1. SURFACES
ALICIA COX

2. 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.

3. 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.

4. 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.”

6. 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.

7. 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

11. Types of Surfaces
Compact
Complete
Flat
Minimal
Riemann
Smooth
Solid

12. Contour Lines

13. 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.

14. Conclusion
How can these surfaces be used in our classrooms?