- Vector calculus - From Wikipedia, the free encyclopedia
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics. Vector analysis has its origin in quaternion analysis, and was formulated by the American scientist, J. Willard Gibbs [1]. It concerns vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.
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Note: The summary of gradient, curl and divergence is particularly good. - Differential Equations - Wikibooks, collection of open-content textbooks
This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. It is hoped that this book will provide information about the whole area of differential equations, but for the moment it will concentrate on the simpler equations.
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Note: This stuff I found to be rather good: http://en.wikibooks.org/wiki/Differential_Equations/First_Order - Proof: "any topological space with the fixed point property is connected" - PlanetMath
Theorem Any topological space with the fixed-point property is connected. Proof. We will prove the contrapositive. ....
Note: A decent explanation of the fixed point property: http://planetmath.org/encyclopedia/FixedPointProperty.html - Bijection - wikipedia
In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f(x) = y. Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective).[1] (See also Bijection, injection and surjection.)
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- Bijection, injection and surjection - Wikipedia
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
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Note: Need to learn this. - Fixed point property - Wikipedia
In mathematics, a topological space X has the fixed point property if all continuous mappings from X to X have a fixed point.
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Note: In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function... http://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29 - Fixed Point Theorem Finite-Closed - Topology Q+A Board
Does a space which has the finite closed topology have the fixed-point property? I really don't know how to go about this, but my initial thoughts are: - This should be related to continuous functions and connectedness.
- Fixed-point theorem - Wikipedia
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
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- Implicit function theorem
In multivariable calculus of mathematics the implicit function theorem says that for a suitable set of equations, some of the variables are defined as functions of the others.
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- Range (mathematics) - Wikipedia
In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes it is called the image, or more precisely, the image of the domain of the function....The range should not be confused with the codomain B. The range is a subset of the codomain, but is not necessarily equal to the codomain, since there may be elements of the codomain which are not elements of the range. The codomain is sometimes taken to be the range, but more often is some standard set, such as the real numbers or the complex numbers, which contains the range. A function whose range equals its codomain is called onto or surjective.
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- Separation axiom - Wikipedia
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are axioms only in the sense that, when defining the notion of topological space, you could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces.
Note: See also: The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms... http://en.wikipedia.org/wiki/History_of_the_separation_axioms - Smooth function - Wikipedia, the free encyclopedia
In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i.e., has derivatives of all finite orders: * A function is called C, or more commonly C0, if it is a continuous function. * A function is called C1 if it has a derivative that is continuous; such functions are also called continuously differentiable. * A function is called Cn for n ≥ 1 if it can be differentiated n times, leaving a continuous n-th derivative: such functions are also called finitely differentiable. * The smooth functions are those that lie in the class Cn for all n; they are often referred to as C∞ functions.
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- Topology
Wikibooks, collection of open-content textbooks
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