INTRODUCTION TO VECTORS AND TENSORS

Transcription

1 INTRODUCTION TO VECTORS AND TENSORS Vector and Tensor Analyss Volume 2 Ray M. Bowen Mechancal Engneerng Texas A&M Unversty College Staton, Texas and C.-C. Wang Mathematcal Scences Rce Unversty Houston, Texas Copyrght Ray M. Bowen and C.-C. Wang (ISBN (v. 2))

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3 PREFACE To Volume 2 Ths s the second volume of a two-volume work on vectors and tensors. Volume s concerned wth the algebra of vectors and tensors, whle ths volume s concerned wth the geometrcal aspects of vectors and tensors. Ths volume begns wth a dscusson of Eucldean manfolds. The prncpal mathematcal entty consdered n ths volume s a feld, whch s defned on a doman n a Eucldean manfold. The values of the feld may be vectors or tensors. We nvestgate results due to the dstrbuton of the vector or tensor values of the feld on ts doman. Whle we do not dscuss general dfferentable manfolds, we do nclude a chapter on vector and tensor felds defned on hypersurfaces n a Eucldean manfold. Ths volume contans frequent references to Volume. However, references are lmted to basc algebrac concepts, and a student wth a modest background n lnear algebra should be able to utlze ths volume as an ndependent textbook. As ndcated n the preface to Volume, ths volume s sutable for a one-semester course on vector and tensor analyss. On occasons when we have taught a one semester course, we covered materal from Chapters 9,, and of ths volume. Ths course also covered the materal n Chapters,3,4,5, and 8 from Volume. We wsh to thank the U.S. Natonal Scence Foundaton for ts support durng the preparaton of ths work. We also wsh to take ths opportunty to thank Dr. Kurt Rencke for crtcally checkng the entre manuscrpt and offerng mprovements on many ponts. Houston, Texas R.M.B. C.-C.W.

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5 CONTENTS Vol. 2 Vector and Tensor Analyss Contents of Volume v PART III. VECTOR AND TENSOR ANALYSIS Selected Readngs for Part III 296 CHAPTER 9. Eucldean Manfolds Secton 43. Eucldean Pont Spaces Secton 44. Coordnate Systems 36 Secton 45. Transformaton Rules for Vector and Tensor Felds. 324 Secton 46. Anholonomc and Physcal Components of Tensors. 332 Secton 47. Chrstoffel Symbols and Covarant Dfferentaton Secton 48. Covarant Dervatves along Curves CHAPTER. Vector Felds and Dfferental Forms Secton 49. Le Dervatves Secton 5O. Frobenus Theorem 368 Secton 5. Dfferental Forms and Exteror Dervatve Secton 52. The Dual Form of Frobenus Theorem: the Poncaré Lemma.. 38 Secton 53. Vector Felds n a Three-Dmensona Eucldean Manfold, I. Invarants and Intrnsc Equatons Secton 54. Vector Felds n a Three-Dmensona Eucldean Manfold, II. Representatons for Specal Class of Vector Felds. 399 CHAPTER. Hypersurfaces n a Eucldean Manfold Secton 55. Normal Vector, Tangent Plane, and Surface Metrc 47 Secton 56. Surface Covarant Dervatves. 46 Secton 57. Surface Geodescs and the Exponental Map Secton 58. Surface Curvature, I. The Formulas of Wengarten and Gauss 433 Secton 59. Surface Curvature, II. The Remann-Chrstoffel Tensor and the Rcc Identtes Secton 6. Surface Curvature, III. The Equatons of Gauss and Codazz 449 v

6 v CONTENTS OF VOLUME 2 Secton 6. Surface Area, Mnmal Surface Secton 62. Surfaces n a Three-Dmensonal Eucldean Manfold. 457 CHAPTER 2. Elements of Classcal Contnuous Groups Secton 63. The General Lnear Group and Its Subgroups Secton 64. The Parallelsm of Cartan. 469 Secton 65. One-Parameter Groups and the Exponental Map 476 Secton 66. Subgroups and Subalgebras. 482 Secton 67. Maxmal Abelan Subgroups and Subalgebras 486 CHAPTER 3. Integraton of Felds on Eucldean Manfolds, Hypersurfaces, and Contnuous Groups Secton 68. Arc Length, Surface Area, and Volume Secton 69. Integraton of Vector Felds and Tensor Felds 499 Secton 7. Integraton of Dfferental Forms. 53 Secton 7. Generalzed Stokes Theorem.. 57 Secton 72. Invarant Integrals on Contnuous Groups INDEX. x

7 CONTENTS Vol. Lnear and Multlnear Algebra PART BASIC MATHEMATICS Selected Readngs for Part I 2 CHAPTER Elementary Matrx Theory. 3 CHAPTER Sets, Relatons, and Functons 3 Secton. Sets and Set Algebra... 3 Secton 2. Ordered Pars" Cartesan Products" and Relatons. 6 Secton 3. Functons. 8 CHAPTER 2 Groups, Rngs and Felds 23 Secton 4. The Axoms for a Group. 23 Secton 5. Propertes of a Group.. 26 Secton 6. Group Homomorphsms.. 29 Secton 7. Rngs and Felds.. 33 PART I VECTOR AND TENSOR ALGEBRA Selected Readngs for Part II 4 CHAPTER 3 Vector Spaces.. 4 Secton 8. The Axoms for a Vector Space.. 4 Secton 9. Lnear Independence, Dmenson and Bass Secton. Intersecton, Sum and Drect Sum of Subspaces. 55 Secton. Factor Spaces Secton 2. Inner Product Spaces Secton 3. Orthogonal Bases and Orthogonal Complments 69 Secton 4. Recprocal Bass and Change of Bass 75 CHAPTER 4. Lnear Transformatons 85 Secton 5. Defnton of a Lnear Transformaton. 85 Secton 6. Sums and Products of Lnear Transformatons 93

8 v CONTENTS OF VOLUME 2 Secton 7. Specal Types of Lnear Transformatons 97 Secton 8. The Adont of a Lnear Transformaton.. 5 Secton 9. Component Formulas... 8 CHAPTER 5. Determnants and Matrces 25 Secton 2. The Generalzed Kronecker Deltas and the Summaton Conventon 25 Secton 2. Determnants. 3 Secton 22. The Matrx of a Lnear Transformaton 36 Secton 23 Soluton of Systems of Lnear Equatons.. 42 CHAPTER 6 Spectral Decompostons Secton 24. Drect Sum of Endomorphsms 45 Secton 25. Egenvectors and Egenvalues.. 48 Secton 26. The Characterstc Polynomal. 5 Secton 27. Spectral Decomposton for Hermtan Endomorphsms.. 58 Secton 28. Illustratve Examples. 7 Secton 29. The Mnmal Polynomal.. 76 Secton 3. Spectral Decomposton for Arbtrary Endomorphsms CHAPTER 7. Tensor Algebra. 23 Secton 3. Lnear Functons, the Dual Space 23 Secton 32. The Second Dual Space, Canoncal Isomorphsms. 23 Secton 33. Multlnear Functons, Tensors Secton 34. Contractons Secton 35. Tensors on Inner Product Spaces. 235 CHAPTER 8. Exteror Algebra Secton 36. Skew-Symmetrc Tensors and Symmetrc Tensors Secton 37. The Skew-Symmetrc Operator 25 Secton 38. The Wedge Product Secton 39. Product Bases and Strct Components Secton 4. Determnants and Orentatons. 27 Secton 4. Dualty.. 28 Secton 42. Transformaton to Contravarant Representaton. 287 INDEX.x

