UNIT 7. THE FUNDAMENTAL EQUATIONS OF HYPERSURFACE THEORY

Similar documents
Estimation of a proportion under a certain two-stage sampling design

A A Non-Constructible Equilibrium 1

Improvements on Waring s Problem

Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder

Scattering of two identical particles in the center-of. of-mass frame. (b)

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction

and decompose in cycles of length two

Design of Recursive Digital Filters IIR

Harmonic oscillator approximation

Variable Structure Control ~ Basics

Chapter 5: Root Locus

RICCI TYPE IDENTITIES FOR BASIC DIFFERENTIATION AND CURVATURE TENSORS IN OTSUKI SPACES 1

corresponding to those of Heegaard diagrams by the band moves

Root Locus Techniques

Additional File 1 - Detailed explanation of the expression level CPD

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted

Statistical Properties of the OLS Coefficient Estimators. 1. Introduction

we have E Y x t ( ( xl)) 1 ( xl), e a in I( Λ ) are as follows:

Improvements on Waring s Problem

Monica Purcaru and Nicoleta Aldea. Abstract

Dmitry A. Zaitsev Odessa National Telecommunication Academy Kuznechnaya, 1, Odessa, Ukraine

WEYL MANIFOLDS WITH SEMI-SYMMETRIC CONNECTION. Füsun Ünal 1 and Aynur Uysal 2. Turkey.

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

APPENDIX A Some Linear Algebra

Power-sum problem, Bernoulli Numbers and Bernoulli Polynomials.

a new crytoytem baed on the dea of Shmuley and roved t rovably ecure baed on ntractablty of factorng [Mc88] After that n 999 El Bham, Dan Boneh and Om

Quantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry

Small signal analysis

Integral Formula of Minkowski Type and New Characterization of the Wulff Shape

Pythagorean triples. Leen Noordzij.

6. Hamilton s Equations

On the U-WPF Acts over Monoids

CHAPTER 9 LINEAR MOMENTUM, IMPULSE AND COLLISIONS

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

ESCI 341 Atmospheric Thermodynamics Lesson 13 Phase Changes Dr. DeCaria

Weak McCoy Ore Extensions

1 Matrix representations of canonical matrices

Iterative Methods for Searching Optimal Classifier Combination Function

Linear Approximation with Regularization and Moving Least Squares

Nonextensibility of energy in Tsallis statistics and the zeroth law of

Supervised Learning. Neural Networks and Back-Propagation Learning. Credit Assignment Problem. Feedforward Network. Adaptive System.

Introduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015

Two Approaches to Proving. Goldbach s Conjecture

CS 468 Lecture 16: Isometry Invariance and Spectral Techniques

From Biot-Savart Law to Divergence of B (1)

The 7 th Balkan Conference on Operational Research BACOR 05 Constanta, May 2005, Romania

Note 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Andre Schneider P622

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Lower bounds for the Crossing Number of the Cartesian Product of a Vertex-transitive Graph with a Cycle

Valores propios de la matriz de truncamiento asociados al operador de transición de la máquina sumadora en la base 2

First day August 1, Problems and Solutions

Polynomials. 1 More properties of polynomials

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

Research on Complex Networks Control Based on Fuzzy Integral Sliding Theory

Some results on a cross-section in the tensor bundle

LECTURE 21 Mohr s Method for Calculation of General Displacements. 1 The Reciprocal Theorem

New Cartan s Tensors and Pseudotensors in a Generalized Finsler Space

G : Statistical Mechanics

Affine and Riemannian Connections

Quick Visit to Bernoulli Land

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.

PHYS 705: Classical Mechanics. Calculus of Variations II

On the Calderón-Zygmund lemma for Sobolev functions

GEOMETRIC INTERPRETATIONS OF CURVATURE. Contents 1. Notation and Summation Conventions 1

Physics 111. CQ1: springs. con t. Aristocrat at a fixed angle. Wednesday, 8-9 pm in NSC 118/119 Sunday, 6:30-8 pm in CCLIR 468.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

Lecture 21: Numerical methods for pricing American type derivatives

Chapter 6. Rotations and Tensors

Effects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

LECTURE 5: FIBRATIONS AND HOMOTOPY FIBERS

3. Stress-strain relationships of a composite layer

8. INVERSE Z-TRANSFORM

Physics 443, Solutions to PS 7

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference

Mixture of Gaussians Expectation Maximization (EM) Part 2

Modelli Clamfim Equazione del Calore Lezione ottobre 2014

7. Products and matrix elements

( ) 2 ( ) ( ) Problem Set 4 Suggested Solutions. Problem 1

VIII. Addition of Angular Momenta

More metrics on cartesian products

Electric and magnetic field sensor and integrator equations

Pre-Talbot ANSS. Michael Andrews Department of Mathematics MIT. April 2, 2013

Tensors for matrix differentiation

Algorithms for factoring

E-Companion: Mathematical Proofs

Introduction. Modeling Data. Approach. Quality of Fit. Likelihood. Probabilistic Approach

9 Characteristic classes

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis

Projectile Motion. Parabolic Motion curved motion in the shape of a parabola. In the y direction, the equation of motion has a t 2.

