Posts Tagged ‘form’
Friday, December 18th, 2009
Finding the Canonical Form of an Object
Description
A detailed tutorial of finding the canonical form of an object. Step by step tutorial including several examples of finding the canonical form of an object for reference.
Overview
Canonical form is also referred to as normal form or standard form. The canonical form of an object is a standard way of presenting that object. The process of finding a canonical form of something is referred to as canonization. Sometimes the word canonicalization is used instead. Canonical forms of objects are closly linked to differential forms of equations and numbers, and equivalence relations.
Tags: canonical, canonicalization, canonization, differential, discrete math, equation, equivalence, finding, form, normal, number, object, presenting, process, relations, standard
Posted in Discrete Math | No Comments »
Thursday, December 10th, 2009
Inverse Image of Sets
Description
A detailed tutorial on the inverse image of sets. Step by step tutorial on the inverse image of sets for reference. Knowledge of the inverse image of sets is important in advanced discrete mathematics courses.
Overview
Say that you have a function f: A –> B. Then, X is a subset of A and Y is a subset of B. The image of X or the image set of X is f(X) = {y belongs to B: y = f(x) for some x belonging to X}. The inverse image of Y is defined as f^-1(Y) = {x belongs to A: f(x) belongs to Y}. The inverse image is simply a reversed form of the image. Often when asked to find the inverse image, it will help to set up a drawing of the image of the function, connecting everything where it needs to go. Then to find the inverse you simply reverse your work.
Tags: a, b, connect, diagram, discrete math, form, function, image, image set, inverse, mapping, picture, reverse, set, subset, x, y
Posted in Discrete Math | No Comments »
Thursday, November 5th, 2009
Overview of the Dominated Convergence Theorem
Description
A detailed tutorial on the dominated convergence theorem. Step by step tutorial including several examples of the dominated convergence theorem for reference.
Overview
Unlike the monotone convergence theorem, the dominated convergence theorem only has one form. The official name of the theorem is Lebesgue’s Dominated Convergence Theorem, but most people just call it the dominated convergence theorem. It is considered to be a special version of the Fatou-Lebesque theorem, so Fatou’s lemma is used in direct proofs of this theorem. This theorem is also closely related to the bounded convergence theorem.
Tags: bounded proof, Calculus, convergence, direct, dominated, Fatou, form, Lebesque, lemma, monotone, special, theorem, version
Posted in Calculus | No Comments »
Thursday, November 5th, 2009
Overview of the Monotone Convergence Theorem
Description
A detailed tutorial on the monotone convergence theorem. Step by step tutorial including several examples of the monotone convergence theorem for reference.
Overview
There are several different theorems that the term “monotone convergence” can apply to. However, the most important one, and the one most common called the monotone convergence theorem, is the Lebesgue Monotone Convergence Theorem. This particular monotone convergence theorem deals with calculus, and with integrals and limits specifically. It is a more general form of the other two monotone convergence theorems, which is why it is considered to be the most important.
Tags: Calculus, converge, convergence, form, general, integral, Lebesgue, limit, monotone, number, real, sequence, series, theorem
Posted in Calculus | No Comments »
Friday, October 9th, 2009
Lissajous Curve Explained
Description
A detailed tutorial of a lissajous curve. Step by step tutorial including several visual examples of lissajous curves for reference.
Overview
A Lissajous curve represents the graph of a system of parametric equations, which can be mathematically expressed as 
. This also decribes complex harmonic motion. The way that the figure appears is very sensitive to the ratio a / b, so the figure can appear in many different forms.
Tags: Bowditch, Calculus, complex, curve, equation, figure, form, graph, harmonic, Lissajous, Math, motion, paramentric, ratio, system
Posted in Calculus | No Comments »
