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01:34, 11 December 2021 Khanh talk contribs created page Equivalents Fractions (Created page with "==Equivalent fractions== Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fractio...")
01:20, 11 December 2021 Khanh talk contribs created page Prime Numbers (Created page with "== Introduction == A prime number (or prime for short) is a natural number that has exactly two divisors: itself and the number 1. Since 1 has only one divisor — itself —...")
00:59, 11 December 2021 Khanh talk contribs created page Patterns (Created page with "= Arithmetic sequence = An '''arithmetic progression''' or '''arithmetic sequence''' is a sequence of numbers such that the difference between the consecutive terms is consta...")
15:34, 27 November 2021 Khanh talk contribs created page The Cauchy Criterion (Created page with "Recall that a sequence of real numbers <math>(a_n)</math> is said to be a Cauchy Sequence if <math>\forall \epsilon > 0</math> there exists an <math>N \in \ma...")
15:16, 27 November 2021 Khanh talk contribs created page Limit of a Sequence in the Real Numbers (Created page with "==Definition== Sequences occur frequently in analysis, and they appear in many contexts. While we are all familiar with sequences, it is useful to have a formal definition....")
22:53, 21 November 2021 Khanh talk contribs created page Triangle Inequality (Created page with "== Licensing == Content obtained and/or adapted from: * [http://mathonline.wikidot.com/the-triangle-inequality The Triangle Inequality, mathonline.wikidot.com] under a CC BY-...")
22:50, 21 November 2021 Khanh talk contribs created page Absolute Value and the Real Line (Created page with "== The Real Line == thumb|right|382x382px|The real line In mathematics, the '''real line''', or '''real number line''' is the line whose points...")
22:38, 21 November 2021 Khanh talk contribs created page Differentiation of Vector-valued Functions (Created page with "==Derivative of a three-dimensional vector function== Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the componen...")
22:32, 21 November 2021 Khanh talk contribs created page Subsequences (Created page with " In mathematics, a '''subsequence''' of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order o...")
22:23, 21 November 2021 Khanh talk contribs created page The Cauchy Criterion for Convergence (Created page with "Recall that a sequence of real numbers <math>(a_n)</math> is said to be a Cauchy Sequence if <math>\forall \epsilon > 0</math> there exists an <math>N \in \ma...")
15:34, 21 November 2021 Khanh talk contribs created page Sequences and Their Limits (Created page with "Consider the sequence <math>\left \{ \frac{1}{n} \right \}_{n=1}^{\infty} = \left \{ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \right \}</math>. As <math>n \to \infty</mat...")
14:33, 21 November 2021 Khanh talk contribs created page Monotone Sequences (Created page with "==Monotone Sequences of Real Numbers== We will now look at two new types of sequences, '''increasing sequences''' and '''decreasing sequences'''. <blockquote style="backgroun...")
13:40, 19 November 2021 Lila talk contribs created page Abstract Algebra: Groups (Created page with "Recall that an operation <math>\cdot</math> on <math>S</math> is said to be associative if for all <span c...")
01:08, 19 November 2021 Khanh talk contribs created page Functions(The Cartesian Product Definition) (Created page with " File:Cartesian Product qtl1.svg|thumb|Cartesian product <math>\scriptstyle A \times B</math> of the sets <math>\scriptstyle A=\{x,y,z\}</math> and <math>\scriptstyle B=\{1,...")
00:27, 19 November 2021 Khanh talk contribs created page Solutions of Linear Systems (Created page with "==Fundamental Solutions to Linear Homogenous Differential Equations== <blockquote style="background: white; border: 1px solid black; padding: 1em;"> :'''Theorem 1:''' Let <m...")
23:37, 18 November 2021 Khanh talk contribs created page Determinant (Created page with "== Wronskian Determinants of Two Functions == We are going to look more into second order linear homogenous differential equations, but before we do, we need to first learn a...")
08:43, 18 November 2021 Lila talk contribs created page Abstract Algebra: Preliminaries (Created page with "==Natural numbers== ==Integers== ==Division algorithm== ==Congruence modulo m== ==Algebra on <math>\Z_m</math>== ==GCD, LCM, and Bézout's identity== ==Primes== ===Eucli...")
00:02, 18 November 2021 Khanh talk contribs created page Linear Dependence of Vectors (Created page with "==Linear Independence and Dependence== <blockquote style="background: white; border: 1px solid black; padding: 1em;"> :'''Definition:''' A set of vectors <math>V = \{ \mathbf...")
23:31, 17 November 2021 Khanh talk contribs created page Direct Integration (Created page with "==The Method of Direct Integration== The simplest differential equations to solve are those in the form of <math>\frac{dy}{dt} = f(t)</math>. These differential equations can...")
15:44, 17 November 2021 Lila talk contribs created page Bounded Functions (Created page with "Image:Bounded and unbounded functions.svg|right|thumb|A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded f...")
14:55, 17 November 2021 Lila talk contribs created page Cosets and Lagrange’s Theorem (Created page with "==Left and Right Cosets of Subgroups== <blockquote style="background: white; border: 1px solid black; padding: 1em;"> <td>Definition: Let <span class="math-in...")
14:39, 17 November 2021 Lila talk contribs created page Linear Independence of Vectors (Created page with "thumb|right|Linearly independent vectors in <math>\R^3</math> File:Vec-dep.png|thumb|right|Linearly dependent vectors in a plane in <math>\R^3.</math>...")
13:57, 17 November 2021 Lila talk contribs created page Permutation Groups (Created page with "A '''permutation group''' is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which ar...")
11:17, 17 November 2021 Lila talk contribs created page Cyclic Groups (Created page with "A '''cyclic group''' or '''monogenous group''' is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary o...")
16:13, 16 November 2021 Lila talk contribs created page Groups (Created page with "Recall that an operation <math>\cdot</math> on <math>S</math> is said to be associative if for all <span c...")
14:29, 16 November 2021 Lila talk contribs created page The Integers (Created page with "An '''integer''' (from the Latin ''integer'' meaning "whole"){{efn|''Integer'' 's first literal meaning in Latin is "untouched", from ''in'' ("not...")
13:24, 16 November 2021 Lila talk contribs created page Unions and Intersections of Sets (Created page with "thumb|200px|Union of two sets: <math>~A \cup B</math> [[File:Venn 0111 1111.svg|thumb|200px|Union of three sets: <math>~A \cup B \cup C</math>]...")
20:54, 13 November 2021 Khanh talk contribs created page Sequences:Subsequences (Created page with "==Subsequence== In mathematics, a '''subsequence''' of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without chang...")
20:37, 13 November 2021 Khanh talk contribs created page Separation Properties (Created page with "In topology and related branches of mathematics, '''separated sets''' are pairs of subsets of a given topological space that are related to each other in a certain way: roughl...")
17:52, 13 November 2021 Khanh talk contribs created page Totally Bounded Metric Spaces (Created page with "In topology and related branches of mathematics, '''total-boundedness''' is a generalization of compactness for circumstances in which a set is not n...")
17:26, 13 November 2021 Khanh talk contribs created page Bases of Open Sets (Created page with "A set <math>S \subseteq \mathbb{R}^n</math> is said to be open if <math>S =\mathrm{int} (S)</math>, that is, for every point <math>\mathbf{a} \in S</math> we have that there e...")
16:55, 13 November 2021 Khanh talk contribs created page A Topology Given By A Metric (Created page with "==Metric Space== ===Definition=== A metric space is a Cartesian pair <math>(X,d)</math> where <math>X</math> is a non-empty set and <math>d:X\times X\to[0,\infty)</math>, is a...")
15:43, 13 November 2021 Khanh talk contribs created page The Nested Interval Theorem in Higher Dimensions (Created page with "== Nested Intervals == <blockquote style="background: white; border: 1px solid black; padding: 1em;"> :'''Definition:''' A sequence of intervals <math>I_n</math> where <math>...")
12:50, 13 November 2021 Khanh talk contribs created page Iterated Integrals and Fubini's Theorem (Created page with "==Resources== * [https://tutorial.math.lamar.edu/classes/calciii/IteratedIntegrals.aspx Iterated Integrals, Paul's Online Notes] == Licensing == Content obtained and/or adap...")
