Units, Dimensions and Dimensional Analysis
... in physics only certain equations are acceptable because for instance a length cannot equal a mass. The basic rules are 1) two physical quantities can only be equated if they have the same dimensions 2) two physical quantities can only be added if they have the same dimensions 3) the dimensions of t ...
... in physics only certain equations are acceptable because for instance a length cannot equal a mass. The basic rules are 1) two physical quantities can only be equated if they have the same dimensions 2) two physical quantities can only be added if they have the same dimensions 3) the dimensions of t ...
Lecture 11: High Dimensional Geometry, Curse of Dimensionality, Dimension Reduction
... A good approximation to picking a random point on the surface of Bn is by choosing random xi ∈ {−1, 1} independently for i = 1..n and normalizing to get √1n (x1 , ..., xn ). A better approximation is to pick each coordinate as a gaussian with mean 0 and variance 1/n. To get a point inside the ball, ...
... A good approximation to picking a random point on the surface of Bn is by choosing random xi ∈ {−1, 1} independently for i = 1..n and normalizing to get √1n (x1 , ..., xn ). A better approximation is to pick each coordinate as a gaussian with mean 0 and variance 1/n. To get a point inside the ball, ...
1 Dimension 2 Dimension in linear algebra 3 Dimension in topology
... no guarantee at all that the same value is found for each definition. It is a game in topology to construct topological spaces that have dimensions 0, 1, 2 for three different concepts of dimension. A dimension does not have to be an integer. In cases where one has a concept of area or volume or con ...
... no guarantee at all that the same value is found for each definition. It is a game in topology to construct topological spaces that have dimensions 0, 1, 2 for three different concepts of dimension. A dimension does not have to be an integer. In cases where one has a concept of area or volume or con ...
counting degrees of freedom of the electromagnetic field
... is “pure gauge”. This part of the vector potential obviously cannot be determined from J~ and any initial data by the field equation, since it is entirely at our whim. (Even if the Lorenz gauge condition is imposed, we can still perform a gauge transformation with χ a solution of the scalar wave eq ...
... is “pure gauge”. This part of the vector potential obviously cannot be determined from J~ and any initial data by the field equation, since it is entirely at our whim. (Even if the Lorenz gauge condition is imposed, we can still perform a gauge transformation with χ a solution of the scalar wave eq ...
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, and the state-space of quantum mechanics is an infinite-dimensional function space.The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.