Advanced Calculus: Difference between revisions

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Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book].

This is content covered in MATH410 and MATH411 at UMD.

==Sequences==

==Topology==

===Closed===

The folllowing definitions of Closed Sets are equivalent.

* (Order)

* (Sequences) A set <math>S</math> is clsoed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>.

* (Topology)

; Notes

* Union of infinitely many closed sets can be open.

* Intersection of infinitely many open sets can be closed.

* <math>\{\}</math> and <math>\mathbb{R}</math> are both open and closed

===Compact===

Compactness is a generalization of closed and bounded.<br>

;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

==~~Sequences~~==

===Metric Space===

==Continuity==

===Definitions of Continuity===

The following 3 definitions of Continuity are equivalent.

The following definitions of Continuity are equivalent.

* (Order) A function <math>f</math> is continuous at <math>x_0</math> if for all <math>\epsilon</math> there exists <math>\delta</math> such that <math>|x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon</math>

* (Sequences) A function <math>f</math> is continuous at <math>x_0</math> if <math>\{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)</math>

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==Implicit Function Theorem==

==Line and Surface Integrals==

==Derivatives with respect to vectors and matrices==

Not typically covered in undergraduate analysis and calculus classes but necessary for machine learning.<br>

See [https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf The Matrix Cookbook]

* <math>\partial_{x} x^t x = \partial_{x} \operatorname{Tr}(x^t x) = \operatorname{Tr}( (\partial x)^t x + x^t (\partial x)) = 2 * \operatorname{Tr}(x^t (\partial x)) = 2x</math>

@@ Line 1: / Line 1: @@
 Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book].
+This is content covered in MATH410 and MATH411 at UMD.
+==Sequences==
+==Topology==
+===Closed===
+The folllowing definitions of Closed Sets are equivalent.
+* (Order)
+* (Sequences) A set <math>S</math> is clsoed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>.
+* (Topology)
+; Notes
+* Union of infinitely many closed sets can be open.
+* Intersection of infinitely many open sets can be closed.
+* <math>\{\}</math> and <math>\mathbb{R}</math> are both open and closed
+===Compact===
+Compactness is a generalization of closed and bounded.<br>
+;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
-==Sequences==
+===Metric Space===
 ==Continuity==
 ===Definitions of Continuity===
-The following 3 definitions of Continuity are equivalent.
+The following definitions of Continuity are equivalent.
 * (Order) A function <math>f</math> is continuous at <math>x_0</math> if for all <math>\epsilon</math> there exists <math>\delta</math> such that <math>|x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon</math>
 * (Sequences) A function <math>f</math> is continuous at <math>x_0</math> if <math>\{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)</math>
@@ Line 22: / Line 44: @@
 ==Implicit Function Theorem==
 ==Line and Surface Integrals==
+==Derivatives with respect to vectors and matrices==
+Not typically covered in undergraduate analysis and calculus classes but necessary for machine learning.<br>
+See [https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf The Matrix Cookbook]
+* <math>\partial_{x} x^t x = \partial_{x} \operatorname{Tr}(x^t x) = \operatorname{Tr}( (\partial x)^t x + x^t (\partial x)) = 2 * \operatorname{Tr}(x^t (\partial x)) = 2x</math>