**Math 3593H Honors Mathematics II Spring Semester 2004
**

Exercises:

Seciont 3.2 (pages 321-323): 1*, 2, 3, 4, 5a, 6*

Extra questions:

A. Find the equation for the tangent line at the point (2,4,8) to the curve parametrized in question 3.1.11a

B. Find the equation for the tangent plane at the point (cos(1), sin(1), 1) to the surface parametrized in question 3.1.12.

C*. Find the equation for the tangent plane at the point (sin(2)+1, 3, 2) to the surface parametrized in question 3.1.28.

The definition in 3.1 specifies that a manifold should be given as a subset of some ambient space. In general manifolds are defined more abstractly without specifying a particular embedding of the manifold into a larger space. Although the abstract definition is actually easier to work with in some ways, it does raise the question of whether every manifold can be embedded in R^n, and the definition of the tangent space becomes more difficult. The book avoids these issues, and probably rightly so. The course where this approach is taught is the graduate level course, 'Manifolds and Topology'.

As it is, we need to be able to work with manifolds given by means of parametrizations, and also as the set of solutions of an equation, and this is what most of the exercises to 3.1 are about.