9 PART III VECTOR AND TENSOR ANALYSIS

10 Selected Readng for Part III BISHOP, R. L., and R. J. CRITTENDEN, Geometry of Manfolds, Academc Press, New York, 964 BISHOP, R. L., and S. I. GOLDBERG, Tensor Analyss on Manfolds, Macmllan, New York, 968. CHEVALLEY, C., Theory of Le Groups, Prnceton Unversty Press, Prnceton, New Jersey, 946 COHN, P. M., Le Groups, Cambrdge Unversty Press, Cambrdge, 965. EISENHART, L. P., Remannan Geometry, Prnceton Unversty Press, Prnceton, New Jersey, 925. ERICKSEN, J. L., Tensor Felds, an appendx n the Classcal Feld Theores, Vol. III/. Encyclopeda of Physcs, Sprnger-Verlag, Berln-Gottngen-Hedelberg, 96. FLANDERS, H., Dfferental Forms wth Applcatons n the Physcal Scences, Academc Press, New York, 963. KOBAYASHI, S., and K. NOMIZU, Foundatons of Dfferental Geometry, Vols. I and II, Interscence, New York, 963, 969. LOOMIS, L. H., and S. STERNBERG, Advanced Calculus, Addson-Wesley, Readng, Massachusetts, 968. MCCONNEL, A. J., Applcatons of Tensor Analyss, Dover Publcatons, New York, 957. NELSON, E., Tensor Analyss, Prnceton Unversty Press, Prnceton, New Jersey, 967. NICKERSON, H. K., D. C. SPENCER, and N. E. STEENROD, Advanced Calculus, D. Van Nostrand, Prnceton, New Jersey, 958. SCHOUTEN, J. A., Rcc Calculus, 2nd ed., Sprnger-Verlag, Berln, 954. STERNBERG, S., Lectures on Dfferental Geometry, Prentce-Hall, Englewood Clffs, New Jersey, 964. WEATHERBURN, C. E., An Introducton to Remannan Geometry and the Tensor Calculus, Cambrdge Unversty Press, Cambrdge, 957.

11 Chapter 9 EUCLIDEAN MANIFOLDS Ths chapter s the frst where the algebrac concepts developed thus far are combned wth deas from analyss. The man concept to be ntroduced s that of a manfold. We wll dscuss here only a specal case caled a Eucldean manfold. The reader s assumed to be famlar wth certan elementary concepts n analyss, but, for the sake of completeness, many of these shall be nserted when needed. Secton 43 Eucldean Pont Spaces Consder an nner produce space V and a set E. The set E s a Eucldean pont space f there exsts a functon f : E E V such that: and (a) f( xy, ) = f( xz, ) + f( zy, ), xyz,, E (b) For every x E and v V there exsts a unque element y E such that f ( x, y) = v. The elements of E are called ponts, and the nner product space V s called the translaton space. We say that f ( x, y ) s the vector determned by the end pont x and the ntal pont y. Condton b) above s equvalent to requrng the functon fx : E V defned by fx( y) = f( x, y ) to be one to one for each x. The dmenson of E, wrtten dme, s defned to be the dmenson of V. If V does not have an nner product, the set E defned above s called an affne space. A Eucldean pont space s not a vector space but a vector space wth nner product s made a Eucldean pont space by defnng f ( v, v2) v v 2 for all v V. For an arbtrary pont space the functon f s called the pont dfference, and t s customary to use the suggestve notaton f ( x, y) = x y (43.) In ths notaton (a) and (b) above take the forms 297

12 298 Chap. 9 EUCLIDEAN MANIFOLDS x y = x z+ z y (43.2) and x y = v (43.3) Theorem 43.. In a Eucldean pont space E () x x = ( ) x y = ( y x) ( ) f x y = x ' y ', then x x' = y y' (43.4) Proof. For () take x = y = z n (43.2); then x x = x x+ x x whch mples x x =. To obtan () take y = x n (43.2) and use (). For () observe that x y ' = x y+ y y' = x x' + x' y ' from (43.2). However, we are gven x y = x ' y ' whch mples (). The equaton x y = v has the property that gven any v and y, x s unquely determned. For ths reason t s customary to wrte x = y + v (43.5)

13 Sec. 43 Eucldean Pont Spaces 299 for the pont x unquely determned by y E and v V. The dstance from x to y, wrtten d( x, y ), s defned by {( ) ( )} /2 d ( x, y) = x y = x y x y (43.6) It easly folows from the defnton (43.6) and the propertes of the nner product that d( x, y) = d( y, x ) (43.7) and d( x, y) d( xz, ) + d( zy, ) (43.8) for all x, y, and z n E. Equaton (43.8) s smply rewrtten n terms of the ponts x, y, and z rather than the vectors x y, x z, and z y. It s also apparent from (43.6) that d( x, y) and d( x, y) = x = y (43.9) The propertes (43.7)-(43.9) establsh that E s a metrc space. There are several concepts from the theory of metrc spaces whch we need to summarze. For smplcty the defntons are sated here n terms of Eucldean pont spaces only even though they can be defned for metrc spaces n general. In a Eucldean pont space E an open ball of radus ε > centered at x E s the set { d ε} B( x, ε ) = x ( x, x ) < (43.) and a closed ball s the set

14 3 Chap. 9 EUCLIDEAN MANIFOLDS { d ε} B( x, ε ) = x ( x, x ) (43.) A neghborhood of x E s a set whch contans an open ball centered at x. A subset U of E s open f t s a neghborhood of each of ts ponts. The empty set s trvally open because t contans no ponts. It also follows from the defntons that E s open. Theorem An open ball s an open set. Proof. Consder the open ball B( x, ε ). Let x be an arbtrary pont n B( x, ε ). Then ε d( xx, ) > and the open ball B( x, ε d( x, x )) s n B( x, ε ), because f y B( x, ε d( x, x )), then d( y, x) < ε d( x, x ) and, by (43.8), d( x, y) d( x, x) + d( x, y ), whch yelds d( x, y ) < ε and, thus, y B( x, ε ). A subset U of E s closed f ts complement, EU, s open. It can be shown that closed balls are ndeed closed sets. The empty set,, s closed because E = E s open. By the same logc E s closed snce EE = s open. In fact and E are the only subsets of E whch are both open and closed. A subset U E s bounded f t s contaned n some open ball. A subset U E s compact f t s closed and bounded. Theorem The unon of any collecton of open sets s open. Proof. Let { U α α I} be a collecton of open sets, where I s an ndex set. Assume that x α IUα. Then x must belong to at least one of the sets n the collecton, say U α. Snce U α s open, there exsts an open ball B( x, ε ) Uα α IUα. Thus, B( x, ε ) α IUα. Snce x s arbtrary, s open. U α I α Theorem43.4. The ntersecton of a fnte collecton of open sets s open. Proof. Let { },..., α n U U be a fnte famly of open sets. If U = s empty, the asserton s trval. Thus, assume n U s not empty and let x be an arbtrary element of n U. Then = = x U for =,..., n and there s an open ball B( x, ε ) U for =,..., n. Let ε be the smallest of the postve numbers,..., n n ε ε n. Then x B( x, ε ) = U. Thus U = s open.

