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Faculty of Science Mathematics & Statistics
Faculty of Science

# MATH 3B03, Fall 2018

GEOMETRY

Math 3B03 will focus on the geometry of surfaces. Throughout high school and the first two years of university, geometric concepts/facts that students come across and use are based on Euclidean geometry. These include (a) the shortest path between two points is the line segment joining them, (b) the sum of the angles of a triangle equals 180 degrees, and (c) given a line and an external point, there is exactly one line passing through the given point that does not intersect the given line. Yet, we have known for centuries that the earth is round (i.e., not flat), and in spherical geometry, (a)-(c) above are all false. What facts are their replacements? In fact, the earth is also not perfectly round. So spherical geometry is not exactly applicable and we need to learn more general tools which allow us to deal with and analyse the geometry of arbitrary surfaces. Besides the surface of the earth, interesting surfaces occur in abundance in nature and by human creation. These include interfaces between liquids and gases, or solids and liquids that arise from physical (e.g., geothermal) processes, the surfaces of body parts and organs, membranes and cell walls, as well as curved portions of buildings and the surfaces of aircrafts and missiles. A fundamental concept that will be explored in Math 3B3 is the curvature of a surface.  A point on a surface has it positive curvature if em near the point the surface lies on only one side of the tangent plane at the point in question. The point has it negative curvature if all sufficiently small pieces of the surface containing the point lie on both sides of the tangent plane. Thus the Sphere x^2 + y^2 + z^2 = 1 has positive curvature everywhere while the saddle z = x^2 - y^2 has negative curvature near the origin. For a given surface are we free to bend it (without tearing it) into any shape we want? For example, if we let a little bit of air out of a soccer ball, which starts off as round, we can make some points have negative curvature. But can we bend the soccer ball so that it has negative curvature everywhere? Can we bend the surface of a doughnut within R^3 so that it has zero curvature everywhere?  The answers to these and similar questions will be discussed in Math 3B3. The only prerequisite is a firm grasp of advanced calculus and linear algebra.

INSTRUCTOR:  M. Wang

Course Website

Calendar Description
3 Units
Selected topics from: affine and projective geometry, Euclidean, spherical and hyperbolic geometry, differential geometry of curves and surfaces.
Three lectures; one term
Prerequisite(s): One of MATH 2A03, MATH 2X03, ISCI 2A18 A/B; and MATH 2R03

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Department of Mathematics & Statistics
McMaster University

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1280 Main Street West
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Department of Mathematics & Statistics
McMaster University

Hamilton Hall, Room 218
1280 Main Street West