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of

ON AUTOMATIC CONTINUITY OF DERIVATIONS FOR BANACH ALGEBRAS WITH INVOLUTION

1 cos. where v v sin. Range Equations: for an object that lands at the same height at which it starts. v sin 2 i. t g. and. sin g

Interest rates parity and no arbitrage as equivalent equilibrium conditions in the international financial assets and goods markets

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0

Transcription:

UNIT 7. THE FUNDAMENTAL EQUATIONS OF HYPERSURFACE THEORY ================================================================================================================================================================================================================================================ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Gau frame of a arameterzed hyerurface, formuae for the arta dervatve of the Gau frame vector fed, Chrtoffe ymbo, Gau and Codazz-Manard equaton, fundamenta theorem of hyerurface, "Theorema Egregum", comonent of the curvature tenor, tenor n near agebra, tenor fed over a hyerurface, curvature tenor. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Now we derve ome formuae for hyerurface. Conder a reguar arameterzed hyerurface r:w----------l R. The arta dervatve r,...,r 1 n defne a ba of the tangent ace of the hyerurface at each ont. If we add to thee vector the norma vector of the hyerurface, we get a ba of R at each ont of the hyerurface. The ytem of the vector fed r,...,r,n aong r caed the Gau frame of the hyerurface. Gau 1 n ------------------------------------------------------- frame ay mar roe n the theory of hyerurface a Frenet frame doe n curve theory. Smarty not comete however, nce a Gau frame much more deendent on the arameterzaton. Neverthee, n the ame way a for Frenet frame, t mortant to now how the dervatve of the frame vector fed wth reect to the arameter can be exreed a a near combnaton of the frame vector. For th we have to determne the coeffcent G, a, b, g n the exreon j j r = S G r + a N, N = S b r + g N. () j j j Let u begn wth the me obervaton that nce N nown to be j tangenta, and N = -L(r ), where L the Wengarten ma, j j u--------------------------------------------------------------------------------o g = 0 for a j j m--------------------------------------------------------------------------------. n -1 and (-b ) the matrx L = B G of the Wengarten ma wth reect to j j,=1 the ba r,...,r. Denote by g and b the entre of the frt and 1 n econd fundamenta form a uua, and denote by g the comonent of the nvere matrx of the matrx of the frt fundamenta form. (Attenton! -1 Entre of G and G are dtnguhed by the oton of ndce.) Then u-------------------------------------------------------------------------------------o b = - S b g. j j m-------------------------------------------------------------------------------------. Tang the dot roduct of the frt equaton of () wth N we gan the 1

equaty <r,n > = a and nce <r,n > = b, u-------------------------------------------------------------------------------------------------------------------o m-------------------------------------------------------------------------------------------------------------------. a = b for a,j. There ony one queton eft: what are the coeffcent G equa to? Let u tae the dot roduct of the frt equaton of () wth r <r,r > = S G <r,r > = S G g, or denotng the dot roduct <r,r > horty by G, G = S G g. The coeffcent G and G are caed the Chrtoffe ymbo of frt and ---------------------------------------------------------------------------------------------------------------------------------------------------------------- econd tye reectvey. The at equaton how how to exre Chrtoffe ------------------------------------------------------- ymbo of econd tye wth the he of Chrtoffe ymbo of frt tye. It can ao be ued to exre Chrtoffe ymbo of frt tye n term of econdary Chrtoffe ymbo. Indeed, mutyng the equaton wth g, tang um for and ung S g g = d (J Kronecer deta), we get S G g = S S G g g = S G d = G. Now et u try to determne Chrtoffe ymbo of econd tye. Dfferentatng the equaty g = <r,r > wth reect to the -th varabe j and then ermutng the roe of ndce,j, we get the equate g, = <r,r > + <r,r > j j g j, = <r,r > + <r,r > j j g,j = <r,r > + <r,r >. j Sovng th near ytem of equaton for the econdary Chrtoffe ymbo tandng on the rght hand de, we obtan and 1 G = <r,r > = ----- (g + g - g ) 2,j j,, u--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------o 1 G = S G g = S ----- (g + g - g ) g. 2,j j,, m--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Oberve that the Chrtoffe ymbo deend ony on the frt fundamenta form of the hyerurface. 2 Now we a the foowng queton. Suoe we are gven 2n mooth functon g, b,j=1,2,...,n on an oen doman W of R. When can we fnd a arameterzed hyerurface r:w----------l R wth fundamenta form G = (g ) and B = (b ). We have ome obvou retrcton on the functon g and b. Frt, g = g, b = b, and nce G the matrx of a otve j j defnte bnear form, the determnant of the corner ubmatrce (g ),j=1 2

