General: The test will (roughly 33% each) consist of:
- old homework problems
- asking for defintions
- some easy new problems.
Important things you ought to know:
1) Recall three different norms on R^n, and that they all three satisfy Theorem 8.6.
2) Comparison between the different norms: see remark 8.7 (page 231)
3) Cauchy Schwartz (proof!)
4) Linear Transformations and the fact that they are bounded (Theorem 8.17)
5) Definition of open sets and closed sets (Definition 8.20)
6) Open sets are closed under arbitrary unions and finite intersections
Closed sets are closed under finite unions and arbitrary intersections.
(Theorem 8.24).
Give examples which show that open sets/ closed sets are not
closed under arbitrary intersections/unions.
7) Relative open/closed sets (Def 8.26)
8) Connected sets (Def 8.27 and Remark 8.29).
Know how to prove that a set in R is connected if and only if it is an interval.
Know how to show that the union of two dijoint closed balls is not conntected.
9) Interior and closure of a set (Def 8.31) Know how to show main properties (Theorems 8.32, 8.37)
10) Boundary; Definition 8.34 and Theorem 8.36.
11) Convergence in R^n: Definition 9.1. Know how to prove convergence theorems 9.2 - 9.7. Most of
them are like in the one dimensional case. Bolzano - Weierstrass in R^n.
12) Equivalent description of closed set: Theorem 9.8 (proof).
13) Open coverings and compact sets (Definitions 9.9). Important Theorem: 9.11 (proof)
14) Limit of functions: Definition 9.13 and Theorem 9.14 (similar to one dimensional case).
Know how to verify limits (Exercises 269/1,2,3).
15) Continuous Functions. Definition 9.22 and equivalent conditions: Theorems 9.25 and 9.26 (proof).
16) Compactness and Continuity: Theorems 9.29, 9.32 and 9.33 (proof).
18) Differentiability:
- Partially differentiable (pages 321)
- Differentiable
- Continuously partially differentiable 11.12
- Higher order partially differentiable Def 11.1
19) Examples concerning different types of differentiability:
Example 11.3, Example 11.18, f(x,y)=xy/(x^2+y^20, Exercises 2, 5, 6, 7,9.
20) Differentiablity - Partial Differentiability and how to compute derivatives: Theorem 11.15 (proof)
21) Interchanging derivatives: Theorem 11.2.
22) Interchanging limits/ derivatives with integration: 11.4 and 11.5.