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 \(\displaystyle S\) is clsoed if it contains all its limit points. That is \(\displaystyle \forall \{x_i\} \subseteq S\), \(\displaystyle \{x_i\} \rightarrow x_0 \implies x_0 \in S\).
  • (Topology)


Union of infinitely many closed sets can be open. Intersection of infinitely many open sets can be closed.

Compact

Metric Space

Continuity

Definitions of Continuity

The following definitions of Continuity are equivalent.

  • (Order) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all \(\displaystyle \epsilon\) there exists \(\displaystyle \delta\) such that \(\displaystyle |x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon\)
  • (Sequences) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if \(\displaystyle \{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)\)
  • (Topology) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all open sets \(\displaystyle V\) s.t. \(\displaystyle f(x_0) \in V\), \(\displaystyle f^{-1}(V)\) 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