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Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions , to describe the sizes or locations of objects in the everyday world.
Definition. A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces.
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian ...
An associate of the School of Paris —a group of avant-gardists including Pablo Picasso, Guillaume Apollinaire, Max Jacob, Jean Metzinger, and Marcel Duchamp —Princet is credited with introducing the work of Henri Poincaré and the concept of the "fourth dimension" to the cubists at the Bateau-Lavoir during the first decade of the 20th century.
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b ...
Laplace transform. In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace ( / ləˈplɑːs / ), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane ).
The McDonnell Douglas F-4 Phantom II [N 1] is an American tandem two-seat, twin-engine, all-weather, long-range supersonic jet interceptor and fighter-bomber originally developed by McDonnell Aircraft for the United States Navy. [2] Proving highly adaptable, it entered service with the Navy in 1961 [3] before it was adopted by the United States ...