:: wikimiki.org ::

Tangent Vector

Tangent vector

The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.

Informal description

In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces have the same dimension, equal to the dimension of the manifold. For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that point and is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space one can picture the tangent space in this literal fashion. In algebraic geometry, in contrast, there is an intrinsic definition of tangent space at a point P of a variety V, that gives a vector space of dimension at least that of V. The points P at which the dimension is exactly that of V are called the non-singular points; the others are singular points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of V are those where the 'test to be a manifold' fails. See Zariski tangent space. Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle of the manifold.

Formal definitions

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition as directions of curves

Suppose M is a C^k manifold (k ≥ 1) and p is a point in M. Pick a chart φ : U → Rⁿ where U is an open subset of M containing p. Suppose two curves γ₁ : (-1,1) → M and γ₂ : (-1,1) → M with γ₁(0) = γ₂(0) = p are given such that φ o γ₁ and φ o γ₂ are both differentiable at 0. Then γ₁ and γ₂ are called tangent at 0 if the ordinary derivatives of φ o γ₁ and φ o γ₂ at 0 coincide. This is an equivalence relation, and the equivalence classes are known as the tangent vectors of M at p. The equivalence class of the curve γ is written as γ'(0). The tangent space of M at p, denoted by T_pM, is defined as the set of all tangent vectors; it does not depend on the choice of chart φ. To define the vector space operations on T_pM, we use a chart φ : U → Rⁿ and define the map (dφ)_p : T_pM → Rⁿ by (dφ)_p(γ'(0)) = (φ o γ)'(0). It turns out that this map is bijective and can thus be used to transfer the vector space operations from Rⁿ over to T_pM, turning the latter into an n-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not.

Definition via derivations

Suppose M is a C^∞ manifold. A real-valued function g : M → R belongs to C^∞(M) if g o φ^-1 is infinitely often differentiable for every chart φ : U → Rⁿ. C^∞(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication. Pick a point p in M. A derivation at p is a linear map D : C^∞(M) → R which has the property that for all g, h in C^∞(M): :D(gh) = D(g)·h(p) + g(p)·D(h) modeled on the product rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space T_pM. The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is D(g) = (g o γ)'(0) (where the derivative is taken in the ordinary sense, since g o γ is a function from (-1,1) to Rⁿ).

Definition via the cotangent space

Again we start with a C^∞ manifold M and a point p in M. Consider the ideal I in C^∞(M) consisting of all functions g such that g(p) = 0. Then I and I² are real vector spaces, and T_pM may be defined as the dual space of the quotient space I / I². This latter quotient space is also known as the cotangent space of M at p. While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry. If D is a derivation, then D(g) = 0 for every g in I², and this means that D gives rise to a linear map I / I² → R. Conversely, if r : I / I² → R is a linear map, then D(g) = r((g - g(p)) + I²) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space.

Properties

If M is an open subset of Rⁿ, then M is a C^∞ manifold in a natural manner (take the charts to be the identity maps), and the tangent spaces are all naturally identified with Rⁿ.

Tangent vectors as directional derivatives

One way to think about tangent vectors is as directional derivatives. Given a vector v in Rⁿ one defines the directional derivative of a smooth map f : Rⁿ→R at a point p by :

D_v f(p) = \frac\bigg|_f(p+tv)=\sum_^v^i\frac(p).

This map is naturally a derivation. Moreover, it turns out that every derivation of C^∞(Rⁿ) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations. Since tangent vectors to a general manifold can be defined as derivations it is natural to think of them as directional derivatives. Specifically, if v is a tangent vector of M at a point p (thought of as a derivation) then define the directional derivative in the direction v by :D_v(f) = v(f) where f : M → R is an element of C^∞(M). If we think of v as the direction of a curve, v = γ'(0), then we write :D_v(f) = (f o γ)'(0).

The derivative of a map

Main article: pushforward Every differentiable map f : M → N between C^k manifolds induces natural linear maps between the corresponding tangent spaces: :(df)_p : T_pM → T_f(p)N defined by :(df)_p(γ'(0)) = (f o γ)'(0) if the tangent space is defined via curves and by :(df)_p(D)(g) = D(g o f) if the tangent space is defined via derivations. The linear map (df)_p is called variously the derivative, total derivative, differential, or pushforward of f at p. It is frequently expressed using a variety of other notations : df_p, Df_p, f_∗, f′(p). In a sense, the derivative is the best linear approximation to f near p. Note that when N = R, the map (df)_p : T_pM→R coincides with the usual notion of the differential of the function f. In local coordinates the derivative of f is given by the Jacobian. An important result regarding the derivative map is the following: :Theorem. If f : M → N is a local diffeomorphism at p in M then (df)_p : T_pM → T_f(p)N is a linear isomorphism. Conversely, if (df)_p is an isomorphism then there is an open neighborhood U of p such that f maps U diffeomorphically onto its image. This is a generalization of the inverse function theorem to maps between manifolds. Category:Differential topology Category:Differential geometry ja:接ベクトル空間

Differential geometry

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using calculus. These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical systems.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one (see the Nash embedding theorem).

Technical requirements

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature. A differential manifold is a topological space with a collection of homeomorphisms from open sets of the space to open subsets in Rⁿ such that the open sets cover the space, and if f, g are homeomorphisms then the function f o g ^-1 from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every homeomorphism results in an infinitely differentiable function from the open unit ball to R. At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rⁿ. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability. A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point. An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^- of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

Differential topology

Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). Smooth manifolds are 'softer' than manifolds with extra geometric stuctures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume. Conversely, smooth manifolds are more rigid than the topological manifolds. Certain topological manifolds have no smooth structures at all (see Donaldson's theorem) and others have more than one inequivalent smooth structure (such as exotic spheres). Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.

Branches of differential geometry

Contact geometry

Contact geometry is an analog of symplectic geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form

\alpha

such that

\alpha\wedge (d\alpha)^n

does not vanish anywhere.

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is much more general structure than a Riemannian metric.

Riemannian geometry

Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space. These allow one to generalise the notion from Euclidean geometry and analysis such as gradient of a function, divergence, length of curves and so on; without assumptions that the space is globally so symmetric. The Riemannian curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.

Symplectic topology

Symplectic topology is the study of symplectic manifolds, which can occur only in even dimensions. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form). Unlike in Riemannian geometry, all symplectic manifolds are locally isomorphic, so the only invariants of a symplectic manifold are global in nature.

External links

- [http://rsp.math.brandeis.edu/3D-XplorMath/Surface/a/bk/curves_surfaces_palais.pdf A Modern Course on Curves and Surface, Richard S Palais, 2003]
- [http://rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery.html Richard Palais's 3DXM Surfaces Gallery]
- [http://www.cs.elte.hu/geometry/csikos/dif/dif.html Balázs Csikós's Notes on Differential Geometry]

Reference books

1. A Comprehensive Introduction to Differential Geometry (5 Volumes), 3rd Edition by Michael Spivak (1999) 2. Differential Geometry of Curves and Surfaces by Manfredo Do Carmo (1976). A classical geometric approach to differential geometry without the tensor machinery. 3. Riemannian Geometry by Manfredo Perdigao do Carmo, Francis Flaherty (1994) 4. Geometry from a Differentiable Viewpoint by John McCleary (1994) 5. A First Course in Geometric Topology and Differential Geometry by Ethan D. Bloch (1996) 6. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. by Alfred Gray (1998) ja:微分幾何学

Vector Space

Vector space

Dimension of a vector space

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension. All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the dimension of a vector space is uniquely defined. The dimension of the vector space V over the field F can be written as dim_F(V) or as [V : F], read "dimension of V over F". We say V is finite-dimensional if the dimension of V is finite.

Examples

E.g. The vector space R³ has as a basis, and therefore we have dim_R(R³) = 3. More generally, dim_R(Rⁿ) = n. And more generally still, dim_F(Fⁿ) = n. The complex numbers C are both a real and complex vector space; we have dim_R(C) = 2 and dim_C(C) = 1. So the dimension depends on the base field. The only vector space with dimension 0 is , the vector space consisting only of its zero element.

Facts

If W is a linear subspace of V, then dim(W) ≤ dim(V). To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite-dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V. Any two vector spaces over F having the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If B is some set, a vector space with dimension |B| over F can be constructed as follows: take the set F^(B) of all functions f : B → F such that f(b) = 0 for all but finitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the desired F-vector space. An important result about dimensions related to a linear transformation is given by the rank-nullity theorem. If F/K is a field extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also a K-vector space. The dimensions are related by the formula :dim_K(V) = dim_K(F) dim_F(V). In particular, every complex vector space of dimension n is a real vector space of dimension 2n. Some simple formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field F then, denoting the dimension of V by dimV, we have: :If dimV is finite, then |V| = |F|^dimV. :If dimV is infinite, then |V| = max(|F|, dimV).

Generalizations

One can see a vector space as a particular case of a matroid, and in the latter there is a well defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces.

Perpendicular

:For the Deep Purple album, see Purpendicular. right Perpendicular is a geometric term that may be used as a noun or adjective. The fundamental meaning pertains to the position of straight lines relative to one another. Two lines are said to be perpendicular if they meet at a right angle. Note that two line segments positioned at 90° to one another are perpendicular only if they meet. Two lines are considered perpendicular if the product of their slopes is -1. Naturally, if a line is given, then a perpendicular is any line at a ninety-degree angle to that line. This is an important property in geometry and trigonometry since important properties accrue to line systems containing right angles. When graphing, the convention is to use either an X and Y axis, or to use an X, Y, and Z axis, which are defined as being mutually perpendicular. Right triangles, too, include two perpendicular lines and so have special properties, which are the foundation of trigonometry. Compare to parallel.

Formula

When given 2 straight lines A and B, with A: y = ωx + a; B: y = ω'x + b A and B are perpendicular in an orthonormal base (where X-axis and Y-axis are perpendicular and the distance between (0,0) and (1,0) is equal to the distance between (0,0) and (0,1)) if ω
- ω'=-1. This fact can also lead to funny (but correct) results with imaginary lines: e.g. the line y = ix is perpendicular with a line with ω = i, we see now that the line y = ix is perpendicular with itself! This seems odd but is nevertheless correct. For people interested in this strange fact and who like to give this a little thought, it needs to be said that imaginary lines can be seen as cubes in a 4-dimension space.

Euclidean space

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finite-dimensional, real, inner product space. A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness. An inner product space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within functional analysis. Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry. One mathematical motivation for defining a distance function is the ability to define an open ball around points in the space. This fundamental concept justifies a differential calculus between a Euclidean space and other manifolds. Differential geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of non-Euclidean manifolds.

Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rⁿ. An element of Rⁿ is written x = (x₁, x₂, …, x_n) where each x_i is a real number. The vector space operations on Rⁿ are defined by :

\mathbf + \mathbf = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)

a\,\mathbf = (a x_1, a x_2, \ldots, a x_n)

Real coordinate space Rⁿ comes with a standard basis: :

\mathbf_1 = (1, 0, \ldots, 0)

\mathbf_2 = (0, 1, \ldots, 0)

\vdots

\mathbf_n = (0, 0, \ldots, 1)

An arbitrary vector in Rⁿ can then be written in the form :

\mathbf = \sum_^n x_i \mathbf_i

Real coordinate space is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rⁿ. This isomorphism is not canonical however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rⁿ in V). The reason for working with arbitrary vector spaces instead of Rⁿ is that it is often preferable to work in a coordinate-free manner (i.e. without choosing a preferred basis).