mut be otve for = 1,...,n. However, the exame we have how that thee condton are not enough to guarantee the extence of a hyerurface. For exame, f G the dentty matrx everywhere, whe B = f G for ome functon on W, then the hyerurface (f ext) cont of umbc. We now however that f a urface cont of umbc, then the rnca curvature are contant, o athough our choce of B and G atfe a the condton we have ted o far, t doe not correond to a hyerurface une f contant. So there mut be ome further reaton between the comonent of B and G. Our an to fnd ome of thee correaton the foowng. Let u exre r and r a a near combnaton of the Gau frame vector. The coeffcent we get are functon of the entre of the frt and econd fundamenta form. For r = r, the correondng coeffcent n the exreon for thee vector mut be equa and t can be hoed that th way we arrve at further non-trva reaton between G and B. Th wa the hoohy, and now et u get down to wor. r = ( S G r + b N ) = S (G r + G r ) + b N + b N =,,, = S (G r + G (S G r + b N)) + b N - b SS b g r =,, = S (G + S G G - b S b g ) r + (b + S G b )N.,, Comarng the coeffcent of r n r and r, we obtan u----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------o G - G + S (G G - G G ) = S (b b - b b ) g,,j j j m----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. whe comaron of the coeffcent of N gve u-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------o b - b = S G b - S G b.,,j j m-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. 4 The frt n equaton (we have an equaton for a,j,,), are the 3 Gau equaton for the hyerurface. The econd famy of n equaton are --------------------------------------------------------------------------- the Codazz-Manard equaton. ---------------------------------------------------------------------------------------------------------------------------------- Exerce. Exre the econd order dervatve N and N a a near ---------------------------------------- j combnaton of the Gau frame vector. Comare the correondng coeffcent and rove that ther equaty foow from the Gau and Codazz-Manard equaton. The exerce ont out that a mar try to derve new reaton between G and B doe not ead to reay new reut. Th no wonder, nce the Gau and Codazz-Manard equaton together wth the revouy ted obvou condton on G and B form a comete ytem of neceary and 3

uffcent condton for the extence of a hyerurface wth fundamenta form G and B. n Theorem. (Fundamenta theorem of hyerurface). Let W C R be an oen ----------------------------------- n connected and my connected ubet of R (e.g. an oen ba or cube), and uoe that we are gven two mooth n by n matrx vaued functon G and B on W uch that G = (g ) and B = (b ) agn to every ont a ymmetrc matrx, G gve the matrx of a otve defnte bnear form. In th cae, f the functon G derved from the comonent of G accordng to the above formuae atfy the Gau and Codazz-Manard equaton, then there ext a reguar arameterzed hyerurface r:w----------l R for whch the matrx rereentaton of the frt and econd fundamenta form are G and B reectvey. Furthermore, th hyerurface unque u to rgd moton of the whoe ace. Namey, f r and r are two uch hyerurface, then there 1 2 ext an ometry (=dtance reervng becton) F:R ----------L R for whch r = Fqr. 2 1 Let u denote the exreon tandng on the eft hand de of the Gau equaton by u-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------o R := G - G + S (G G - G G ).,,j j m-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Then Gau equaton can be abbrevated wrtng R = S (b b - b b ) g. j Let u muty th equaton by g and tae a um for m S R g = S S (b b - b b ) g g = m j m = S (b b - b b ) d = (b b - b b ). j m m jm u--------------------------------------------------------------------------------------------------------o 1 1 mj m m--------------------------------------------------------------------------------------------------------. u-------------------------------------------------------------------------------------------------------------------------------------------------o Introducng the functon R := S R g, we may wrte R = (b b - b b ). mj m jm m-------------------------------------------------------------------------------------------------------------------------------------------------. Let u oberve, that the functon R can be exreed n term of the mj frt fundamenta form G. Coroary. (Theorema Egregum) The Gauan curvature of a reguar --------------------------------------------- 3 arameterzed urface n R can be exreed n term of the frt fundamenta form a foow R 1212 K = ------------------------------------. det G Theorema Egregum one of thoe theorem of Gau he wa very roud of. 4

