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• mathonline.wikidot.com

  Permutations as Products of Cycles - Mathonline - Wikidot

  http://mathonline.wikidot.com/permutations-as-products-of-cycles

  So the product of cycles from is necessarily a permutation of , though, can all permutations of be expressed as a product of cycles? The answer is yes, and in fact, every permutation can be expressed as a product of disjoint cycles as we will prove on the Decomposition of Permutations as Products of Disjoint Cycles page.
• mathonline.wikidot.com

  Mathonline

  http://mathonline.wikidot.com/calculus

  Calculus Topics. 1. Single-Variable Functions, Limits and Continuity. Different Types of Functions. Symmetric Functions. Increasing and Decreasing Functions. Introduction to …
• mathonline.wikidot.com

  The Principle of Superposition - Mathonline - Wikidot

  http://mathonline.wikidot.com/the-principle-of-superposition

  The Principle of Superposition. Suppose that we have a linear homogenous second order differential equation d2y dt2 + p(t)dy dt + q(t)y = 0 and that y = y1(t) and y = y2(t) are both solutions. The following theorem says that any linear combination y = Cy1(t) + Dy2(t) is also a solution to this differential equation for any constants C and D.
• mathonline.wikidot.com

  The Ideal of a Set of Points is a Radical Ideal - Mathonline - Wikidot

  http://mathonline.wikidot.com/the-ideal-of-a-set-of-points-is-a-radical-ideal

  The Ideal of a Set of Points is a Radical Ideal. Recall from the The Ideal of a Set of Points page that if K is a field then the ideal of X is defined as: I(X) = {F ∈ K[x1,x2,...,xn]: F(p) = 0, ∀p ∈ X} We will shortly observe that the ideals I(X) are always radical ideals. We first define what it means for an ideal to be a radical ideal.
• mathonline.wikidot.com

  Theorems on The Properties of The Real Numbers - Mathonline

  http://mathonline.wikidot.com/theorems-on-the-properties-of-the-real-numbers

  Theorem 2: If a and b are real numbers such that a ⋅ b = a, then b = 1. We will manipulate both sides of this equation to arrive at the conclusion that. Theorem 3: If is a real number then . Theorem 4: If and are real numbers where , then if , then . Theorem 5: If and are real numbers and then or or both .
• mathonline.wikidot.com

  Archive of Math Online News - Mathonline

  http://mathonline.wikidot.com/archive-of-math-online-news

  Archive of Math Online News - Mathonline. An archive of some older announcements made on the main page can be found here. JANUARY 5TH, 2015: The …
• mathonline.com

  MathOnline - How It Works

  https://www.mathonline.com/how-it-works

  MathOnline - How It Works. Join more than 217,000 students now confident in math because finally they can do it! Learn at your pace, not somebody else's. Stop and rewind …
• mathonline.wikidot.com

  Real Analysis - Mathonline

  http://mathonline.wikidot.com/real-analysis

  Mathonline. Welcome to the Real Analysis page. Here you can browse a large variety of topics for the introduction to real analysis. This hub pages outlines many useful topics …
• mathonline.wikidot.com

  The Existence/Uniqueness of Solutions to Second Order Lin

  http://mathonline.wikidot.com/the-existence-uniqueness-of-solutions-to-second-order-linear

  Determine the largest interval for which the second order linear differential equation with the initial conditions and has a unique solution. In order to apply Theorem 1, we will need to rewrite our differential equation as follows. (2) Therefore , , and . We have that is continuous for , is continuous for and , and is continuous everywhere.
• mathonline.wikidot.com

  The Group of Symmetries of the Square - Mathonline - Wikidot

  http://mathonline.wikidot.com/the-group-of-symmetries-of-the-square

  We will now see that the group of symmetries of the square also form a group with respect to the operation of composition . Consider a square and label the vertices , , , and : One type of symmetry we can define are once again, rotational symmetries of , , and which produce: Another type of symmetry we can define are axial flips along ...
• mathonline.wikidot.com

  The Dirichlet Convolution of Two Arithmetic Functions

  http://mathonline.wikidot.com/the-dirichlet-convolution-of-two-arithmetic-functions

