Advanced Calculus: Difference between revisions
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Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book]. | 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 | |||
== | ===Metric Space=== | ||
==Continuity== | ==Continuity== | ||
===Definitions of Continuity=== | ===Definitions of Continuity=== | ||
The following | 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> | * (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> | * (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> | ||
* (Topology) A function <math>f</math> is continuous at <math>x_0</math> if for all open sets <math>V</math> s.t. <math>f(x_0) \in V</math>, <math>f^{-1}(V)</math> is an open set. | * (Topology) A function <math>f</math> is continuous at <math>x_0</math> if for all open sets <math>V</math> s.t. <math>f(x_0) \in V</math>, <math>f^{-1}(V)</math> is an open set. | ||
** The preimage of an open set is open. | ** The preimage of an open set is open. | ||
** | ** Continuous functions map compact sets to compact sets. | ||
==Differentiation== | ==Differentiation== | ||
| Line 22: | Line 44: | ||
==Implicit Function Theorem== | ==Implicit Function Theorem== | ||
==Line and Surface Integrals== | ==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> | |||
Latest revision as of 17:41, 13 August 2020
Advanced Calculus as taught in 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 \(\displaystyle S\) 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)
- Notes
- Union of infinitely many closed sets can be open.
- Intersection of infinitely many open sets can be closed.
- \(\displaystyle \{\}\) and \(\displaystyle \mathbb{R}\) 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 \(\displaystyle f\) 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
Derivatives with respect to vectors and matrices
Not typically covered in undergraduate analysis and calculus classes but necessary for machine learning.
See The Matrix Cookbook
- \(\displaystyle \partial_{x} x^t x = \partial_{x} \operatorname{Tr}(x^t x) = \operatorname{Tr}( (\partial x)^t x + x^t (\partial x)) = 2 * \operatorname{Tr}(x^t (\partial x)) = 2x\)