Posts Tagged ‘bounded’
Thursday, December 10th, 2009
Overview of the Bounded Monotone Sequence Theorem
Description
A detailed tutorial on the bounded monotone sequence theorem. Step by step tutorial including several examples of the bounded monotone sequence theorem for reference.
Overview
The bounded monotone sequence theorem actually has several parts to it. First, you need to find out if something is bounded above or bounded below. The sequence is bounded above if there exists a real number B such that x sub n is less than or equal to B. The sequence is bounded below if there exists a real number B such that x sub n is greater than or equal to B. If something is a bounded sequence, that means it is bounded both above and below. Absolute values are also very important in determining the bounded sequence. The bounded monotone sequence theorem states that for every bounded monotone sequence x, there is a real number L such that x sub n implies L.
Tags: above, absolute, algebra, below, bounded, boundedness, equal to, greater than, implies, less than, monotone, number, real, sequence, theorem, value
Posted in Algebra | No Comments »
Thursday, October 15th, 2009
Definition of Open and Closed Intervals
Description
A detailed tutorial on open and closed intervals. Step by step tutorial including several examples of open and closed intervals for reference.
Overview
An interval is a set of real numbers, expressed by an ordered pair. There are two types of intervals, open intervals and closed intervals. An open interval is an interval written with parenthesis. It implies that the endpoint is not included in the set. A closed interval is an interval written with brackets. It implies that the endpoint is included in the set. It is possible for one endpoint of an interval to be closed, and for the other to be open.
Tags: algebra, bounded, brackets, closed, coordinates, element, endpoint, interval, Math, open, ordered pair, parenthesis, real numbers, set
Posted in Algebra | No Comments »
Tuesday, October 6th, 2009
How to Solve a Dirichlet Problem
Description
A detailed tutorial of solving Dirichlet problems. Step by step tutorial including several examples of how to solve Dirichlet problems for reference.
Overview
A Dirichlet problem is a problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region. It was originally supposed to be used for Laplace’s equation, although other equations can use it as well. The Dirichlet problem can be stated as: given a function f that has values everywhere on the boundary of a region in R^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary? A mathematical solution can be expressed as:
Tags: bounded, continuous, differential equations, Dirichlet, equation, harmonic, interior, Laplace, Math, partial differential equation, problem, region, solution, value
Posted in Differential Equations | No Comments »
Thursday, October 1st, 2009
Boundedness Theorem Explained
Description
A detailed tutorial of the boundedness theorem. Step by step tutorial including an explanation of the boundedness theorem for reference. Knowledge of the boundedness theorem is required in calculus.
Overview
The boundedness theorem is a theorem that is very closely linked to the extreme value theorem. The boundedness theorem states that a continuous function f in the closed interval [a, b] is bounded on that interval. In mathematical terms, this means that there exist real numbers m and M such that ![m \le f(x) \le M\quad\text{for all }x \in [a,b].\,](http://s.wordpress.com/latex.php?latex=m%20%5Cle%20f%28x%29%20%5Cle%20M%5Cquad%5Ctext%7Bfor%20all%20%7Dx%20%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "m \le f(x) \le M\quad\text{for all }x \in [a,b].\,")
This translates to mean “m is less than or equal to f(x) which is less than or equal to M for all x belonging to [a, b]“.
Tags: a, b, bounded, boundedness theorem, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, m, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Extreme Value Theorem Explained
Description
A detailed tutorial of the extreme value theorem. Step by step tutorial including an explanation of the extreme value theorem for reference. Knowledge of the extreme value theorem is required in calculus.
Overview
The extreme value theorem states that if a real valued function f is continuous in the closed and bounded interval [a, b], then f must attain its maximum and minimum value at least once. In mathematical terms, this means that there exist numbers c and d in [a, b] such that ![f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,](http://s.wordpress.com/latex.php?latex=f%28c%29%20%5Cge%20f%28x%29%20%5Cge%20f%28d%29%5Cquad%5Ctext%7Bfor%20all%20%7Dx%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,")
The translation of that formula is “f(c) is greater than or equal to f(x) which is greater than or equal to f(d), for all x belonging to [a, b]“. In order for something to belong to an interval, it must be found in the interval.
Tags: a, b, bounded, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Friday, September 25th, 2009
The Heine-Borel Theorem Explained
Description
A detailed tutorial of the Heine-Borel theorem. Step by step tutorial including several examples of the Heine-Borel theorem for reference.
Overview
The Heine-Borel theorem is a concept in math that has to do with metric spaces. It states that for a subset S of Euclidian space R^n, the following two statements are equivalent: S is closed and bounded, and every open cover of S has a finite subcover, that is, S is compact. A more simple way of writing this theorem is that a subset of metric space is compact if and only if it is complete and totally bounded. Written in that form it is a biconditional statement.
Tags: biconditional, bounded, closed, compact, complete, discrete math, Eduard Heine, Emile Borel, equivalent, Euclidian space, finite subcover, Heine-Borel Theorem, Math, metric, spaces, subset, totally
Posted in Discrete Math | No Comments »