  The Dirichlet Convolution of f and g is defined to be the function h = ∑d|n f(d)g(n d) and is denoted by h = f ∗ g. Theorem 1: Let f and g be arithmetic functions. If f and g are multiplicative then h = f ∗ g is multiplicative. Theorem 1 is significant. It says that if two arithmetic functions f and g are multiplicative then their ...
• mathonline.wikidot.com

  Changing The Order of Integration in Triple Integrals

  http://mathonline.wikidot.com/changing-the-order-of-integration-in-triple-integrals

  Then we will need to evaluate the triple integral ∭E f(x, y, z)dV in terms of triple iterated integrals. There will be six different orders of evaluating the triple iterated integrals. We can: Integrate with respect to z first, and then with respect to x and then y (Type 1 Region). Integrate with respect to z first, and then with respect to y ...
• mathonline.wikidot.com

  Path Connectedness of Open and Connected Sets in Euclidean …

  http://mathonline.wikidot.com/path-connectedness-of-open-and-connected-sets-in-euclidean-s

  Theorem 1: If A is an open and connected subset of Rn (with the usual topology) then A is path connected. Proof: Let Rn have the usual topology and let A ⊆Rn be an open and connected subset of Rn. Let x, y ∈ A and let R denote the collection of all open balls contained in A: (1) R = {B = B(x, r): x ∈ A, r > 0, B(x, r) ⊆ A}
• mathonline.wikidot.com

  The Method of Undetermined Coefficients Examples 1

  http://mathonline.wikidot.com/the-method-of-undetermined-coefficients-examples-1

  Example 1. Solve the following second order linear nonhomogeneous differential equation d2y dt2 + dy dt − 6y = 12e3t + 12e−2t using the method of undetermined coefficients. The corresponding second order homogeneous differential equation is d2y dt2 + dy dt − 6y = 0 and the characteristic equation is . The roots to the characteristic ...
• mathonline.wikidot.com

  Techniques for Solving Counting Problems - Mathonline

  http://mathonline.wikidot.com/techniques-for-solving-counting-problems

  One way to solve this counting problem is by assigning three placeholders, say , and replacing each with the number of options/choices we have for each of those positions. For the problem posed above, we first write: (1) In the first position we must choose one element in . There are elements in to choose from.
• mathonline.wikidot.com

  The Trinomial Theorem - Mathonline - Wikidot

  http://mathonline.wikidot.com/the-trinomial-theorem

  \begin{align} \quad (x + y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1} x^{n-1}y^1 + ... + \binom{n}{n-1}x^1y^{n-1} + \binom{n}{n} x^0y^n = \sum_{k=0}^{n} \binom{n}{k} x^{n ...
• mathonline.wikidot.com

  Parameterization of Curves in Three-Dimensional Space

  http://mathonline.wikidot.com/parameterization-of-curves-in-three-dimensional-space

  Example 4. Parameterize the curve of intersection of and in two different ways. We first note that represents a cylinder parallel to the -axis and with radius , while represents a plane that passes through the point , and so the intersection will be an ellipse. The easiest way to parameterize this curve of intersection is by letting and .
• mathonline.wikidot.com

  Topologies on Sets - Mathonline - Wikidot

  http://mathonline.wikidot.com/topologies-on-sets

  Topologies on Sets. Definition: Let X be a set. A Topology on X is a collection τ of subsets of X that satisfies the following properties: 1) X, ∅ ∈ τ. 2) If {Ui: i ∈ I} is any arbitrary collection of subsets of X such that Ui ∈ τ for all i ∈ I then the union ⋃i∈IUi ∈ τ. 3)) If {U1,U2,...,Un} is any finite collection of ...
• mathonline.wikidot.com

  Open and Closed Balls in Metric Spaces - Mathonline - Wikidot

  http://mathonline.wikidot.com/open-and-closed-balls-in-metric-spaces

  Now notice that ∥x −a∥ is simply the Euclidean distance function. d: R n → [ 0, ∞) for the metric space . We can extend the concept of open and closed balls to any metric space with its own defined metric as defined below. Definition: If is a metric space, , and then the Open Ball centered at with radius is defined to be the set .
• mathonline.wikidot.com

  The Algorithm for Doolittle's Method for LU Decompositions

  http://mathonline.wikidot.com/the-algorithm-for-doolittle-s-method-for-lu-decompositions

