Search results
Results from the WOW.Com Content Network
Download as PDF; Printable version; In other projects ... This is a partial list of notable homeschooling curricula and programmes that are popularly used in the ...
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. [1]: 172 To prove this one shows that u + iv satisfies the Cauchy–Riemann equations exactly when u + iv is locally an analytic function of x + iy. Of course an analytic function w(z) = u + iv is the local derivative of something (namely ∫w(z ...
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
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 ...
The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties. [2] [3] [4]
It is a straightforward consequence of standard elliptic regularity results that any Ricci-flat Riemannian metric on a smooth manifold is analytic, in the sense that harmonic coordinates define a compatible analytic structure, and the local representation of the metric is real-analytic. This also holds in the broader setting of Einstein ...
of weight 2 is a harmonic Maass form of weight 2. Zagier's Eisenstein series E 3/2 of weight 3/2 [2] is a harmonic Maass form of weight 3/2 (for the group Γ 0 (4)). Its image under / is a non-zero multiple of the Jacobi theta function =.