# Vector Calculus Independent Study Path

## Unit 7: Surfaces

We studied surfaces before when we investigated the graphs of scalar
valued functions of two variables. In this unit, we generalize to
the notion of parameterized surfaces. Parameterized surfaces are
defined by a mapping from two space to three space -- think of the
straight longitude and latitude ``lines'' on a flat map being sent to
the curved longitude and latitude lines on a sphere. While
parameterized surfaces are a bit awkward at first, they soon allow you
to do all sorts of things, including integrating scalar functions
and vector fields over the surface.
In this unit, you will learn:
- How to find the normal vector to an implicity defined surface.
- How to describe a surface parametrically.
- How to convert an implicitly defined surface to a parametrically
defined surface, and vice-versa.
- How to find a normal vector to a parametrically defined surface.
- How to find the area of a surface.
- How to find the integral of a scalar function over a surface.
- How to find the flux of a vector field through a surface.

For more detailed instructions, see the Xdvi
or PDF pages.
## Suggested Procedure

- Read and do some problems from
- Rogers Chapters 23 and 24, or
- Marsden and Tromba sections 7.3, 7.4, 7.5, and 7.6

- Take the Sample Test, Xdvi or PDF.
- Take a unit test.

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watko@athena.mit.edu
Last modified July 28, 1998