  Doolittle's method provides an alternative way to factor A into an LU decomposition without going through the hassle of Gaussian Elimination. Recall that for a general n × n matrix A, we assume that an decomposition exists, and write the form of and explicitly. We then systematically solve for the entries in in and from the equations that ...
• mathonline.wikidot.com

  Cantor's Theorem - Mathonline - Wikidot

  http://mathonline.wikidot.com/cantor-s-theorem

  Cantor's Theorem. We will now look at an prove a very important theorem regarding a set, its power set, and surjectivity. Theorem 1 (Cantor): If A is a set and P(A) represents the power set of A then there exists no function f: A → P(A) that is surjective. Proof of Theorem (by Contradiction): Suppose that there exists a surjection f: A → P(A).
• mathonline.wikidot.com

  Convergence of Filters and Filter Bases in a Topological Space

  http://mathonline.wikidot.com/convergence-of-filters-and-filter-bases-in-a-topological-spa

  Convergence of Filters and Filter Bases in a Topological Space. Definition: Let E be a topological space. If F is a filter, we say that F Converges to a, and write F → a, if every neighbourhood of a contains a set in F. Definition: Let E be a topological space. If B is a filter base, we say that B Converges to a, and write. B → a.
• mathonline.wikidot.com

  Limits of Functions of Two Variables - Mathonline - Wikidot

  http://mathonline.wikidot.com/limits-of-functions-of-two-variables

  For that reason, we establish the following definition for the non-existence of a limit.
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  calculus - On the differentiability of increasing functions

  https://math.stackexchange.com/questions/4632367/on-the-differentiability-of-increasing-functions

  I'm trying to follow the proof of this website http://mathonline.wikidot.com/lebesgue-s-theorem-for-the-differentiability-of …
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  Abstract Algebra Topics - Mathonline

  http://mathonline.wikidot.com/abstract-algebra

  Algebraic and Transcendental Elements in a Field Extension. The Minimal Polynomial of an Algebraic Element in a Field Extension. The Minimal Polynomial of √2 + √3 over Q. Extension Fields Generated by Elements. Splitting Fields of Polynomials. 6. Subnormal Series in a Group. 6.1. Subnormal Series.
• mathonline.wikidot.com

  Series - Mathonline - Wikidot

  http://mathonline.wikidot.com/series

  Definition: A series is said to be Convergent to the Sum if sequence of partial sums converges to , that is and so . A series is said to be Divergent if the sequence of partial …
• mathonline.wikidot.com

  Divisibility Rules for 1 to 10 - Mathonline - Wikidot

  http://mathonline.wikidot.com/divisibility-rules-for-1-to-10

  Divisibility Rules for 1 to 10. Before we discuss some important divisibility rules we will make precise what it means for an integer n to be divisible by another integer m. Definition 1: Let n and m be integers. n is said to be Divisible by m if there exists an integer k such that n = km. Equivalently, we can say that n is divisible by m if m ...
• mathonline.wikidot.com

  The Fundamental Theorem of Space Curves - Mathonline - Wikidot

  http://mathonline.wikidot.com/the-fundamental-theorem-of-space-curves

  We are now going to look at an extremely important theorem known as The Fundamental Theorem of Space Curves which says that two curves C1 and C2 with the same non-vanishing curvature function κ(s) and the same torsion function τ(s) will be congruent - that is each curve can be rigidly shifted so that all points on the curve coincide.
• mathonline.wikidot.com

  Unbounded Linear Functionals - Mathonline - Wikidot

  http://mathonline.wikidot.com/unbounded-linear-functionals

  \begin{align} \quad \lim_{n \to \infty} \| x - z_n \|_X &= \lim_{n \to \infty} \left \| x - \left (x - \frac{f(x)}{f(y_n)}y_n\right) \right \|_X \\ &= \lim_{n \to ...
• mathonline.wikidot.com

  Continuous Functions Between Topological Spaces - Mathonline

  http://mathonline.wikidot.com/continuous-functions-between-topological-spaces

  Continuous Functions. Definition: Let E and F be topological spaces. A function f: E → V is Continuous at x if for every neighbourhood Vf(x) of f(x) in F there exists a neighbourhood Ux of x in E such that f(u) ∈ V for all , or equivalently, . The function is said to be Continuous on if it is continuous at every point .

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