15 Sec. 43 Eucldean Pont Spaces 3 It should be noted that arbtrary ntersectons of open sets wll not always lead to open sets. n, n, The standard counter example s gven by the famly of open sets of R of the form ( ) n =,2,3... The ntersecton ( ) s the set { } whch s not open. n n, n = By a sequence n E we mean a functon on the postve ntegers {,2,3,..., n,...} wth values n E. The notaton { x, x2, x3..., x n,... }, or smply { x n }, s usually used to denote the values of the sequence. A sequence { x n } s sad to converge to a lmt x E f for every open ball B( x, ε ) centered at x, there exsts a postve nteger n ( ) ε such that x B( x, ε ) whenever n n ( ε ). Equvalently, a sequence { x n } converges to x f for every real number ε > there exsts a postve nteger n ( ) ε such that d( x, x ) < ε for all n> n ( ε ). If { } x converges to x, t s conventonal to wrte n n n x = lm x or x x as n n n n Theorem If { x n } converges to a lmt, then the lmt s unque. Proof. Assume that x = lm x, y = lm x n n n n Then, from (43.8) d( x, y) d( xx, ) + d( x, y ) n n for every n. Let ε be an arbtrary postve real number. Then from the defnton of convergence of { x n }, there exsts an nteger n ( ε ) such that n n ( ε ) mples (, ) d xx n < ε and d( xn, y ) < ε. Therefore, d( xy, ) 2ε for arbtrary ε. Ths result mples d ( xy, ) = and, thus, x = y.

16 32 Chap. 9 EUCLIDEAN MANIFOLDS A pont x E s a lmt pont of a subset U E f every neghborhood of x contans a pont of U dstnct from x. Note that x need not be n U. For example, the sphere { x d( x, x ) = ε} are lmt ponts of the open ball B( x, ε ). The closure of U E, wrtten U, s the unon of U and ts lmt ponts. For example, the closure of the open ball B( x, ε ) s the closed ball B( x, ε ). It s a fact that the closure of U s the smallest closed set contanng U. Thus U s closed f and only f U = U. The reader s cautoned not to confuse the concepts lmt of a sequence and lmt pont of a subset. A sequence s not a subset of E ; t s a functon wth values n E. A sequence may have a lmt when t has no lmt pont. Lkewse the set of pnts whch represent the values of a sequence may have a lmt pont when the sequence does not converge to a lmt. However, these two concepts are related by the followng result from the theory of metrc spaces: a pont x s a lmt pont of a set U f and only f there exsts a convergent sequence of dstnct ponts of U wth x as a lmt. A mappng f : U E ', where U s an open set n E and E ' s a Eucldean pont space or an nner product space, s contnuous at x U f for every real number ε > there exsts a real number δ ( x, ε ) > such that d( xx, ) < δ ( ε, x ) mples d'( f( x), f( x )) < ε. Here d ' s the dstance functon for E '. When f s contnuous at x, t s conventonal to wrte lm f( x) or f( x) f( x ) as x x x x The mappng f s contnuous on U f t s contnuous at every pont of U. A contnuous mappng s called a homomorphsm f t s one-to-one and f ts nverse s also contnuous. What we have ust defned s sometmes called a homeomorphsm nto. If a homomorphsm s also onto, then t s called specfcally a homeomorphsm onto. It s easly verfed that a composton of two contnuous maps s a contnuous map and the composton of two homomorphsms s a homomorphsm. A mappng f : U E ', where U and E ' are defned as before, s dfferentable at there exsts a lnear transformaton A LV ( ; V ') such that x x U f f( x+ v) = f( x) + A v+ o( x, v ) (43.2) x

17 Sec. 43 Eucldean Pont Spaces 33 where lm o( x, v ) = v v (43.3) In the above defnton V ' denotes the translaton space of E '. Theorem The lnear transformaton A x n (43.2) s unque. Proof. If (43.2) holds for A x and A x, then by subtracton we fnd By (43.3), ( x x) ( Ax Ax) v = o( x, v ) o( x, v ) A A e must be zero for each unt vector e, so that A = A. x x If f s dfferentable at every pont of U, then we can defne a mappng grad f : U L ( V ; V '), called the gradent of f, by grad f ( x) = A, x U (43.4) x If grad f s contnuous on U, then f s sad to be of class C. If grad f exsts and s 2 r tself of class C, then f s of class C. More generally, f s of class C, r >, f t s of class r C r and ts ( r ) st gradent, wrtten grad f contnuous on U. If f s a r s called a C dffeomorphsm., s of class r C one-to-one map wth a C. Of course, f s of class r C nverse C f t s f defned on f ( U ), then f If f s dfferentable at x, then t follows from (43.2) that f( x+ τ v) f( x) d Av x = lm = f ( x+ τ v ) (43.5) τ τ dτ τ =

18 34 Chap. 9 EUCLIDEAN MANIFOLDS for all v V. To obtan (43.5) replace v by τ v, τ > n (43.2) and wrte the result as f( x+ τ v) f( x) o( x, τ v ) Av x = (43.6) τ τ By (43.3) the lmt of the last term s zero as τ, and (43.5) s obtaned. Equaton (43.5) holds for all v V because we can always choose τ n (43.6) small enough to ensure that x+ τ v s n U, the doman of f. If f s dfferentable at every x U, then (43.5) can be wrtten d grad ( x) v = f( x+ τ v ) (43.7) d ( f ) τ τ = A functon f : U R, where U s an open subset of E, s called a scalar feld. Smlarly, f : U V s a vector feld, and f : U T ( V ) s a tensor feld of order q. It should be noted that the term feld s defned here s not the same as that n Secton 7. q Before closng ths secton there s an mportant theorem whch needs to be recorded for later use. We shall not prove ths theorem here, but we assume that the reader s famlar wth the result known as the nverse mappng theorem n multvarable calculus. r Theorem Let f : U E ' be a C mappng and assume that grad f ( x ) s a lnear somorphsm. Then there exsts a neghborhood U of x such that the restrcton of f to U s a r C dffeomorphsm. In addton ( ) x x (43.8) grad f ( f( )) = grad f( ) Ths theorem provdes a condton under whch one can asert the exstence of a local nverse of a smooth mappng. Exercses 43. Let a sequence { x n } converge to x. Show that every subsequence of { n } converges to x. x also

19 Sec. 43 Eucldean Pont Spaces Show that arbtrary ntersectons and fnte unons of closed sets yelds closed sets Let f : U E ', where U s open n E, and E ' s ether a Eucldean pont space or an nner produce space. Show that f s contnuous on U f and only f f ( D ) s open n E for all D open n f ( U ) Let f : U E ' be a homeomorphsm. Show that f maps any open set n U onto an open set n E ' If f s a dfferentable scalar valued functon on LV ( ; V ), show that the gradent of f at A LV ( ; V ), wrtten f ( A) A s a lnear transformaton n LV ( ; V ) defned by f T df tr ( ) = + A AB d A B ( τ ) τ τ = for all B LV ( ; V ) 43.6 Show that μ μ ( ) = and N ( ) = ad A A I A A A ( ) T