12:29, 13 November 2021 Khanh talk contribs created page Lebesque Theorem for Riemann Integrability on the Real Line (Created page with "In mathematics, the '''Lebesgue differentiation theorem''' is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the...")
15:52, 12 November 2021 Khanh talk contribs created page The Inverse Function Theorem and the Implicit Function Theorem (Created page with "In this chapter, we want to prove the inverse function theorem (which asserts that if a function has invertible differential at a point, then it is locally invertible itself)...")
15:40, 12 November 2021 Khanh talk contribs created page Taylor's Formula in Several Variables (Created page with "Let <math>S \subseteq \mathbb{R}^n</math> be open, <math>\mathbf{x} \in S</math>, and <math>f : S \to \mathbb{R}</math>. If <math>\mathbf{t} \in \mathbb{R}^n</math> then we kn...")
14:27, 12 November 2021 Khanh talk contribs created page Mean-Value Theorems for Vector Valued Functions (Created page with "==The Mean Value Theorem for Differentiable Functions from Rn to Rm== Recall that if <math>f</math> is a continuous function on the closed interval <math>[x, y]</math> and di...")
20:04, 11 November 2021 Khanh talk contribs created page Rules for Differentiation and Tangent Planes (Created page with "We will now look at a bunch of rules for differentiating vector-valued function, all of which are analogous to that of differentiating real-valued functions. We will not prove...")
17:07, 11 November 2021 Khanh talk contribs created page Limits of Sequences in the Euclidean space and the Bolzano-Weierstrass Theorem (Created page with "File:Archimedes pi.svg|350px|right|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|The sequence given by the perimeters of regular ''n''-sided pol...")
12:42, 11 November 2021 Khanh talk contribs created page Laplace Transform to Systems of ODEs (Created page with "This is an example to illustrate how Laplace transforms can be used to solve systems of differential equations. Solve the following system. <div style=...")
15:05, 10 November 2021 Lila talk contribs created page Divergence Theorem (Created page with "The '''divergence theorem''', also known as '''Gauss's theorem''' or '''Ostrogradsky's theorem''', is a theorem which relates the flux of a vector field through a closed surfa...")
14:13, 10 November 2021 Lila talk contribs created page The Hilbert Space L2 and the Hilbert Cube (Created page with "==Definition and illustration of Hilbert Space== ===Motivating example: Euclidean vector space=== One of the most familiar examples of a Hilbert space is the Euclidean vect...")
12:51, 10 November 2021 Lila talk contribs created page Surface Integrals (Created page with "A '''surface integral''' is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. G...")
12:17, 10 November 2021 Lila talk contribs created page Partial Derivatives and Integrals (Created page with "===Partial Derivatives of Functions from Rn to Rm=== One of the core concepts of multivariable calculus involves the various differentiations of functions from <span class=...")
16:07, 9 November 2021 Lila talk contribs created page The Substitution and Composition Theorems (Created page with "== Substitution Theorem e == === Definite integrals === Let {{math|''φ'' : [''a'', ''b''] → ''I''}} be a differentiable function with a continuous derivative, where {{math...")
15:44, 9 November 2021 Lila talk contribs created page The Additivity Theorem (Created page with "<blockquote style="background: white; border: 1px solid black; padding: 1em;"> Let <math> f </math> be a real-valued function on the interval...")
15:08, 9 November 2021 Lila talk contribs created page Riemann Integrable Functions (Created page with "== Overview == Let {{mvar|f}} be a non-negative real-valued function on the interval {{math|[''a'', ''b'']}}, and let :<math>S = \left \{ (x, y) \, : \ a \leq x \leq b, 0 <...")
14:18, 9 November 2021 Lila talk contribs created page The Topology of Higher Dimensions: interior, closure and boundary (Created page with "We are now going to discuss some important classifications of points regarding a subset <math>S</math> of <math>\mat...")
14:03, 9 November 2021 Lila talk contribs created page Bounded Sets and Bounded Functions in a Metric Space (Created page with "<blockquote style="background: white; border: 1px solid black; padding: 1em;"> <td>Definition: Let <math>(M, d)</math> be a m...")

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