The urrng fact not the actua form of th formua but the mere extence of a formua that exree the Gauan curvature n term of the frt fundamenta form. The geometrca meanng of the extence of uch a formua that the Gauan curvature doe not change when we bend the urface (athough rnca curvature do change n genera!). Defnton. Let r:w----------l R be a hyerurface. Conder the mang R that -------------------------------------------------- agn four tangenta vector fed X = S X r, Y = S Y r, Z = S Z r, W = S W r a functon accordng to the formua m j R(X,Y;Z,W) = S S S S R X Y Z W. mj m j We ha ca R the curvature tenor of the hyerurface, the functon R -------------------------------------------------------------------------------- mj the comonent of the curvature tenor. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Let u brefy reca ome defnton from near agebra, concernng tenor. Let V be a vector ace (over R). The et V of near functon form a vector ace wth reect to the oeraton ( + )(v) := (v) + (v), ( )(v) := ((v)). 1 2 1 2 The vector ace V of near functon on V caed the dua ace of V. -------------------------------------------------- If V fnte dmenona and e,...,e a ba of V, then we may 1 n 1 n conder the near functon e,...,e e V defned by e (e ) = d. It j j not dffcut to rove that thee near functon form a ba of V caed the dua ba of the ba e,...,e. A a conequence we get that dm V = -------------------------------------------------- 1 n dm V for fnte dmenona vector ace. A tenor of vaency (/order /tye) (,) over V a mutnear functon T : V x...xv x V x...x V ---------------L R defned on the Cartean roduct of coe of V and coe of V. "Mutnear" mean that fxng a but one varabe, we obtan a near (,) functon of the free varabe. Denote by T V the et of tenor of vaency (,). The um of two tenor of order (,) and the caar mute of a tenor are tenor of the ame order, hence the et of tenor of a gven vaency form a vector ace. If e,...,e a ba of V, then every 1 n tenor T unquey determned by t vaue on ba vector combnaton,.e. by the number... 1 1 T = T(e,...,e ;e,...,e ), j...j j j 1 1 whch are caed the comonent of the tenor T wth reect to the ba ------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------- 5

... (+) 1 e,...,e. Snce any (dm V) number T correond to a tenor, 1 n j...j 1 (,) (+) dm T V = (dm V). Now et u conder a reguar arameterzed hyerurface M, r:w ----------L R. ----- A tenor fed of vaency (,) over M a mang T that agn to ---------------------------------------------------------------------------------------------------------------------------------- every ont u e W a tenor of vaency (,) over the tangent ace of M at -----... 1 r(u). T(u) unquey determned by t comonent T (u) wth reect ----- ----- ----- j...j ----- 1... 1 to the ba r (u),...,r (u). The functon u 9-----LT (u) are caed the 1 ----- n ----- ----- j...j 1 ----- comonent of the tenor fed T. T ------------------------------------------------------------------------------------------------------------------------------------------------------ ad to be a mooth tenor fed --------------------------------------------------------------------------------------------------------- f t comonent are mooth. Exame. - Functon on M are tenor fed of vaency (0,0). - Tangenta vector fed are tenor fed of vaency (1,0) (V omorhc to V n a natura way). - The frt and econd fundamenta form of a hyerurface are tenor fed of vaency (0,2). - The mang that agn to every ont of a hyerurface the Wengarten ma at that ont a tenor of vaency (1,1). (The near ace of (1,1) V----------L V near mang omorhc to T V n a natura way.) - Let f be a mooth functon on M. Conder the tenor fed of vaency (0,1) defned on a tangent vector X to be the dervatve of f n the drecton X. Th tenor fed the dfferenta of f. - The curvature tenor a tenor fed of vaency (0,4). The curvature tenor one of the mot bac object of tudy n dfferenta geometry. In the revou comutaton the curvature tenor came acro e a rabbt from a cynder. To undertand t rea meanng, we ha ntroduce the curvature tenor n a more natura way n a more genera framewor, n the framewor of Remannan manfod. For th uroe, we have to get acquanted wth ome fundamenta defnton and contructon. Th w be the goa of the foowng unt. 6

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback