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In the theory of vector measures, Lyapunov's theorem states that the range of a finite-dimensional vector measure is closed and convex. [ 1 ] [ 2 ] [ 3 ] In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes ). [ 2 ]
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [ 1 ] [ 2 ] It is typically formulated as the product of a unit of measurement and a vector numerical value ( unitless ), often a Euclidean vector with magnitude and direction .
Vector quantities Derived quantity Symbol Description SI derived unit Dimension Comments Absement: A: Measure of sustained displacement: the first integral with respect to time of displacement m⋅s L T: vector Acceleration: a →: Rate of change of velocity per unit time: the second time derivative of position m/s 2: L T −2: vector Angular ...
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space .
Velocity is a physical vector quantity: both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or m⋅s −1). For example, "5 metres per second" is a scalar, whereas "5 metres per ...
A simple example is a volume (how big an object occupies a space) as a measure. In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and ...
Loosely speaking, the measure of a vector is a length, the measure of a bivector is an area, and the measure of a trivector is a volume. If a vector is factored directly into projective and rejective terms using the geometric product v = 1 u ( u ⋅ v + u ∧ v ) {\displaystyle v={\frac {1}{u}}(u\cdot v+u\wedge v)} , then it is not necessarily ...
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