PDF-Differentiable manifolds

iiPrefaceThepurposeofthesenotesistointroduceandstudydierentiablemanifoldsDierentiablemanifoldsarethecentralobjectsindierentialgeometryandtheygeneralizetohigherdimensionsthecurvesandsurfacesknown. [40]G.AnandaSwarup.Onembeddedspheresin3-manifolds.Math.Ann.,203:89{102,1973.[41]G.AnandaSwarup.Projectiveplanesinirreducible3-manifolds.Math.Z.,132:305{317,1973.[42]G.AnandaSwarup.Pseudo-isotopiesofS3 a a i = L a two a space. a a a a a a = Z a a a finite sketched I a a a (dr) a T(E p 4SIDDHARTHAGADGILToconstructnon-orientable3-manifolds,onegluesnon-orientablehandlebodiesofthesamegenusalongtheirboundaries.Afundamentaltheoremassertsthattheseconstructionsgiveall3-manifolds.Theorem2.E Differentiability. A function is differentiable at point . c . if and only if. the derivative from the left of . c. equals the derivative from the right of . c. .. AND. if . c. is in the domain of . Chapter 3.2. How . Might Fail to Exist.  . A function will not have a derivative at a point . where the slopes of the secant lines . fail to approach a limit as . Some of the common ways where a function fails to have a derivative:. AKA “Shortcuts”. Review from 3.2. 4 places derivatives do not exist:. Corner. Cusp. Vertical tangent (where derivative is undefined). Discontinuity (jump, hole, vertical asymptote, infinite oscillation). Four-Manifolds Constructed via Plumbing ~ . a ~ k + 1 = + 1 ag _ 1 = _ 1 + 1 F 1 1 (in fact one can reduce to the +t cases RI and RII with F1 = • F o a . = S E 2 L~ L 2 E 2 L 2 as required. requ VALUE THEOREMS. Derivability of a function :. A function . f . defined on [. a, b. ] is said to be derivable or differentiable at if exists. This limit is called derivative of . analysis and its applications on Riemannian manifolds. S. . Hosseini. FSDONA 2011, Germany.. Nonsmooth analysis. However, in many aspects of mathematics such as . control theory . and . matrix analysis, . 1. , . Piotr. DACKO. 2. & . Cengizhan. MURATHAN. 1 . . 1 . Uludağ. University, Art and Science Faculty, Department of Mathematics, Bursa-TURKEY. ,. 2. . Wroclaw. , POLAND . 1. . Preliminaries. are . Continuous. Connecting Differentiability . and . Continuity. Differentiability and Continuity. Continuous functions . are . not necessarily differentiable. . For instance, start with . Theorem and the Mean Value Theorem.  .  . Mean Value Theorem. The Mean Value Theorem can be interpreted geometrically as follows:. Is the slope of the line segment joining the points where . x. =. Daniel Dreibelbis. University of North Florida. USA. Umbilic Bracelet. Outline. Define duals and dual generalizations.. Describe the singularities of duals of hypersurfaces.. Define dual sphere bundles, and connect their singularities.. HANDBOOKOFKNOTTHEORYEditedbyWilliamMenascoandMorwenThistlethwaite2005ElsevierB.V.Allrightsreserved In1926,Artin[3]describedtheconstructionofcertainknotted2-spheresin.Theintersectionofeachoftheseknotsw