20 36 Chap. 9 EUCLIDEAN MANIFOLDS Secton 44 Coordnate Systems r Gven a Eucldean pont space E of dmenson N, we defne a C -chart at x E to be a N r U, xˆ, where U s an open set n E contanng x and xˆ : U R s a C dffeomorphsm. U, xˆ, there are N scalar felds xˆ : U R such that par ( ) Gven any chart ( ) N ( ) xˆ( x) = xˆ ( x),..., xˆ ( x ) (44.) for all x U. We call these felds the coordnate functons of the chart, and the mappng ˆx s also called a coordnate map or a coordnate system on U. The set U s called the coordnate neghborhood. N N Two charts xˆ : U R and yˆ : U2 R, where U U 2, yeld the coordnate transformaton yˆ xˆ : xˆ( U ˆ U2) y( U U2) and ts nverse xˆ yˆ : yˆ( U ˆ U2) x( U U2). Snce N ( ) yˆ( x) = yˆ ( x),..., yˆ ( x ) (44.2) The coordnate transformaton can be wrtten as the equatons ( ˆ ˆ N ) ˆ ˆ ( ),..., ( ) ( ˆ ( ),..., ˆ N y x y x = y x x x x ( x) ) (44.3) and the nverse can be wrtten ( ˆ ˆ N ) ˆ ˆ ( ),..., ( ) ( ˆ ( ),..., ˆ N x x x x = x y y x y ( x) ) (44.4) The component forms of (44.3) and (44.4) can be wrtten n the smplfed notaton N k y = y ( x,..., x ) y ( x ) (44.5)

21 Sec. 44 Coordnate Systems 37 and N k x = x ( y,..., y ) x ( y ) (44.6) N N The two N-tuples ( y,..., y ) and ( x,..., x ), where y = yˆ ( x ) and x = xˆ ( x ), are the coordnates of the pont x U U 2. Fgure 6 s useful n understandng the coordnate transformatons. N R U U 2 U U 2 xˆ( U U ) 2 ˆx ŷ yˆ( U U ) 2 yˆ ˆ x xˆ ˆ y Fgure 6 It s mportant to note that the quanttes N x,..., x, N y,..., y are scalar felds,.e., realvalued functons defned on certan subsets of E. Snce U and U 2 are open sets, U U 2 s open, r and because ˆx and ŷ are ˆx ŷ U U are open subsets of C dffeomorphsms, ( U U ) and ( ) N R. In addton, the mappng yˆ xˆ and xˆ yˆ are dffeomorphsm, equaton (44.) wrtten n the form 2 2 r C dffeomorphsms. Snce ˆx s a N ˆ( ) (,..., ) x x = x x (44.7) can be nverted to yed

22 38 Chap. 9 EUCLIDEAN MANIFOLDS (,..., N x x ) ( x ) (44.8) x = x = x where x s a dffeomorphsm x : xˆ ( U) U. r r A C -atlas on E s a famly (not necessarly countable) of - where I s an ndex set, such that {, x I α α α } C charts ( ˆ ) U, E = Uα (44.9) α I Equaton (44.9) states that E s covered by the famly of open sets { U α α I}. A r C -Eucldean r manfold s a Eucldean pont space equpped wth a C -atlas. A C -atlas and a C -Eucldean manfold are defned smlarly. For smplcty, we shall assume that E s C. A C curve n E s a C mappng λ : ( ab, ) E, where ( ab, ) s an open nterval of R. A C curve λ passes through x E f there exsts a c ( a, b ) such that λ( c ) = x. Gven a chart ( U, xˆ ) and a pont x U, the th coordnate curve passng through x s the curve λ defned by + N λ ( t) = x ( x,..., x, x + t, x,..., x ) (44.) + N k for all t such that ( x ˆ,..., x, x + t, x,..., x ) x( U ), where ( x ˆ ) = x( x ). The subset of U obtaned by requrng x = xˆ ( x ) = const (44.) s called the th coordnate surface of the chart Eucldean manfolds possess certan specal coordnate systems of maor nterest. Let be an arbtrary bass, not necessarly orthonormal, for V. We defne N constant vector {,..., N } felds : E V, =,..., N, by the formulas

23 Sec. 44 Coordnate Systems 39 ( x) =, x U (44.2) The use of the same symbol for the vector feld and ts value wll cause no confuson and smplfes the notaton consderably. If E denotes a fxed element of E, then a Cartesan coordnate system 2 N on E s defned by the N scalar felds zˆ, zˆ,..., z ˆ such that z = zˆ ( x) = ( x E ), x E (44.3) If the bass {,..., N } s orthonormal, the Cartesan system s called a rectangular Cartesan system. The pont E s called the orgn of the Cartesan coordnate system. The vector feld defned by rx ( )= x E (44.4) for all If { },..., N x E s the poston vector feld relatve to E. The value rx ( ) s the poston vector of x. s the bass recprocal to {,..., N }, then (44.3) mples x = x = ˆ x = (44.5) E N ( z,..., z ) E z ( ) z Defnng constant vector felds,..., N as before, we can wrte (44.5) as ˆ r = z (44.6) The product of the scalar feld z ˆ wth the vector feld n (44.6) s defned pontwse;.e., f f s a scalar feld and v s a vector feld, then fv s a vector feld defned by fvx ( ) = f( xvx ) ( ) (44.7) for all x n the ntersecton of the domans of f and v. An equvalent verson of (44.3) s ˆ z = r (44.8)

24 3 Chap. 9 EUCLIDEAN MANIFOLDS where the operator r between vector felds s defned pontwse n a smlar fashon as n 2 N,,..., be another bass for V related to the orgnal bass (44.7). As an llustraton, let { } by k = Q k (44.9) and let E be another fxed pont of E. Then by (44.3) z = ( x ) = ( x ) + ( ) E E E E = Q ( x ) + ( ) = Q z + c k k k E E E k (44.2) where (44.9) has been used. Also n (44.2) the constant scalars k c, k,..., = N, are defned by k k c = ( E E ) (44.2) 2 If the bases {,,..., N 2 N } and {,,..., } are both orthonormal, then the matrx orthogonal. Note that the coordnate neghborhood s the entre space E. Q k s curve The th coordnate curve whch passes through E of the Cartesan coordnate system s the λ() t = t + E (44.22) 2 N Equaton (44.22) follows from (44.), (44.5), and the fact that for x = E, z = z = = z =. As (44.22) ndcates, the coordnate curves are straght lnes passng through E. Smlarly, the th coordnate surface s the plane defned by

25 Sec. 44 Coordnate Systems 3 ( x ) = const E 3 z 3 E 2 2 z z Fgure 7 Geometrcally, the Cartesan coordnate system can be represented by Fgure 7 for N=3. The coordnate transformaton represented by (44.2) yelds the result n Fgure 8 (agan for N=3). Snce every nner product space has an orthonormal bass (see Theorem 3.3), there s no loss of generalty n assumng that assocated wth every pont of E as orgn we can ntroduce a rectangular Cartesan coordnate system.

