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Differential Geometry Intro


Differential Geometry Intro Catalog Description MAT 4140 3 hours [On-Demand] Differential Geometry Bulletin Description: This is an introductory course in the ... – PowerPoint PPT presentation

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Title: Differential Geometry Intro

Differential Geometry Intro
Catalog Description
  • MAT 4140 3 hours On-Demand
  • Differential Geometry
  • Bulletin Description This is an introductory
    course in the differential geometry of curves and
    surfaces in space, presenting both theoretical
    and computational components, intrinsic and
    extrinsic viewpoints, and numerous applications.
    The geometry of space-time will also be
    considered. Prerequisite MAT 2130.

  • Geometry Riemannian Geometry PhD
  • Physics Special relativity with Ralph Alpher,
    one of the creators of the big bang.
  • Research Geometry of Orbifolds, Mathematics and
    Society Popular Culture, Women Minorities

Course Goals
  • To develop geometric problem solving skills and
    3-D spatial visualization skills.
  • To develop a greater appreciation for connections
    between various disciplines of mathematics,
    including geometry, linear algebra, complex
    analysis, and differential equations, along with
    an introduction to these subjects as they apply
    to differential geometry.
  • To understand the importance of differential
    geometry in various scientific fields, including

  • Geometry of curves in space, including Frenet
  • Theory of surfaces, including curvature,
    geodesics, and metrics
  • Geometry of space-time and applications to
    general relativity (as time allows)

Culmination of Early Ideas
  • At a very early age, children develop a very rich
    visual intelligence' in terms of perception and
    experiences. They have questions and lots of
    these questions and explorations can be connected
    to geometry if we use the right types of physical
    and visual presentations. They have developed
    advanced skills for which the precise vocabulary
    is 3-D differential geometry and differential
    topology 28 (Hoffmann, 1998), 36 (Koenderink,
    1990)...we should connect with these abilities.
    (Whiteley, 1999)

Rich History and New Relevance
  • Machinery Design
  • Classification of Spaces
  • Fundamental Forces in the Universe
  • Study of DNA
  • Medical Imaging
  • Computer Graphics

  • Intuitive
  • Calculable
  • Useful
  • Interdisciplinary unification of many topics
    including geometry, spatial visualization,
    calculus, linear algebra, differential equations,
    and complex variables along with various topics
    from the sciences, including physics

Smokestack Problem
  • Frank Morgan received a call from a company
    constructing a huge smokestack, which required
    the attachment of a spiraling strip or stake for
    structural support.

  • Strake cut out of flat metal pieces
  • What inner radius would make it fit best?

Soap Bubbles
  • Pressure versus surface tension
  • What shape does it take on, and why?

Inconsistency in Mercurys Orbit
  • Newton used solar gravitational attraction and
    calculus to explain Keplers elliptical planetary
  • The orbit rotated (precessed) at an unexpected
  • General relativity and the geometry of space-time

  • Some of the results were motivated by the desire
    for more accurate clocks
  • I have been occupied with a new discovery... In
    order to make my clock even more exact.... What,
    however, I never had expected I would discover, I
    have now hit upon, the undoubtedly true shape of
    curves... I determined it by geometric
    reasoning. (C. Huygens December 1659)
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