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In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold ...
This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1 ...
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
6.7 Volume form. 6.8 Hodge operator on ... is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor ... (and volume element)
The Holmes–Thompson volume can be defined without coordinates: if is a measurable set in an n-dimensional real normed space (, ‖ ‖), then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form! ⏞ over the set ,
The absolute value ... the magnitude of a vector is the value of the quadratic form for that vector. ... A simple example is a volume (how big an object occupies a ...
Image source: The Motley Fool. Equinix (NASDAQ: EQIX) Q4 2024 Earnings Call Feb 12, 2025, 5:30 p.m. ET. Contents: Prepared Remarks. Questions and Answers. Call ...
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).