Euclidean structure

Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distance between points and the angles between lines or vectors. The natural way in which to do this is to introduce what is called an inner product or dot product on Rⁿ. This product is defined by :

\mathbf\cdot\mathbf = \sum_^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

The dot product of any two vectors x and y gives a real number. This product allows us to define the "length" of a vector x in the following way :

\|\mathbf\| = \sqrt = \sqrt

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rⁿ. The (interior) angle θ between x and y is then given by :

\theta = \cos^\left(\frac\right)

where cos⁻¹ is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on Rⁿ in the following manner :

d(\mathbf, \mathbf) = \|\mathbf - \mathbf\| = \sqrt.

The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric. Real coordinate space together with the above Euclidean structure (dot product and the associated norm and metric) is called Euclidean space often denoted by Eⁿ. (Many authors refer to Rⁿ itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on Eⁿ gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Euclidean topology

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on Eⁿ is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rⁿ considered as a product of n copies of the real line R (with its standard topology). An important result on the topology of Rⁿ, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rⁿ (with its subspace topology) which is homeomorphic to another open subset of Rⁿ is itself open. An immediate consequence of this is that R^m is not homeomorphic to Rⁿ if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove. Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rⁿ is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces. Euclidean space is also known as linear manifold. An m-dimensional linear submanifold of Rⁿ is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space.

Algebraic variety

:This article is about algebraic varieties. For varieties of algebras, and an explanation of the difference, see variety (universal algebra). An affine algebraic variety is essentially the set of common zeroes of a set of polynomials, and is one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory.

Definition

Let k be a field and let kⁿ be affine n-space over k. The polynomials f in k[x₁, ..., x_n] can be viewed as k-valued functions on kⁿ by evaluating f at the points in kⁿ. This gives a ring of k-valued functions on kⁿ called the coordinate ring of kⁿ. By considering the set of common zeros of a set of functions, each subset S of the coordinate ring determines a subset Z(S) of affine space. Let I(S) be the ideal of all functions vanishing on the subset S of affine space. The quotient of the polynomial ring by this ideal is the coordinate ring of the affine algebraic variety. A subset V of kⁿ is called an affine algebraic set (or simply an algebraic set) if V = Z(S) for some subset S of the coordinate ring. There are a number of close connections between an algebraic set and its corresponding coordinate ring that we will not detail here. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is called an affine variety.

Basic results

- An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
- Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties (where none of the sets in the decomposition are subsets of each other).
- Let k[V] be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k[V] over k.

Discussion and generalizations

The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The current notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals. This definition works over any field K. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to problems since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. These are usually not considered varieties, and we get rid of them by requiring the schemes underlying a variety to be separated. (There is strictly speaking also a third condition, namely, that in the definition above one needs only finitely many affine patches.) Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety simply mean that the affine charts have trivial nilradical. Here are some interesting subclasses of varieties. A projective variety is a variety which admits an embedding into projective space. A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa. These varieties have been called 'varieties in the sense of Serre', since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way. One way that leads to generalisations is to allow reducible algebraic sets (and fields K that aren't algebraically closed), so the rings R may not be integral domains. This is not a big step technically. More serious is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in co-ordinate rings aren't seen as co-ordinate functions. From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products). Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of schemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety. There are further generalizations called stacks and algebraic spaces.

References

- Robin Hartshorne, Algebraic Geometry, Springer-Verlag (1977)
- David Cox, John Little, Donal O'Shea, Ideals, Varieties and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, (1997), Springer-Verlag ISBN 0-387-94680-2.
- David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, Springer.
- David Dummit and Richard M. Foote. Abstract Algebra, Wiley. ko:대수다양체
-

Zariski tangent space

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations. For example, suppose given a plane curve C defined by a polynomial equation :F(X,Y) = 0 and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get a 'linearised' equation reading :L(X,Y) = 0 in which all terms X^aY^b have been discarded if :a + b > 1. We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.) It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2. The definition given generalises directly to higher dimensions, in which case a number of equations may be involved in defining a variety V. The non-linear terms are dropped from all of them, giving a system of linear equations that define the tangent space. The definition of singular point is then that the dimension of the tangent space is the dimension of V. For more abstract theory, one notes that for any commutative local ring R, with maximal ideal M, there is the definition :M/M² of an R-module, in terms of which the previous definitions can be recovered. For R coming from geometry over a field K, this will be a vector space over K. It therefore serves as an abstract analogue, and is also called the Zariski tangent space. It has an interpretation in terms of homomorphisms to the dual numbers for K, :K[t]/[t²] which (thinking about affine schemes) allows one to speak in geometric terms, talking about tangent vectors. Category:Algebraic geometry Category:Differential algebra

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of the manifold's tangent bundle.

Definition

Given a subset S in Rⁿ a vector field is represented by a vector-valued function

V: S \to \mathbf^n

in standard Euclidean coordinates (x₁, ..., x_n). If there is another coordinate system y, then

V_y := \frac V

is the expression for the same vector field in the new coordinates. In particular a vector field is not a bunch of scalar fields. We say V is a C^k vector field if V is k times continuously differentiable. A point p in S is called stationary if the vector at that point is zero (

V(p) = 0

). A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point. Given two C^k-vector fields V, W defined on S and a real valued C^k-function f defined on S, the two operations scalar multiplication and vector addition :

(fV)(p) := f(p)V(p)

(V+W)(p) := V(p) + W(p)

define the module of C^k-vector fields over the ring of C^k-functions.

