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Since every unit vector can be thought of as a point on a unit sphere, and since a versor can be thought of as the quotient of two vectors, a versor has a representative great circle arc, called a vector arc, connecting these two points, drawn from the divisor or lower part of quotient, to the dividend or upper part of the quotient. [20] [21]
The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the trivial group 0, the infinite cyclic group Z, b) the finite cyclic groups of order n (written as Z n), or c) the direct products of such groups (written, for example, as Z 24 ×Z ...
The table usually lists only one name and symbol that is most commonly used. The final column lists some special properties that some of the quantities have, such as their scaling behavior (i.e. whether the quantity is intensive or extensive ), their transformation properties (i.e. whether the quantity is a scalar , vector , matrix or tensor ...
energy efficiency, economics (ratio of energy input to kinetic motion) Damping ratio = mechanics, electrical engineering (the level of damping in a system) Decibel: dB: acoustics, electronics, control theory (ratio of two intensities or powers of a wave) Elasticity : E
Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space R 3 . {\displaystyle \mathbb {R} ^{3}.} [ b ]
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
A quotient space of a locally compact space need not be locally compact. Dimension. The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
In mathematics, the Rayleigh quotient [1] (/ ˈ r eɪ. l i /) for a given complex Hermitian matrix and nonzero vector is defined as: [2] [3] (,) =. For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric , and the conjugate transpose x ∗ {\displaystyle x^{*}} to the usual transpose x ...