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More generally, if g is C k, α (with k larger than one) and Ric(g) is C l, α relative to some coordinate charts, then the transition function to a harmonic coordinate chart will be C k + 1, α, and so Ric(g) will be C min(l, k), α in harmonic coordinate charts. So, by the previous result, g will be C min(l, k) + 2, α in harmonic coordinate ...
Euler's Tonnetz. The Tonnetz originally appeared in Leonhard Euler's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae.Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a ...
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general ...
The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten partial differential equations in which the metric tensor can be solved for.
The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations – which determine the geometry of spacetime in the presence of matter – contain the Ricci tensor. Since the Ricci tensor ...
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5-limit Tonnetz. Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2 −3 ·3 1 ·5 1 = 15/8.
A woman got routine bloodwork during her second pregnancy and was shocked when the test results came back: It suggested her baby was healthy, but there was something unusual about her own health.