Notes

Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space (or every point of some manifold). The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. The first of these operations is defined in any number of dimensions (that is, for any value of n). The curl however, is defined only for n=3, but it can be generalized to an arbitrary dimension using the exterior product and exterior derivative.

Examples

- A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
- Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid.
- There are 3 types of lines that can be made from vector fields. They are : ::streaklines — as revealed in wind tunnels using smoke. ::fieldlines — as a line depicting the instantaneous field at a given time. ::pathlines — showing the path that a given particle (of zero mass) would follow.
- Magnetic fields. The fieldlines can be revealed using small iron filings.
- Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.

Gradient field

Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on X(a scalar field) such that

V = \nabla f.

The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero. :

\int_\gamma \langle V(x), \mathrmx \rangle = \int_\gamma \langle \nabla f(x), \mathrmx \rangle = f((\gamma)(1)) - f((\gamma)(0))

Central field

A C^∞-vector field over Rⁿ \ is called a central field if :

V(T(p)) = T(V(p)) \qquad (T \in \mathrm(n, \mathbf))

where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called the center of the field. Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Curve integral

A common technique in physics is to integrate a vector field along a curve: a path integral. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path. The curve integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector field V and a curve γ parametrized by [0, 1] the curve integral is defined as :

\int_\gamma \langle V(x), \mathrmx \rangle = \int_0^1 \langle V(\gamma(t)), \gamma'(t)\;\mathrmt \rangle

Flow curves

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations. Given a vector field V defined on S, we can try to define curves γ on S such that for each t in an interval I :

\gamma'(t) = V(\gamma(t))

If V is Lipschitz continuous we can find a unique C¹-curve γ_x for each point x in X so that :

\gamma_x(0) = x

\gamma'_x(t) = V(\gamma_x(t)) \qquad ( t \in (-\epsilon, +\epsilon) \subset \mathbf)

The curves γ_x are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (-ε, +ε) to the whole real number line. The flow may for example reach the edge of S in a finite time. In two or three dimensions one can visualize the vector field as given rise to a flow on S. If we drop a particle into this flow at a point p it will move along the curve γ_p in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p. Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups.

Difference between scalar and vector field

The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates.

Example 1

This example is about 2-dimensional Euclidean space (R²) where we examine Euclidean (x, y) and polar (r, θ) coordinates (which are undefined at the origin). Thus x = r cos θ and y = r sin θ and also r² = x² + y², cos θ = x/(x² + y²) and sin θ = y/(x² + y²). Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the r-direction with length 1 to each point. More precisely, they are given by the functions :

s_:(r, \theta) \mapsto 1, \quad v_:(r, \theta) \mapsto (1, 0).

Let us convert these fields to Euclidean coordinates. The vector of length 1 in the r-direction has the x coordinate cos θ and the y coordinate sin θ. Thus in Euclidean coordinates the same fields are described by the functions :

s_:(x, y) \mapsto 1, \quad v_:(x, y) \mapsto (\cos \theta, \sin \theta) = \left(\frac, \frac\right).

We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by the next example.

Example 2

Consider the 1-dimensional Euclidean space R with its standard Euclidean coordinate x. Also consider the coordinate ξ := 2x. Suppose we have a scalar field and a vector field which are both given in the ξ coordinates by the constant function 1, :

s_:\xi \mapsto 1, \quad v_:\xi \mapsto 1.

Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then x changes 2 units. Then, this vector field has a magnitude of 2 in units of x. Therefore, in the x coordinate the scalar field and the vector field are described by the functions :

s_:x \mapsto 1, \quad v_:x \mapsto 2,

which are different.

Example 3

In 1D, an example of a scalar field is the electric potential V, which is e.g. 20 volt at a particular point. This is a scalar, not depending on the coordinate system. An electric field at that point of 5 volt/metre in some coordinate system is −5 volt/metre in an inverse coordinate system. Since a physical quantity is not just a number, but a number times a unit, there is no change of coordinate system that gives any other than one of these two values for the electric field at the point.

External links

- [http://mathworld.wolfram.com/VectorField.html Vector field] -- Mathworld
- [http://planetmath.org/encyclopedia/VectorField.html Vector field] -- PlanetMath
- [http://www.amasci.com/electrom/statbotl.html 3D Magnetic field viewer]
- [http://www-solar.mcs.st-and.ac.uk/~alan/MT3601/Fundamentals/node2.html Vector fields and field lines]
- [http://www.vias.org/simulations/simusoft_vectorfields.html Vector field simulation] An interactive application to show the effects of vector fields Category:Differential topology Category:Vector calculus Category:Infographics

Ordinary differential equation

:ODE redirects here. For the real-time physics engine, see open dynamics engine. In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is :

f' = f \,

, where

f \,

is an unknown function, and

f'\,

is its derivative. See differential calculus and integral calculus for basic calculus background. An ODE may be thought of as an equation depending on a single spatial variable. An ODE is called autonomous if it is time-independent, and nonautonomous otherwise. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Bendixson theorem.

Definition

Let y represent an unknown function of x, and let :

y', y

,\ \dots,\ y^ denote the derivatives : $\frac,\ \frac,\ \dots,\ \frac.$ An ordinary differential equation (ODE) is an equation involving : $x,\ y,\ y',\ y$ ,\ \dots . The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of the equation is the highest power of the highest derivative. A solution of an ODE is a function y(x) whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique. A general solution of an nth-order equation is a solution containing

n

arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. A singular solution is a solution that can't be derived from the general solution. When a differential equation of order n has the form :

F\left(x, y', y

,\ \dots,\ y^\right) = 0 it is called an implicit differential equation whereas the form : $F\left(x, y', y$ ,\ \dots,\ y^\right) = y^ is called an explicit differential equation. A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

General application

An important special case is when the equations do not involve

x

. These differential equations may be represented as vector fields. This type of differential equation has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also symplectic topology for abstract discussion.) In the case where the equations are linear, the original equation can be solved by breaking it down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see numerical ordinary differential equations). Ordinary differential equations are to be distinguished from partial differential equations where

y

is a function of several variables, and the differential equation involves partial derivatives.

Existence and nature of solutions

The problem of solving a differential equation is to find the function

y

whose derivatives satisfy the equation. For example, the differential equation :

y

+ y = 0 \, has the general solution : $y = A \cos + B \sin \,$ , where A, B are constants determined from boundary conditions. In general, an n-th order equation allows both $x$ and $y$ to be fixed, as well as all the $n-1$ lower order derivatives of $y$ ; the remaining equation can be solved (at least conceptionally) for $y^$ . If the equation has finite degree $d$ , then we now have a polynomial equation in $y^$ with at most $d$ roots. Therefore there can be as many as $d$ possible values for $y^$ at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A Lipschitz condition must also be satisfied for a solution to exist. Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of $y$ exists for any value of

y

). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one. Lipschitz condition Consider now :

(y')^2 + xy' - y = 0 \,

with general solution :

y = Ax + A^2 \,

This is a first-order, second-degree equation, thus any point can have at most two solutions passing through it, corresponding to the two roots of

y'

in the quadratic equation that would result after fixing

x

and

y

. Studying the quadratic equation's discriminant (

x^2 + 4y

) leads to the conclusion that only a single solution exists along the parabola

y = - \frac x^2

(where the discriminant is zero) and that no solution exists below this parabola (where both roots are complex). The parabola in this problem is an example of a cusp locus; a curve along which two or more roots of the differential equation are identical. Along such a locus it is possible to move from one general solution to another while still obeying the differential equation; thus the presence of cusp loci introduce the possibility of singular solutions. In this example, the parabola

y = - \frac x^2

is such a singular solution; it satisfies the original differential equation, and a full set of solutions must include such possibilities as the hybrid solution:

y = \begin x + 1, & \mbox x < -2 \\ - \frac x^2, & \mbox -2 <= x < 2 \\ -x + 1, & \mbox x >= 2 \end

where the cusp locus has been used to connect two particular solutions; note that the first derivative (the only derivative to appear in the differential equation) is continuous at the transitions. ([http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 Johnson], Chapter 5)

Types of differential equations with some history

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients.

Homogeneous linear ODEs with constant coefficients

The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form

e^

, for possibly-complex values of

z

. Thus :

\frac   + A_\frac   + \cdots + A_y = 0

has the form :

z^n e^ + A_1 z^ e^ + \cdots + A_n e^ = 0

so dividing by

e^

gives the nth-order polynomial :

F(z) = z^ + A_z^ + \cdots + A_n = 0

In short the terms :

\frac  \quad\quad(k = 1, 2, \cdots, n).

of the original differential equation are replaced by z^k. Solving the polynomial gives n values of z,

z_1, \dots,z_n

. Plugging those values into

e^

gives a basis for the solution; any linear combination of these

e^

will satisfy the differential equation. This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy. If z is a (possibly not real) zero of F(z) of multiplicity m and $k\in\ \,$ then $y=x^ke^ \,$ is a solution of the ODE. These functions make up a basis of the ODE's solutions. If the A_i are real then real-valued solutions are preferable. Since the non-real z values will come in conjugate pairs, so will their corresponding ys; replace each pair with their linear combinations Re(y) and Im(y). A case that involves complex roots can be solved with the aid of Euler's formula.
- Example: Given $y$ -4y'+5y=0 \,. The characteristic equation is

z^2-4z+5=0 \,

which has zeroes 2+i and 2−i. Thus the solution basis

\

\ \,

. Now y is a solution iff

y=c_1y_1+c_2y_2 \,

for

c_1,c_2\in\mathbb C

. Because the coefficients are real,
- we are likely not interested in the complex solutions
- our basis elements are mutual conjugates The linear combinations :

u_1=\mbox(y_1)=\frac=e^\cos(x) \,

and :

u_2=\mbox(y_1)=\frac=e^\sin(x) \,

will give us a real basis in

\

Linear ODEs with constant coefficients

Suppose instead we face :

\frac   + A_\frac   + \cdots + A_y = f(x)

For later convenience, define the characteristic polynomial :

P(v)=v^n+A_1v^+\cdots+A_n

We find the solution basis

\

as in the homogeneous (f=0) case. We now seek a particular solution y_p by the variation of parameters method. Let the coefficients of the linear combination be functions of x: :

y_p=u_1y_1+u_2y_2+\cdots+u_ny_n

Using the "operator" notation

D=d/dx

and a broad-minded use of notation, the ODE in question is

P(D)y=f

; so :

f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\cdots+P(D)(u_ny_n)

With the constraints :

0=u'_1y_1+u'_2y_2+\cdots+u'_ny_n

0=u'_1y'_1+u'_2y'_2+\cdots+u'_ny'_n

:… :

0=u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

the parameters commute out, with a little "dirt": :

f=u_1P(D)y_1+u_2P(D)y_2+\cdots+u_nP(D)y_n+u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