26 32 Chap. 9 EUCLIDEAN MANIFOLDS 3 z 3 z 3 r r 3 z E 2 2 z z E 2 2 z Fgure 8 N Gven any rectangular coordnate system ( zˆ,..., z ˆ ), we can characterze a general or a curvlnear coordnate system as follows: Let (, xˆ ) U be a chart. Then t can be specfed by the coordnate transformaton from ẑ to ˆx as descrbed earler, snce n ths case the overlap of the coordnate neghborhood s U = E U. Thus we have z z = zˆ xˆ x x N N (,..., ) (,..., ) x x = xˆ zˆ z z N N (,..., ) (,..., ) (44.23) where zˆ xˆ : xˆ( U) zˆ( U) and of (44.23) are ˆ ˆ : ˆ( ) ˆ( U) are dffeomorphsms. Equvalent versons x z zu x z = z x x = z x N k (,..., ) ( ) x = x z z = x z N k (,..., ) ( ) (44.24)

27 Sec. 44 Coordnate Systems 33 As an example of the above deas, consder the cylndrcal coordnate system. In ths case N=3 and equatons (44.23) take the specal form ( z, z, z ) ( x cos x, x sn x, x ) = (44.25) In order for (44.25) to qualfy as a coordnate transformaton, t s necessary for the transformaton functons to be C. It s apparent from (44.25) that ẑ xˆ s C 3 on every open subset of R. Also, by examnaton of (44.25), ẑ xˆ s one-to-one f we restrct t to an approprate doman, say 3 (, ) (,2 π ) (, ). The mage of ths subset of R under ẑ xˆ s easly seen to be the set {( z, z, z ) z, z = } R and the nverse transformaton s /2 z 3 ( x, x, x ) = ( z ) + ( z ),tan, z z (44.26) whch s also C. Consequently we can choose the coordnate neghborhood to be any open subset U n E such that { } zˆ( ) ( z, z, z ) z, z = U R (44.27) or, equvalently, xˆ( U ) (, ) (,2 π ) (, ) Fgure 9 descrbes the cylndrcal system. The coordnate curves are a straght lne (for x ), a 2 2 crcle lyng n a plane parallel to the ( z, z ) plane (for x ), and a straght lne concdent wth 3 3 (for x ). The coordnate surface x = const s a crcular cylnder whose generators are the z lnes. The remanng coordnate surfaces are planes. 3 z The computer program Maple has a plot command, coordplot3d, that s useful when tryng to vsualze coordnate curves and coordnate surfaces. The program MATLAB wll also produce useful and nstructve plots.

28 34 Chap. 9 EUCLIDEAN MANIFOLDS z = x z z 2 x Fgure 9 x Returnng to the general transformatons (44.5) and (44.6), we can substtute the second nto the frst and dfferentate the result to fnd y x x y k N N ( x,..., x ) ( y,..., y ) = δ k (44.28) By a smlar argument wth x and y nterchanged, y x x y k N N ( x,..., x ) ( y,..., y ) = δ k (44.29) Each of these equatons ensures that y N det ( x,..., x ) k x = x N det ( y,..., y ) l y (44.3) For example, ( k )( ) k N determnant det ( x y )( y,..., y ) 2 3 det z x x, x, x = x for the cylndrcal coordnate system. The s the Jacoban of the coordnate transformaton (44.6).

29 Sec. 44 Coordnate Systems 35 Just as a vector space can be assgned an orentaton, a Eucldean manfold can be orented by assgnng the orentaton to ts translaton space V. In ths case E s called an orented Eucldean manfold. In such a manfold we use Cartesan coordnate systems assocated wth postve bass only and these coordnate systems are called postve. A curvlnear coordnate system s postve f ts coordnate transformaton relatve to a postve Cartesan coordnate system has a postve Jacoban. Gven a chart (, xˆ ) and obtan a C U for E, we can compute the gradent of each coordnate functon x ˆ vector feld on U. We shall denote each of these felds by g, namely g = grad xˆ (44.3) for =,..., N. From ths defnton, t s clear that g ( x ) s a vector n V normal to the th coordnate surface. From (44.8) we can defne N vector felds g,..., g N on U by [ ] g = grad x (,...,,...,) (44.32) Or, equvalently, g = t x N N x( x,..., x + t,..., x ) x( x,..., x ) x N lm ( x,..., x ) t (44.33) for all Snce x U. Equatons (44.) and (44.33) show that g ( x ) s tangent to the th coordnate curve. xˆ ( ( x,..., x N )) = x x (44.34) The chan rule along wth the defntons (44.3) and (44.32) yeld g ( x) g ( x ) = δ (44.35) as they should.

30 36 Chap. 9 EUCLIDEAN MANIFOLDS The values of the vector felds { g,..., g N } form a lnearly ndependent set of vectors { g x g x } at each x U. To see ths asserton, assume x U and ( ),..., ( ) N N λ g ( x) + λ g ( x) + + λ g ( x) = 2 2 N for N λ,..., λ R. Takng the nner product of ths equaton wth g ( x ) and usng equaton (44.35), we see that λ =, =,..., N whch proves the asserton. Because V has dmenson N, { g ( ),..., ( ) x g N x } forms a bass for V at each Equaton (44.35) shows that { g ( x),..., g N ( x )} s the bass recprocal to { ( ),..., ( ) N } of the specal geometrc nterpretaton of the vectors { } N x U. g x g x. Because g ( ),..., ( ) x g N x and { g ( x ),..., g ( x ) } mentoned above, these bases are called the natural bases of ˆx at x. Any other bass feld whch cannot be determned by ether (44.3) or (44.32) relatve to any coordnate system s called an anholonomc or nonntegrable bass. The constant vector felds {,..., N } and {,..., N } yeld the natural bases for the Cartesan coordnate systems. U are two charts such that U U 2, we can determne the transformaton rules for the changes of natural bases at x U U 2 n the followng way: We shall let the vector felds h, =,..., N, be defned by If ( U, x ) and (, y ) ˆ 2 ˆ h = grad yˆ (44.36) Then, from (44.5) and (44.3), h x = x y = x x y N = ( x,..., x ) g ( x) x N ( ) grad yˆ ( ) ( x,..., x )grad xˆ ( ) (44.37) for all x U U 2. A smlar calculaton shows that

31 Sec. 44 Coordnate Systems 37 x N h ( x) = ( y,..., y ) g( x ) (44.38) y for all x U U 2. Equatons (44.37) and (44.38) are the desred transformatons. Gven a chart (, x ) U, we can defne 2 2N scalar felds g : U R and g : U ˆ R by g ( x) = g ( x) g ( x) = g ( x ) (44.39) and g ( x) = g ( x) g ( x) = g ( x ) (44.4) for all x U. It mmedately follows from (44.35) that g ( x) = g ( x ) (44.4) snce we have g = g g (44.42) and g = g g (44.43) where the produce of the scalar felds wth vector felds s defned by (44.7). If θ s the angle between the th and the th coordnate curves at x U, then from (44.39) g( x) cos θ = g( x) g ( x) /2 (no sum) (44.44)

32 38 Chap. 9 EUCLIDEAN MANIFOLDS Based upon (44.44), the curvlnear coordnate system s orthogonal f g = when. The symbol g denotes a scalar feld on U defned by g( x) = det g ( x ) (44.45) for all x U. At the pont x U, the dfferental element of arc ds s defned by 2 ds = d d x x (44.46) and, by (44.8), (44.33), and (44.39), x x ˆ x x = g ( x) dx dx 2 ds = ( x( x)) ( x( x)) dx dx ˆ (44.47) If ( U, x ) and (, y ) ˆ U are charts where U U 2, then at x U U 2 2 ˆ h ( x) = h ( x) h ( x) x x = y y k l N N ( y,..., y ) ( y,..., y ) gkl ( x) (44.48) Equaton (44.48) s helpful for actual calculatons of the quanttes g ( x ). For example, for the transformaton (44.23), (44.48) can be arranged to yeld k k z N z N g( x ) = ( x,..., x ) ( x,..., x ) (44.49) x x snce k l kl =δ. For the cylndrcal coordnate system defned by (44.25) a smple calculaton based upon (44.49) yelds

33 Sec. 44 Coordnate Systems 39 2 g ( ) ( x ) x = (44.5) Among other thngs, (44.5) shows that ths coordnate system s orthogonal. By (44.5) and (44.4) 2 g ( ) ( x ) x = (44.5) And, from (44.5) and (44.45), g( ) ( ) 2 x = x (44.52) It follows from (44.5) and (44.47) that ds = ( dx ) + ( x ) ( dx ) + ( dx ) (44.53) Exercses 44. Show that x k = and = grad ˆ z k k z k for any Cartesan coordnate system ẑ assocated wth { } Show that I= grad r( x )

34 32 Chap. 9 EUCLIDEAN MANIFOLDS 44.3 Sphercal coordnates 2 3 ( x, x, x ) are defned by the coordnate transformaton z = x sn x cos x 2 3 z = x sn x sn x z = x cos x relatve to a rectangular Cartesan coordnate system ẑ. How must the quantty ( x, x, x ) be restrcted so as to make ẑ xˆ one-to-one? Dscuss the coordnate curves and the coordnate surfaces. Show that 2 g ( ) ( x ) x = 2 2 ( x sn x ) 44.4 Parabolodal coordnates 2 3 ( x, x, x ) are defned by z = x x cos x 2 3 z = x x sn x z = ( x ) ( x ) 2 ( ) Relatve to a rectangular Cartesan coordnate system ẑ. How must the quantty ( x, x, x ) be restrcted so as to make ẑ xˆ one-to-one. Dscuss the coordnate curves and the coordnate surfaces. Show that ( x ) + ( x ) g( x ) = ( x ) + ( x ) 2 2 ( xx) 44.5 A bsphercal coordnate system coordnate system by 2 3 ( x, x, x ) s defned relatve to a rectangular Cartesan

35 Sec. 44 Coordnate Systems 32 z z asn x cos x = cosh x cos x 2 3 asn x sn x = cosh x cos x 2 2 and z 3 asnh x = cosh x cos x where a >. How must ( x, x, x ) be restrcted so as to make ẑ xˆ one-to-one? Dscuss the coordnate curves and the coordnate surfaces. Also show that a 2 ( cosh x cos x ) g( x ) = 2 a 2 ( cosh x cos x ) a (sn x ) 2 ( cosh x cos x ) Prolate spherodal coordnates 2 3 ( x, x, x ) are defned by z = asnh x sn x cos x 2 3 z = asnh x sn x sn x z = acosh x cos x 3 2 relatve to a rectangular Cartesan coordnate system ẑ, where a >. How must 2 3 ( x, x, x ) be restrcted so as to make ẑ xˆ one-to-one? Also dscuss the coordnate curves and the coordnate surfaces and show that

36 322 Chap. 9 EUCLIDEAN MANIFOLDS ( ) a cosh x cos x g( x ) = a ( cosh x cos x ) a snh x sn x 44.7 Ellptcal cylndrcal coordnates coordnate system by 2 3 ( x, x, x ) are defned relatve to a rectangular Cartesan z = acosh x cos x 2 z = asnh x sn x z 2 2 = x where a >. How must ( x, x, x ) be restrcted so as to make ẑ xˆ one-to-one? Dscuss the coordnate curves and coordnate surfaces. Also, show that ( + ) a snh x sn x g( x ) = a ( snh x + sn x ) 44.8 For the cylndrcal coordnate system show that g = (cos x ) + (sn x ) g = x (sn x ) + x (cos x ) g = At a pont x n E, the components of the poston vector rx ( ) = x E wth respect to the bass {,..., N } assocated wth a rectangular Cartesan coordnate system are N z,..., z. Ths observaton follows, of course, from (44.6). Compute the components of rx ( ) wth respect to the bass { g( x), g2( x), g3( x )} for (a) cylndrcal coordnates, (b) sphercal coordnates, and (c) parabolc coordnates. You should fnd that

37 Sec. 44 Coordnate Systems 323 rx ( ) = x g( x) + x g( x) for (a) rx 3 3 ( ) = x g( x) for (b) 2 rx ( ) = x g( x) + x g2( x) for (c) Torodal coordnates system by 2 3 ( x, x, x ) are defned relatve to a rectangular Cartesan coordnate z z 2 3 asnh x cos x = 2 cosh x cos x 3 asnh x sn x = 2 cosh x cos x and z 3 2 asn x = cosh x cos x where a >. How must ( x, x, x ) be restrcted so as to make the coordnate surfaces. Show that ẑ xˆ one to one? Dscuss a 2 ( cosh x cos x ) g( x ) = 2 a 2 ( cosh x cos x ) a snh x ( cos x cos x ) 2

38 324 Chap. 9 EUCLIDEAN MANIFOLDS Secton 45. Transformaton Rules for Vectors and Tensor Felds In ths secton, we shall formalze certan deas regardng felds on E and then nvestgate the transformaton rules for vectors and tensor felds. Let U be an open subset of E ; we shall denote by F ( U ) the set of C functons f : U R. Frst we shall study the algebrac structure of F ( U ). If f and f 2 are n F ( U ), then ther sum f+ f2 s an element of F ( U ) defned by ( f + f )( x) = f ( x) + f ( x ) (45.) 2 2 and ther produce ff 2 s also an element of F ( U ) defned by ( ff)( x) = f( x) f( x ) (45.2) 2 2 for all x U. For any real number λ R the constant functon s defned by λ( x ) = λ (45.3) for all x U. For smplcty, the functon and the value n (45.3) are ndcated by the same symbol. Thus, the zero functon n F ( U ) s denoted smply by and for every f F ( U ) f + = f (45.4) It s also apparent that f = f (45.5) In addton, we defne f = ( ) f (45.6)

39 Sec. 45 Transformaton Rules 325 It s easly shown that the operatons of addton and multplcaton obey commutatve, assocatve, and dstrbutve laws. These facts show that F ( U ) s a commutatve rng (see Secton 7). An mportant collecton of scalar felds can be constructed as follows: Gven two charts 2 U ˆ 2, y, where U U 2, we defne the y x x,..., x N 2 x,..., x xˆ U U. Usng a suggestve notaton, we can defne N C functons ( U, x ) and ( ) ˆ at every ( ) ( ) y x : U U R by 2 2 N partal dervatves ( )( N ) y x y ( x) = xˆ ( x) (45.7) x for all x U U 2. As mentoned earler, a C vector feld on an open set U of E s a C map v : U V, where V s the translaton space of E. The felds defned by (44.3) and (44.32) are specal cases of vector felds. We can express v n component forms on U U 2, v = υ g = υ g (45.8) υ = g υ (45.9) As usual, we can computer υ : U U R by υ ( x) = v( x) g ( x), x U U (45.) 2 and component form of υ by (45.9). In partcular, f (, y) U s another chart such that U U 2, then the 2 ˆ h relatve to ˆx s x h = g (45.) y

40 326 Chap. 9 EUCLIDEAN MANIFOLDS Wth respect to the chart ( U, y) 2 ˆ, we have also k v=υ h k (45.2) k where υ : U U2 R. From (45.2), (45.), and (45.8), the transformaton rule for the components of v relatve to the two charts ( U, x ) and (, y ) ˆ U s 2 ˆ υ x y = υ (45.3) for all x U U2 U. As n (44.8), we can defne an nner product operaton between vector felds. If v: U V and v2: U2 V are vector felds, then v v 2 s a scalar feld defned on U U 2 by v v ( x) = v ( x) v ( x), x U U (45.4) Then (45.) can be wrtten υ = v g (45.5) Now let us consder tensor felds n general. Let T q ( U ) denote the set of all tensor felds of order q defned on an open set U n E. As wth the set F ( U ), the set T q ( U ) can be assgned an algebrac structure. The sum of A : U Tq( V ) and B : U Tq( V ) s a C tensor feld A+ B : U T ( V ) defned by q ( A+ B)( x) = A( x) + B ( x) (45.6) for all x U. If f F ( U ) and A T ( U ), then we can defne fa T ( U ) by q q

