SURFACES ALICIA COX 12.11.03

DEFINITIONS: Surfaces are the analog in space of curves in the plane. It has a definite area as well as defined boundaries, however it is infinitely thin. Can be used to denote a (n-1)-dimensional sub-manifold of an n-dimensional manifold.

DEFINITIONS: A surface (without boundary) is a ‘space’ which is locally like a piece of the plane. For those with boundary, some points are like a distorted flat unit half disc. Surfaces are the analog in space of curves in the plane.

DEFINITIONS: “A line is the trace of the movement of a point. A surface is the trace of the movement of a line. A solid is the trace of the movement of a surface.” “We regard a line as an infinite number of points, a surface as an infinite number of lines, a solid as an infinite number of surfaces.”

                                                                                                                                                                           

Coordinates In 3-space, setting any coordinates equal to a constant defines a plane. In cylindrical coordinates, r=c is a cylinder. In spherical coordinates, we get spheres by setting p=c. Setting phi=c defines a cone.

Types of Surfaces Sphere- x^2+y^2+z^2=p^2 Ellipsoid- ax^2+by^2+cz^2=p^2 Parabloid- x^2+y^2=z Saddle- x^2-y^2=z Hyperboloid(1 sheet)- x^2+y^2-z^2=1 Hyperboloid(1 sheet)- x^2-y^2-z^2=1 Cylinder- x^2+y^2=r^2

Types of Surfaces Compact Complete Flat Minimal Riemann Smooth Solid

Contour Lines

Orientability A surface is non-orientable if and only if it contains a subset that is topologically different to the Mobius band, otherwise it is orientable.

Conclusion How can these surfaces be used in our classrooms?