In physics and mathematics, the size of the space mathematics (or object) is randomly defined as the minimum number of links needed to clarify any point within it. A line is therefore one-dimensional (1D) because only one link is required to specify a point on it – for example, 5 points on a number line. A plane-like or cylinder surface is a two-dimensional (2D) dimension because two links are needed to determine a point – for example, a length and length are required to locate a point on a local surface. The inside of a cube, cylinder or globe has three dimensions (3D) because three links are needed to get a point in these spaces.

In ancient machines, space and time are of various stages and refer to space and time. That concept of the earth is a four-dimensional being but not one that has been found to be necessary to explain electromagnetism. The (4D) size of the space contains events that are not fully defined in terms of temporarily and temporarily, but are instead known in relation to the movement of the viewer. Minkowski’s space first measures the universe without gravity; the many Riemannian myths of ordinary relationships describe the time of space in a story and gravity. Size 10 is used to describe superstring theory (6D hyperspace + 4D), magnitude 11 can describe vitality and M-theory (7D hyperspace + 4D), and quantum mechanical position space unlimited work space.

The concept of grandeur is not limited to material things. Extreme gaps often occur in mathematics and science. They can be parameter spaces or stop spaces such as Lagrangian or Hamiltonian machines; these are mysterious, independent spaces in the physical realm in which we live.

Define the dimension of a vector space

Dimension

In mathematics, the magnitude of a vector space V is the cardinality (that is, the number of vectors) of the fundamental V in its base field. It is sometimes called the Hemel dimension (after Georg Howell) or the algebraic dimension.

Every vector space has a basis, [a] and all bases of a vector space have the same cardinality; [b] Consequently, the size of a vector space is determined individually. We say that if the amplitude of V is finite then V is of finite dimension and if its amplitude is infinite then it is of infinite dimension.

The magnitude of the vector space V in the F field can be written as DMF(V) or [V: F], read “V oversize f”. When f can be inferred from the context, it is usually written as dim(v).

as a standard basis, and therefore we have dimR(R3) = 3. More generally, dimR(Rn) = n, and even more generally, dimF(Fn) = n for any field F.

The complex numbers C are both a real and complex vector space; we have dimR(C) = 2 and dimC(C) = 1. So the dimension depends on the base field.

The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.

Two-dimensional space

Dimension

A two-dimensional space (also called a 2D space, 2-space or two-dimensional space) is a geometric structure in which two values ​​(called parameters) are needed to determine the position (i.e. point) of an element. it occurs. Pairs of real numbers with the appropriate structure {\displaystyle \mathbb R} ^{2}} \mathbb R} {{2} Pairs often serve as canonical examples of two-dimensional Euclidean spacing. For a generalization of the concept, see Dimensions.

The physical universe can be seen as a projection on a plane of two-dimensional space. In general, this is considered to be Euclidean space and has two dimensions called length and width.

The history two Dimensional Space

Euclid’s books I to IV and VI discuss two-dimensional geometry, the balance of figures, the Pythagorean theorem (Proposition 47), equality of angles and parts, the sum of angles in a triangle, and the three situations in which triangles are “equal”. are (having the same area). in many subjects.

Then, the plane is called the Cartesian coordinate system, which is a coordinate system that defines each point on a plane individually by a pair of number coordinates, which are the same number of signed distances from the point on two fixed perpendicularly directed lines. unit is. Each reference line is called an integral axis or single axis of the system, and the point at which they meet is its origin, usually in the sequence pairs (0, 0). Coordinates can also be defined as the vertical plan position of a point on two axes expressed as a signed distance from the source.

The idea for this system was developed in 1637 in the writings of Descartes and independently of Pierre de Fermat, although Fermat worked in three dimensions, but did not publish the invention. [1] Both authors used the same axis in their treatment and measured a different length defining this axis. The idea of ​​using a pair of axes was introduced later, following a translation into Latin of Descartes’ La Comotri in 1649 by Franz von Schöten and his students. These commentators offered several ideas when trying to elucidate the ideas in Descartes’ work. 

Then, the plane was considered a region in which any two points except 0 could be multiplied and divided. It is called the airport complex. The complex plane is called the arcand plane because it is used in arcand maps. Named after Jean-Robert Arkand (1768–1822), they were first described by the Danish-Norwegian geologist and mathematician Caspar Wessel (1745-1818).  Arcand maps are often used to place the poles and zero positions of a process in a complex plane.

Three Dimensional Space

Dimension

Three-dimensional space (also: 3D space, 3-space or, more rarely, three-dimensional space) is a geometric structure that uses three values ​​(called parameters) to determine the position of an element (i.e., point). is needed. This is an informal definition of a time scale.

In mathematics, a sequence of n numbers can be understood as a space in n-dimensional space. When n = 3, the sum of all such spaces is called a three-dimensional Euclidean space (or Euclidean space if the context is clear). It is usually represented by the symbol 3.  It presents a three-dimensional model of the physical universe (i.e., space, regardless of time), which includes all known objects. This space is a very compelling and effective way to model the world experientially, [3] which is an example of a large variety of places in three dimensions called the 3-Moneybolt. In this classical example, when the three values ​​represent dimensions in different directions (coordinates), any three directions can be chosen, since not all vectors in these coordinates depend on both 2-space (plane) . Also, in this case, these three values ​​can name any combination of the three selected from width, height, depth, and length.

In mathematics, analytic geometry (also called Cartesian geometry) describes each point in three-dimensional space with three combinations. Three coordinate axes are given, each perpendicular to the other two at the point where they cross. They are usually named x, y and z. With respect to these axes, the position of any point in three-dimensional space is given by a triple array of real numbers, each number giving the distance of that point from the source measured on the given axis, which is equal to the distance From the plane determined by the other two axes. Other popular ways of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, although there are several possible ways. See Euclidean for more details.

Below are photos of the above systems.

cartesian integration system

 

cylinder integration system

Spherical Coordinate system

 

Lines and Planes

Lines and Planes are the two distinct points that always define a (straight) line.three different points represent a collinear or a different plane On the other hand, four distinct points can define a collinear, a coplanar, or an entire space.

Two distinct lines may intersect, parallel or diagonally two identical lines, or two flowing lines, lie in a different plane, so spray lines are lines that do not meet and lie in a common plane.

Two different plans may meet on the same line or be identical (that is, do not merge). Three distinct planes, no pair of which are identical, cannot join together on a common line, join at the same common point, or have no points in common. In the latter case, the three lines of intersection of each pair of planes are equidistantly parallel.

A line can lie on a given plane, meet that plane at a unique location, or be parallel to the plane. In the latter case, there will be lines on the plane parallel to the given line.

A hyperplane is a subspace that is one size smaller than the size of the entire space. Supersonic aircraft of three-dimensional space are a subspecies of two-dimensional, that is, aircraft. For Cartesian coordinates, a hyperplane point satisfies a single linear equation, so these 3-position planes are described by linear equations. A line can be described by a pair of independent linear equations – each representing a plane with this line as a common intersection.

Varignon’s principle states that the centers of any quadrilateral form a parallelogram at point 3, so they are cobbled.

balls and balls

 

Perspective view of a sphere from two angles

A sphere in 3-space (also called 2-sphere because it is a 2-dimensional object) is the sum of all points in 3-space at a constant distance r. A ball (or, more accurately a 3-ball) from a center point. the shape of the ball is given

Display\Displaystyle V = {\Frac{4}{3}\Pi R^{3}}V=\Frac 4}{3\Pi R^3.

Another type of sphere comes from a 4-ball, whose three-dimensional surface is a 3-sphere: the number 4 equal to the presence of Euclidean space. If the coordinates of a point are p(x, y, z, w), then x2 + y2 + z2 + w2 = 1 identifies the points in a 3-spherical unit centered at the origin.


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