But

P(D)y_j=0

, therefore :

f=u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

This, with the constraints, gives a linear system in the

u'_j

. This much can always be solved; in fact, combining Cramer's rule with the Wronskian, :

u'_j=(-1)^f\frac

The rest is a matter of integrating

u'_j

. The particular solution is not unique;

y_p+c_1y_1+\cdots+c_ny_n

also satisfies the ODE for any set of constants c_j. See also variation of parameters. Example: Suppose

y

-4y'+5=sin(kx). We take the solution basis found above $\$ . : : : Using the list of integrals of exponential functions : : And so : (Notice that u₁ and u₂ had factors that canceled y₁ and y₂; that is typical.) For interest's sake, this ODE has a physical interpretation as a driven damped harmonic oscillator; y_p represents the steady state, and $c_1y_1+c_2y_2$ is the transient.
Linear ODEs with variable coefficient

Method of undetermined coefficients
The method of undetermined coefficients (MoUC), is useful in finding solution for $y_p$ . Given the ODE $P(D)y = f(x)$ , find another differential operator $A(D)$ such that $A(D)f(x) = 0$ . This operator is called the annihilator, and thus the method of undetermined coefficients is also known as the annihilator method. Applying $A(D)$ to both sides of the ODE gives an homogeneous ODE $\big(A(D)P(D)\big)y = 0$ for which we find a solution basis $\$ as before. Then the original nonhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combinations to satisfy the ODE. Undetermined coefficients is not as general as variation of parameters in the sense that an annihilator does not always exist. Example: Given $y$ -4y'+5=\sin(kx),

P(D)=D^2-4D+5

. The simplest annihilator of

\sin(kx)

A(D)=D^2+k^2

. The zeros of

A(z)P(z)

are

\

, so the solution basis of

A(D)P(D)

\=\

. Setting

y=c_1y_1+c_2y_2+c_3y_3+c_4y_4

we find : giving the system :

i=(k^2+4ik-5)c_3+(-k^2+4ik+5)c_4

0=(k^2+4ik-5)c_3+(k^2-4ik-5)c_4

which has solutions :

c_3=\frac i

c_4=\frac i

giving the solution set :

Method of variation of parameters

As explained above, the general solution to a non-homogeneous, linear differential equation

y

(x) + p(x) y'(x) + q(x) y(x) = g(x) can be expressed as the sum of the general solution $y_h(x)$ to the corresponding homogenous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0 and any one solution

y_p(x)

y

(x) + p(x) y'(x) + q(x) y(x) = g(x). Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x), having already found the general solution to

y

(x) + p(x) y'(x) + q(x) y(x) = 0. Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use. For a second-order equation, the method of variation of parameters makes use of the following fact:
Fact
Let p(x), q(x), and g(x) be functions, and let $y_1(x)$ and $y_2(x)$ be solutions to the homogeneous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0. Further, let u(x) and v(x) be functions such that

u'(x) y_1(x) + v'(x) y_2(x) = 0

and

u'(x) y_1'(x) + v'(x) y_2'(x) = g(x)

for all x, and define

y_p(x) = u(x) y_1(x) + v(x) y_2(x)

. Then

y_p(x)

is a solution to the non-homogeneous, linear differential equation

y

(x) + p(x) y'(x) + q(x) y(x) = g(x).
Proof
$y_p(x) = u(x) y_1(x) + v(x) y_2(x)$ $y_p$ (x) + p(x) y'_p(x) + q(x) y_p(x) = g(x) + u(x) y_1(x) + v(x) y_2(x) + p(x) u(x) y_1'(x) + p(x) v(x) y_2'(x) + q(x) u(x) y_1(x) + q(x) v(x) y_2(x) $= g(x) + u(x) (y_1$ (x) + p(x) y_1'(x) + q(x) y_1(x)) + v(x) (y_2(x) + p(x) y_2'(x) + q(x) y_2(x)) = g(x) + 0 + 0 = g(x)
Usage
To solve the second-order, non-homogeneous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x) using the method of variation of parameters, use the following steps: #Find the general solution to the corresponding homogeneous equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0. Specifically, find two linearly independent solutions $y_1(x)$ and $y_2(x)$ . #Since $y_1(x)$ and $y_2(x)$ are linearly independent solutions, their Wronskian $y_1(x) y_2'(x) - y_1'(x) y_2(x)$ is nonzero, so we can compute $-(g(x) y_2(x))/()$ and $()/()$ . If the former is equal to u(x) and the latter to v(x), then u and v satisfy the two constraints given above: that $u'(x) y_1(x) + v'(x) y_2(x) = 0$ and that $u'(x) y_1'(x) + v'(x) y_2'(x) = g(x)$ . We can tell this after multiplying by the denominator and comparing coefficients. #Integrate $-(g(x) y_2(x))/()$ and $()/()$ to obtain u(x) and v(x), respectively. (Note that we only need one choice of u and v, so there is no need for constants of integration.) #Compute $y_p(x) = u(x) y_1(x) + v(x) y_2(x)$ . The function $y_p$ is one solution of $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x). #The general solution is $c_1 y_1(x) + c_2 y_2(x) + y_p(x)$ , where $c_1$ and $c_2$ are arbitrary constants.
Higher-order equations
The method of variation of parameters can also be used with higher-order equations. For example, if $y_1(x)$ , $y_2(x)$ , and $y_3(x)$ are linearly independent solutions to $y(x) + p(x) y$ (x) + q(x) y'(x) + r(x) y(x) = 0, then there exist functions u(x), v(x), and w(x) such that $u'(x) y_1(x) + v'(x) y_2(x) + w'(x) y_3(x) = 0$ , $u'(x) y_1'(x) + v'(x) y_2'(x) + w'(x) y_3'(x) = 0$ , and $u'(x) y_1$ (x) + v'(x) y_2(x) + w'(x) y_3(x) = g(x). Having found such functions (by solving algebraically for u(x), v(x), and w(x), then integrating each), we have $y_p(x) = u(x) y_1(x) + v(x) y_2(x) + w(x) y_3(x)$ , one solution to the equation $y(x) + p(x) y$ (x) + q(x) y'(x) + r(x) y(x) = g(x).
Example
Solve the previous example, $y$ + y = \sec x Recall $\sec x = \frac = f$ . From technique learned from 3.1, LHS has root of $r = \pm i$ that yield $y_c = C_1 \cos x + C_2 \sin x$ , (so $y_1 = \cos x$ , $y_2 = \sin x$ ) and its derivatives : $\left\$
Curve
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).
Definitions
In mathematics, a (topological) curve is defined as follows. Let $I$ be an interval of real numbers (i.e. a non-empty connected subset of $\mathbb$ ). Then a curve $\!\,\gamma$ is a continuous mapping $\,\!\gamma : I \rightarrow X$ , where $X$ is a topological space. The curve $\!\,\gamma$ is said to be simple if it is injective, i.e. if for all $x$ , $y$ in $I$ , we have $\,\!\gamma(x) = \gamma(y) \rightarrow x = y$ . If $I$ is a closed bounded interval $\,\![a, b]$ , we also allow the possibility $\,\!\gamma(a) = \gamma(b)$ (this convention makes it possible to talk about closed simple curve). If $\gamma(x)=\gamma(y)$ for some $x\ne y$ (other than the extremities of $I$ ), then $\gamma(x)$ is called a double (or: multiple) point of the curve. A curve $\!\,\gamma$ is said to be closed or a loop if $\,\!I = [a, b]$ and if $\!\,\gamma(a) = \gamma(b)$ . A closed curve is thus a continuous mapping of the circle $S^1$ ; a simple closed curve is also called a Jordan curve. A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example.
Conventions and terminology
The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.
Length of curves
If $X$ is a metric space with metric $d$ , then we can define the length of a curve $\!\,\gamma : [a, b] \rightarrow X$ by : $\mbox (\gamma)=\sup \left\$ A rectifiable curve is a curve with finite length. A parametrization of $\!\,\gamma$ is called natural (or unit speed or parametrised by arc length) if for any $t_1$ , $t_2$ in $[a, b]$ , we have : $\mbox (\gamma|_)=|t_2-t_1|$ If $\!\,\gamma$ is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of $\!\,\gamma$ at $t_0$ as : $\mbox(t_0)=\limsup_$ and then : $\mbox(\gamma)=\int_a^b \mbox(t) \, dt$ In particular, if $X = \mathbb^n$ is Euclidean space and $\gamma : [a, b] \rightarrow \mathbb^n$ is differentiable then : $\mbox(\gamma)=\int_a^b \left| \, \, \right| \, dt$
Differential geometry
Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If $X$ is a differentiable manifold, then we can define the notion of differentiable curve in $X$ . This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take $X$ to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to $X$ by means of this notion of curve. If $X$ is a smooth manifold, a smooth curve in $X$ is a smooth map : $\!\,\gamma : I \rightarrow X.$ This is a basic notion. There are less and more restricted ideas, too. If $X$ is a $C^k$ manifold (i.e., a manifold whose charts are $k$ times continuously differentiable), then a $C^k$ curve in $X$ is such a curve which is only assumed to be $C^k$ (i.e. $k$ times continuously differentiable). If $X$ is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and $\!\,\gamma$ is an analytic map, then $\!\,\gamma$ is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two $C^k$ differentiable curves : $\!\,\gamma_1 :I \rightarrow X$ and : $\!\,\gamma_2 : J \rightarrow X$ are said to be equivalent if there is a bijective $C^k$ map : $\!\,p : J \rightarrow I$ such that the inverse map : $\!\,p^ : I \rightarrow J$ is also $C^k$ , and : $\!\,\gamma_(t) = \gamma_(p(t))$ for all $t$ . The map $\!\,\gamma_2$ is called a reparametrisation of $\!\,\gamma_1$ ; and this makes an equivalence relation on the set of all $C^k$ differentiable curves in $X$ . A $C^k$ arc is an equivalence class of $C^k$ curves under the relation of reparametrisation.
Algebraic curve
Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.
History
A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.
See also

- Curvature
- Osculating circle
- List of curves
- List of curve topics
External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html List of famous curves] Category:Curves Category:Metric geometry Category:Topology Category:General topology ko:곡선 ja:曲線

Chart (topology)

: For other uses of "atlas", see Atlas (disambiguation). In topology, an atlas describes how a complicated space is glued together from simpler pieces. Each piece is given by a chart (also known as coordinate chart or local coordinate system). More precisely, an atlas for a complicated space is constructed out of the following pieces of information:
- A list of spaces that are considered simple.
- For each point in the complicated space, a neighborhood of that point that is homeomorphic to a simple space. The homeomorphism is called a chart.
- We require the different charts to be compatible. At the minimum, we require that the composite of one chart with the inverse of another be a homeomorphism (known as a change of coordinates or a transition function), but we usually impose stronger requirements, such as smoothness. This definition of atlas is exactly analogous to the non-mathematical meaning of atlas. Each individual map in an atlas of the world gives a neighborhood of each point on the globe that is homeomorphic to the plane. While each individual map does not exactly line up with other maps that it overlaps with (because of the Earth's curvature), the overlap of two maps can still be compared (by using latitude and longitude lines, for example). Different choices for simple spaces and compatibility conditions give different objects. For example, if we choose for our simple spaces Rⁿ, we get topological manifolds. If we also require the coordinate changes to be diffeomorphisms, we get differentiable manifolds. We call two atlases compatible if the charts in the two atlases are all compatible (or equivalently if the union of the two atlases is an atlas). Usually, we want to consider two compatible atlases as giving rise to the same space. Formally, (as long as our concept of compatibility for charts has certain simple properties), we can define an equivalence relation on the set of all atlases, calling two the same if they are compatible. In fact, the union of all atlases compatible with a given atlas is itself an atlas, called a complete (or maximal) atlas. Thus every atlas is contained in a unique complete atlas (N.B. we don't need Zorn's lemma as is sometimes assumed). By definition, a smooth differentiable structure (or differential structure) on a manifold M is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps. Category:Differential topology

Equivalence relation

In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that # (Reflexivity) a ~ a # (Symmetry) if a ~ b then b ~ a # (Transitivity) if a ~ b and b ~ c then a ~ c A set together with an equivalence relation is called a setoid. Equivalence relations are often used to group together objects that are similar in some sense.

Examples of equivalence relations

- The equality ("=") relation between real numbers or sets.
- The relation "is congruent to (modulo 5)" between integers.
- The relation "is similar to" on the set of all triangles.
- The relation "has the same birthday as" on the set of all human beings.
- The relation of logical equivalence on statements in first-order logic.
- The relation "is isomorphic to" on models of a set of sentences.
- The relation "is in thermal equilibrium with".
- The relation "has the same image under a function" on the elements of the domain of the function.
- Green's relations are five equivalence relations on the elements of a semigroup.

Examples of relations that are not equivalences

- The relation "≥" between real numbers is not an equivalence relation, because although it is reflexive and transitive, it is not symmetric. E.g. 7 ≥ 5 does not imply that 5 ≥ 7. It is, however, a partial order relation.
- The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).
- The empty relation R on a non-empty set X (i.e. a R b is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, it is not reflexive (except when X is also empty).
- The relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (it may seem so at first sight, but many small changes can add up to a big change).
- The relation "is the mother of" on the set of all human beings is not an equivalence relation, because it not reflexive (A is not the mother of A), symmetric (If A is the mother of B, then B is not the mother of A), and is not transitive (if A is the mother of B, and B is the mother of C, it does not necessarily mean A is the mother of C)

Partitioning into equivalence classes

Every equivalence relation on X defines a partition of X into subsets called equivalence classes: all elements equivalent to each other are put into one class. Conversely, if the set X can be partitioned into subsets, then we can define an equivalence relation ~ on X by the rule "a ~ b if and only if a and b lie in the same subset". For example, if G is a group and H is a subgroup of G, then we can define an equivalence relation ~ on G by writing a ~ b if and only if ab^-1 lies in H. The equivalence classes of this relation are the right cosets of H in G. Since every equivalence relation can be identified with a partition and vice versa, the number of equivalence relations on a set X of n elements is given by the nth Bell number, B_n. If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the quotient set of X by ~ and is denoted by X/~.

Generating equivalence relations

If two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relation. This allows for a convenient way of defining equivalence relations: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, the equivalence relation ~ generated by R can be described as follows: a ~ b if and only if there exist elements x₁, x₂,...,x_n in X such that x₁ = a, x_n = b and such that (x_i , x_{i +1}) or (x_{i +1}, x_i) is in R for every i = 1,...,n -1. Note that the resulting equivalence relation can often be trivial! For instance, the equivalence relation ~ generated by the binary relation ≤ has exactly one equivalence class: x~y for all x and y. More generally, the equivalence relation will always be trivial when generated on a relation R having the "antisymmetric" property that, given any x and y, either x R y or y R x must be true. In topology, if X is a topological space and ~ is an equivalence relation on X, then we can turn the quotient set X/~ into a topological space in a natural manner. See quotient space for the details. One often generates equivalence relations to quickly construct new spaces by "gluing things together". Consider for instance the square X = [0,1]x[0,1] and the equivalence relation on X generated by the requirements (a,0) ~ (a,1) for all a in [0,1] and (0,b) ~ (1,b) for all b in [0,1]. Then the quotient space X/~ can be naturally identified with a torus: take a square piece of paper, bend it to glue together the upper and lower edge, then bend the resulting cylinder to glue together the two mouths.

Common notions in Euclid's Elements

The first person who introduced the idea of equivalence relations was Euclid in his book the Elements under Common Notions. Common Notion 1. Things which equal the same thing also equal one another. Nowadays, a binary relation is called Euclidean if it satisfies this property. Unfortunately, he did not mention symmetry or reflexivity. But this suggests an alternative formulation: An equivalence relation is a relation which is Euclidean, symmetric and reflexive. ---- In music see octave equivalency, transpositional equivalency, inversional equivalency, enharmonic equivalency, rotational equivalency. See also: Equivalent weight (chemistry).
Compare: Partial order, Total order, Directed set. Category:Set theory ko:동치관계 ja:同値関係

Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for function. Along these lines, a partial map is a partial function, and a total map is a total function. In many specific branches of mathematics, the term is used for a function with a specific property relevant to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc. In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory. A mapping m which has domain A and codomain B can be denoted symbolically as :

m : A \rightarrow B \

. If element a belongs to the domain A and if element b belongs to the codomain B and if mapping m maps element a to element b, this can be denoted symbolically as :

m : a \mapsto b

, "m maps a to b", which means the same as m(a) = b, "function m of a is b", in function notation. In such case, element b can be referred to as the image of element a. For any element a