Advanced Calculus

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Advanced Calculus as taught in Fitzpatrick's book.


Sequences

Topology

Closed

The folllowing definitions of Closed Sets are equivalent.

  • (Order)
  • (Sequences) A set is clsoed if it contains all its limit points. That is , .
  • (Topology)
Notes
  • Union of infinitely many closed sets can be open.
  • Intersection of infinitely many open sets can be closed.
  • and are both open and closed

Compact

Compactness is a generalization of closed and bounded.

Definitions
  • (Sequence) A set is sequentially compact if for every sequence from the set, there exists a subsequence which converges to a point in the set.
  • (Topology) A set is compact if for every covering by infinitely many open sets, there exists a covering by a finite subset of the open sets.
Notes
  • A set is sequentially compact iff it is closed and bounded

Metric Space

Continuity

Definitions of Continuity

The following definitions of Continuity are equivalent.

  • (Order) A function is continuous at if for all there exists such that
  • (Sequences) A function is continuous at if
  • (Topology) A function is continuous at if for all open sets s.t. , is an open set.
    • The preimage of an open set is open.
    • Continuous functions map compact sets to compact sets.

Differentiation

Integration

Approximation

Series

Inverse Function Theorem

Implicit Function Theorem

Line and Surface Integrals