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The Poincaré lemma states that if B is an open ball in R n, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n. [1] More generally, the lemma states that on a contractible open subset of a manifold (e.g., ), a closed p-form, p > 0, is exact. [citation needed]
In particular, a ball (open or closed) always includes p itself, since the definition requires r > 0. A unit ball (open or closed) is a ball of radius 1. A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross ...
A proof of the recursion formula relating the volume of the n-ball and an (n − 2)-ball can be given using the proportionality formula above and integration in cylindrical coordinates. Fix a plane through the center of the ball. Let r denote the distance between a point in the plane and the center of the sphere, and let θ denote the azimuth.
In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses that orbit each other in space and calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.
It refers to computing tools that help calculating the complex particle interactions as studied in high-energy physics, astroparticle physics and cosmology. The goal of the automation is to handle the full sequence of calculations in an automatic (programmed) way: from the Lagrangian expression describing the physics model up to the cross ...
An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid.
This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for x 0 = 0 and a ball of radius exceeding 1, since the power series 1 − x 2 + x 4 − x 6... diverges for |x| ≥ 1.
A normed vector space is locally compact if and only if the unit ball = {: ‖ ‖} is compact, which is the case if and only if is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional.