162637 en in an nadimensional vector space V generates an isomorphism that connects ER to V Denote the array of coef cients by the row vector The vector v is then represented as v e To gain familiarity With this mixture of vector and matrix notation Write the expression out U161 12252 U363 quotmnen Because the map 5 is an isomorphism it can be inverted to yield a unique map 6 V H ER Thus the array of expansion coef cients is given by v 6 1 It is useful to de ne the column vectors 1 0 0 0 1 0 W1 W2 W 0 0 1 and use them to extract the individual expansion coef cients 0 1 WT 57112WT For example 3 Dual Basis Forms 31 Abstract de nition For each value of the index 7 the function of v 57112WT is a realevalued linear function on the vector space V Thus it is a linear form over V and belongs to the dual space V The expansion coef cients for a vector v are then given by the expressions UT M v 1 Using these expansion coef cients any vector v has the expansion 11211 we 2 Now suppose that a belongs to If we know the value 041 that a assigns to any vector 1 then we know everything there is to know about a From the expansion above7 n 041 a Zwr U Tgt Za f U T r1 r1 r1 so that if we know the numbers O T 0 5T then we can nd 041 for any vector 1 041 ZaTwT v Thus7 we have found an expansion for the arbitrary linear form a n a Z aTwT 4 r1 The linear forms of are therefore a basis for the dual space l7 and are called the basis dual to e 32 Dot and matrix notations Some alternative notations for the relations between forms and their components and vectors and their components may be helpful 07 v WOT a e a In words7 one nds components by dotting things into the dual basis objects Vector components are found by dotting vectors into the form basis and form components are found by dotting forms into the vector basis In terms of ma trices7 one can de ne the column vector of components of the form a as and the row vector of components of the vector v as M v M where w is a row matrix whose elements are the forms of In terms of these matrices7 the expansions of vectors and forms becomes 33 Useful de nition of dual basis forms In general7 we do not know the inverse map 6 1 even though we do know that it has to exist Thus the formula de ning the dual basis forms in terms of 6 1 is reassuring but not of much practical use For a more useful de nition7 use the expansion of an arbitrary vector to expand one of the basis vectors n ES Zwr 636T r1 and note that the linear independence of the basis vectors means that wes6 0quot 5 S 1 7 5 An alternative notation for this expression would be ES wT 6 and7 in terms of the column matrix 6 and the row matrix to where 1 is the identity n X n matrix Written out7 the matrix operation looks like this 61 ml 61 ml 62 w 6n 2 2 2 62 w 61 w 62 w 6n w1 w2 wn 6 w 61 w 62 w 6 so that the relation is w1 61 ml 62 ml 6 1 0 0 w2 61 w2 62 w2 6 0 1 0 w 61 tun62 tun6n 0 0 1 4 Names and Notations 41 Covariant and contravariant vectors Notice that there are automatically the same number of dual basis forms as there are dual basis vectors As a result7 so long as V is nite dimensional7 the dual space 17 is isomorphic to V It is therefore possible to think of the forms as other kinds of vectors and older treatments of tensor analysis do exactly that In the ancient texts of geometrical physics7 the elements of V are called contmvariant vectors and the elements of 17 are called covariant vectors 42 The dual of the dual space and dot products The elements of l are functions that assign real or complex numbers to forms in the dual space Some of these functions are easy to specify Given a vector 17 in V we can assign a number to each form a namely the number a Thus we de ne the form on forms7 17 that is associated with the vector 17 by 17 a 011 Instead of keeping track of two different objects 17 and 17 it is customary to think of one object that happens to have two different actions and de ne the action of 17 on a form by va av 6 The symmetrical relationship between forms and vectors is often emphasized by using a different notation avvaa17 When we wish to emphasize that forms and vectors are equally fundamental but wish to keep them separate we use yet another notation 047v 041 where the form is always placed to the left of the comma and the vector to the right 43 Components and sum conventions From Equation 1 for the components of a vector and Equation 6 which de nes the action of a vector on a form the components of a vector can be found by evaluating the vector acting on the basis forms in just the same way that Equation 3 indicates that the components ofa form can be found by evaluating the form acting on the basis vectors As you discovered when you did the homework each expansion of a vector or a form introduces a new index and a new summation sign Not only do you have to take the time to write each summation sign but you need to explicitly change the order of the summation signs even though that order never really matters The Einstein summation convention takes advantage of the fact that every sum that arises from forms basis vectors and related objects can be summarized by a simple rule Sum over any index that is repeated once as a subscript and once as a superscript The rule works only if you are careful to never use the same index for different sums For example instead of writing 17 E 17 and a E oszT r r we just write 1 UTET and a asws So long as you are careful to avoid coincident names for the summed indexes the summation convention lets you ignore the summations entirely and just do algebra on a typical term of the sum Be sure to try this technique in Homework 04 We will return to this idea when we get to tensors where the number of indexes and summations can become quite large 44 Matrices and automatic symbol manipulation The use of matrix multiplication to replace sums and indexes has a long tradition in differential geometry going back at least to the work of the mathematician Elie Cartan in 1928 It yields simple and elegant expressions that can usually be written in just one way A practical advantage is that it avoids the need to decorate symbols with subscripts and superscripts and avoids the arbitrariness of naming dummy indexes A practical disadvantage is that computations with speci c examples require one to write out all of the terms of each matrix in each expression Another practical disadvantage is that the manual procedure of matching up the row and column entries that are to be multiplied together is both tedious and erroreprone These disadvantages have usually led physicists to revert to index notation to organize their calculations A key feature of matrix expressions is that they obey the associative rule even when they are mixed with dot products Parentheses are never needed in these expressions For example there is no ambiguity about the meaning of the expressions 1 w e and w e a once one knows that v is a vector and a is a form Thus one can regroup these expressions and de ne an object 1d lwl El that acts as an identity operation on both the space V of vectors and the space V of forms according to vId v a Ida Some of the practical disadvantages of using matrix notation in actual calf culations are now less important than they once were Consider the dot product of a vector v 3181 2281 318 and a form a ydac zdy xdz A direct calculation of v a is straightforward but tedious since one must con sider nine possible terms The index expression via is better but requires the additional step of writing out six de nitions for v1v2v3a1a2a3 and then plugging those de nitions into the resulting sum of three terms The matrix version of the calculation y vavay 2z 31 z can be completed in an instant if you are using Scienti c Notebook or Scienti c Workplace by marking the matrix product with your mouse and hitting CTRLe You will nd that symbol manipulation programs are confused by supere scripts which get interepreted as powers by default and by multiple subscripts There are special packages for most symbol manipulation programs that let them handle such notation but those packages are usually a step or two removed from the process of entering and editing expressions Thus the avoidance of indexes is a strong advantage of matrix notation in a symbol manipulating environment such as we are using here 45 The interior trace operation Symbol manipulation routines are also confused by operator symbols such as the dot that we have been using to indicate the action of forms on vectors or vice versa For example7 try evaluating the expression 81 8y dac dy dz 8 2 within the SN or SW environment and you will nd that the software performs a hermitean transpose on the second array and then dots the two arrays together which is not what we want here A workaround is to introduce the interior trace operation Twhich instructs one to let each vector act on the form if any that follows it Evaluating the expression 81 T 8y d1 dy dz 8 2 yields the matrix Taxdac T81 dy T81 dz Taydac Taydy Taydz Tazdac Tazdy Tazdz which is interpreted as exactly what we want 81dac Bagdz Bagdz 1 Bydac Bydy Bydz 0 82dac 82dy 82dz 0 OHO HOD 11