41 Sec. 45 Transformaton Rules 327 fa( x) = f( x) A ( x) (45.7) Clearly ths multplcaton operaton satsfes the usual assocatve and dstrbutve laws wth respect to the sum for all x U. As wth F ( U ), constant tensor felds n T ( U ) are gven the same symbol as ther value. For example, the zero tensor feld s : U T ( V ) and s defned by q q ( x)= (45.8) for all x U. If s the constant functon n F ( U ), then A = ( ) A (45.9) The algebrac structure on the set T q ( V ) ust defned s called a module over the rng F ( U ). scalar felds The components of a tensor feld : U T ( V ) A : U U R defned by... q A wth respect to a chart (, x ) q U are the q N ˆ A... ( x) = A ( x)( g ( x),..., g ( x)) (45.2) q q for all x U U. Clearly we can regard tensor felds as multlnear mappngs on vector felds wth values as scalar felds. For example, A ( g,..., g ) s a scalar feld defned by q A ( g,..., g )( x ) = A ( x )( g ( x ),..., g ( x )) q q for all x U U. In fact we can, and shall, carry over to tensor felds the many algebrac operatons prevously appled to tensors. In partcular a tensor feld A : U T ( V ) has the representaton q q A... g g q A = (45.2)

42 328 Chap. 9 EUCLIDEAN MANIFOLDS U s the coordnate neghborhood for a chart ( ) for all x U U, where U, x. The scalar felds A are the covarant components of A and under a change of coordnates obey the... q transformaton rule ˆ A x x = y y q A k... kq k kq... q (45.22) Equaton (45.22) s a relatonshp among the component felds and holds at all ponts x U U2 U where the charts nvolved are ( U, xˆ ) and ( U ˆ ) 2, y. We encountered an example of (45.22) earler wth (44.48). Equaton (44.48) shows that the g are the covarant components of a tensor feld I whose value s the dentty or metrc tensor, namely I = g g g = g g = g g = g g g (45.23) for ponts U, x. Equatons (45.23) show that the components of a constant tensor feld are not necessarly constant scalar felds. It s only n Cartesan coordnates that constant tensor felds have constant components. x U, where the chart n queston s ( ) ˆ Another mportant tensor feld s the one constructed from the postve unt volume tensor, whch has postve orentaton, E s gven by (4.6), E. Wth respect to an orthonormal bass { }.e., E = = (45.24) N ε N N Gven ths tensor, we defne as usual a constant tensor feld E : E Tˆ ( V ) by N E( x)= E (45.25) for all x E. Wth respect to a chart ( U, x ), t follows from the general formula (42.27) that ˆ E = E = (45.26) N N E g g g N g N

43 Sec. 45 Transformaton Rules 329 where N E and E are scalar felds on U defned by N E = e gε (45.27) N N and E e = g ε (45.28) N N where, as n Secton 42, e s + f { ( )} g x s postvely orented and f { ( )} g x s negatvely orented, and where g s the determnant of g as defned by (44.45). By applcaton of (42.28), t follows that N N N E g g E N = (45.29) An nterestng applcaton of the formulas derved thus far s the dervaton of an expresson for the dfferental element of volume n curvlnear coordnates. Gven the poston vector r U, x, the dfferental of r can be wrtten defned by (44.4) and a chart ( ) ˆ dr = dx = g ( x ) dx (45.3) where (44.33) has been used. Gven N dfferentals of r, dr, dr2,, dr N, the dfferental volume element dυ generated by them s defned by ( r r ) dυ = E d,, d N (45.3) or, equvalently, dυ = dr dr (45.32) N

44 33 Chap. 9 EUCLIDEAN MANIFOLDS If we select N dr = g ( x) dx, dr = g ( x) dx,, dr = g ( x ) dx, we can wrte (45.3)) as N N 2 ( g x g x ) ( ),, ( ) N dυ = E N dx dx dx (45.33) By use of (45.26) and (45.27), we then get ( g ( x),, g ( x) ) E = E = e g (45.34) N 2 N Therefore, 2 N dυ = gdx dx dx (45.35) For example, n the parabolc coordnates mentoned n Exercse 44.4, (( ) ( ) ) 2 2 d x x x x dx dx dx υ = + (45.36) Exercses 45. Let v be a C vector feld and f be a C functon both defned on U an open set n E. We defne v f : U R by v f( x) = v( x) grad f( x), x U (45.37) Show that ( λf + μg) = λ( f ) + μ( g) v v v and

45 Sec. 45 Transformaton Rules 33 ( fg) = ( f ) g+ f ( g) v v v For all constant functons λ, μ and all C functons f and g. In dfferental geometry, an operator on F ( U ) wth the above propertes s called a dervaton. Show that, conversely, every dervaton on F ( U ) corresponds to a unque vector feld on U by (45.37) By use of the defnton (45.37), the Le bracket of two vector felds v : U V and : uv,, s a vector feld defned by u U V, wrtten [ ] [ uv, ] f = u ( v f ) v ( u f ) (45.38) For all scalar felds f F ( U ). Show that [, ] defnes a dervaton on [ uv., ] Also, show that uv s well defned by verfyng that (45.38) [, ] = ( grad ) ( grad ) uv vu u v and then establsh the followng results: (a) [ uv, ] = [ vu, ] v u w + u w v + w v u = (b),[, ],[, ],[, ] for all vector felds uv, and w. ˆ g. Show that g, g =. The results (a) and (b) are known as Jacob s denttes In a three-dmensonal Eucldean space the dfferental element of area normal to the plan formed from dr and dr 2 s defned by (c) Let ( U, x ) be a chart wth natural bass feld { } dσ = dr dr 2 Show that k dσ = E dxdx2 g ( x ) k

46 332 Chap. 9 EUCLIDEAN MANIFOLDS Secton 46. Anholonomc and Physcal Components of Tensors In many applcatons, the components of nterest are not always the components wth g. For defnteness let us call the components of a respect to the natural bass felds { } g and { } tensor feld A T q ( U ) s defned by (45.2) the holonomc components of A. In ths secton, we shall consder brefly the concept of the anholonomc components of A ;.e., the components of A taken wth respect to an anholonomc bass of vector felds. The concept of the physcal components of a tensor feld s a specal case and wll also be dscussed. Let e a denote a set of N vectors felds on U, whch are lnearly ndependent,.e., at each x U, { e a } s a bass of V. If A s a tensor feld n T q ( U ) where U U, then by the same type of argument as used n Secton 45, we can wrte U be an open set n E and let { } a a q A aa 2... a e e q A = (46.) or, for example, b... b q A = A e e (46.2) b bq where { a } e s the recprocal bass feld to { } e defned by a a a e ( x) e ( x ) = δ (46.3) b b for all x U. Equatons (46.) and (46.2) hold on U U, and the component felds as defned by A... = A ( e,..., e ) (46.4) aa 2 aq a aq and b... bq b b q A = A ( e,..., e ) (46.5)

