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The natural "Lebesgue measure" on S 1 is then the push-forward measure f ∗ (λ). The measure f ∗ (λ) might also be called "arc length measure" or "angle measure", since the f ∗ (λ)-measure of an arc in S 1 is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
Let : be a smooth map of smooth manifolds. Given , the differential of at is a linear map : from the tangent space of at to the tangent space of at (). The image of a tangent vector under is sometimes called the pushforward of by .
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
Pushforward measure, measure induced on the target measure space by a measurable function; Pushout (category theory), the categorical dual of pullback; Direct image sheaf, the pushforward of a sheaf by a map; Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres"
The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum ...
In quantum physics, position and momentum are represented by mathematical entities known as Hermitian operators, and the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a ...
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
A Borel measure on a separable Banach space is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional except =, the push-forward measure is a non-degenerate (centered) Gaussian measure on in the sense defined above.