47 Sec. 46 Anholonomc and Physcal Components 333 are scalar felds on U U. These felds are the anholonomc components of A when the bases e are not the natural bases of any coordnate system. { a } e and { } a Gven a set of N vector felds { e a } as above, one can show that a necessary and suffcent condton for { e a } to be the natural bass feld of some chart s [, ] e e a b = for all ab, =,..., N, where the bracket product s defned n Exercse We shall prove ths mportant result n Secton 49. Formulas whch generalze (45.22) to anholonomc components can easly be derved. If { e ˆ a } s an anholonomc bass feld defned on an open set U 2 such that U U 2, then we can express each vector feld e ˆ b n anholonomc component form relatve to the bass { e a }, namely a eˆ = T e (46.6) b b a where the T are scalar felds on U U 2 defned by a b a T = eˆ e b b a The nverse of (46.6) can be wrtten ˆ b e = T e ˆ (46.7) a a ba where ˆ b ( ) a ( ) b T x T x = δ (46.8) a c c and

48 334 Chap. 9 EUCLIDEAN MANIFOLDS a ( ) ˆ c a T x T ( x ) = δ (46.9) c b b for all x U U 2. It follows from (46.4) and (46.7) that A = T T Aˆ (46.) ˆb b ˆ q a... aq a aq b... bq where A ˆ = A ( eˆ,..., eˆ ) (46.) b... bq b bq Equaton (46.) s the transformaton rule for the anholonomc components of A. Of course, (46.) s a feld equaton whch holds at every pont of U U2 U. Smlar transformaton rules for the other components of A can easly be derved by the same type of argument used above. We defne the physcal components of A, denoted by A a,..., aq, to be the anholonomc components of A relatve to the feld of orthonomal bass { g } whose bass vectors g are unt vectors n the drecton of the natural bass vectors g of an orthogonal coordnate system. Let (, x ) U by such a coordnate system wth g =,. Then we defne ˆ g ( x) g ( x) g ( x ) (no sum) (46.2) = at every x U. By (44.39), an equvalent verson of (46.2) s g x = g x g x (46.3) 2 ( ) ( ) ( ( )) (no sum) Snce { } g s orthogonal, t follows from (46.3) and (44.39) that { } g s orthonormal: g g = δ a b ab (46.4)

49 Sec. 46 Anholonomc and Physcal Components 335 as t should, and t follows from (44.4) that g( x) g22( x) g ( x) = gnn ( x) (46.5) Ths result shows that g can also be wrtten g ( x) g x x g x ( g ( x)) 2 ( ) = = ( g ( )) ( ) (no sum) 2 (46.6) Equaton (46.3) can be vewed as a specal case of (46.7), where g g22 ˆ b Ta = g NN By the transformaton rule (46.), the physcal components of A are related to the covarant components of A by (46.7) ( ) 2 ( g, g,..., g ) = aa a... (no sum) aa a a a q aq qaq a aq A A g g A (46.8) Equaton (46.8) s a feld equaton whch holds for all x U. Snce the coordnate system s orthogonal, we can replace (46.8) wth several equvalent formulas as follows:

50 336 Chap. 9 EUCLIDEAN MANIFOLDS aa 2... aq 2 a... aq ( aa ) aqaq ( ) ( ) q q A = g g A 2 2 a2... aq aa aa 2 2 a a a = g g g A ( ) ( ) 2 2 aq aa aq aq aqaq a... aq = g g g A (46.9) In mathematcal physcs, tensor felds often arse naturally n component forms relatve to e are felds of bases, product bases assocated wth several bases. For example, f { } a ˆ b e and { } possbly anholonomc, then t mght be convenent to express a second-order tensor feld A as a feld of lnear transformatons such that bˆ a aeb A e = A ˆ, a =,..., N (46.2) In ths case A has naturally the component form bˆ a b a A = A eˆ e (46.2) a Relatve to the product bass { eˆ b e } formed by { e ˆ b } and { e a } bass of { e a } as usual. For defnteness, we call { ˆ a b } wth the bases { e ˆ b } and { e a }. Then the scalar felds called the composte components of A, and they are gven by, the latter beng the recprocal e e a composte product bass assocated ˆb A a defned by (46.2) or (46.2), may be bˆ a b A = A ( eˆ e ) (46.22) a Smlarly we may defne other types of composte components, e.g., ba ˆ b a Aˆ = A( eˆ, e ), A = A ( eˆ, e ) (46.23) ba b a etc., and these components are related to one another by

51 Sec. 46 Anholonomc and Physcal Components 337 A = A gˆ = A g = A gˆ g (46.24) cˆ c cd ˆ ba ˆ a cb ˆˆ bˆ ca bc ˆˆ da etc. Further, the composte components are related to the regular tensor components assocated wth a sngle bass feld by c ˆ cˆ ˆ ˆ, ba ca b bc ˆˆ a A = A T = A T etc. (46.25) a where T ˆ and { } ˆ b b ˆ a ˆ b T are gven by (46.6) and (46.7) as before, In the specal case where { } e and e are orthogonal but not orthonormal, we defne the normalzed bass vectors e and e ˆ a a as before. Then the composte physcal components A ba of A are gven by ˆ, a A = A ( e ˆ, e ) (46.26) ba ˆ, bˆ a and these are related to the composte components by ba ˆ, bˆ aa ( ˆ ˆˆ) a ( ) 2 2 A = g A g (no sum) (46.27) bb Clearly the concepts of composte components and composte physcal components can be defned for hgher order tensors also. Exercses 46. In a three-dmensonal Eucldean space the covarant components of a tensor feld A relatve to the cylndrcal coordnate system are A. Determne the physcal components of A Relatve to the cylndrcal coordnate system the helcal bass { e a } has the component form

52 338 Chap. 9 EUCLIDEAN MANIFOLDS e = g ( cosα) ( snα) ( snα) ( cosα) e = g + g e = g + g (46.28) where α s a constant called the ptch, and where { g α } s the orthonormal bass assocated wth the natural bass of the cylndrcal system. Show that { e a } s anholonomc. Determne the anholonomc components of the tensor feld A n the precedng exercse relatve to the helcal bass Determne the composte physcal components of A relatve to the composte product bass e g { a b }

53 Sec. 47 Chrstoffel Symbols, Covarant Dfferentaton 339 Secton 47. Chrstoffel Symbols and Covarant Dfferentaton In ths secton we shall nvestgate the problem of representng the gradent of varous tensor felds n components relatve to the natural bass of arbtrary coordnate systems. We consder frst the smple case of representng the tangent of a smooth curve n E. Let λ :( ab, ) E be a smooth curve passng through a pont x, say x = λ ( c). Then the tangent vector of λ at x s defned by dλ() t λ = (47.) x dt = t c Gven the chart (, xˆ ) { g } of ˆx. Frst, the coordnates of the curve λ are gven by U coverng x, we can proect the vector equaton (47.) nto the natural bass N N ( ) = ( λ λ ) ( ) = ( λ λ ) xˆ λ( t) ( t),..., ( t), λ t x ( t),..., ( t) (47.2) for all t such that λ () t U. Dfferentatng (47.2) 2 wth respect to t, we get x dλ λ = (47.3) x x dt t= c By (47.3) ths equaton can be rewrtten as dλ λ = g ( x) (47.4) x dt Thus, the components of λ relatve to { g } are smply the dervatves of the coordnate representatons of λ n ˆx. In fact (44.33) can be regarded as a specal case of (47.3) when λ concdes wth the th coordnate curve of ˆx. t= c An mportant consequence of (47.3) s that