Search Results for ""

Find the surface enclosing the maximum volume per unit surface area, I=V/S. The solution is a sphere, which has I_(sphere)=(4/3pir^3)/(4pir^2)=1/3r. The fact that a sphere ...

An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an ...

In order to integrate a function over a complicated domain D, Monte Carlo integration picks random points over some simple domain D^' which is a superset of D, checks whether ...

Given a point P and a line AB, draw the perpendicular through P and call it PC. Let PD be any other line from P which meets CB in D. In a hyperbolic geometry, as D moves off ...

An Anosov diffeomorphism is a C^1 diffeomorphism phi of a manifold M to itself such that the tangent bundle of M is hyperbolic with respect to phi. Very few classes of Anosov ...

The functions describing the horizontal and vertical positions of a point on a circle as a function of angle (cosine and sine) and those functions derived from them: cotx = ...

A spiral that gives the solution to the central orbit problem under a radial force law r^..=-mu|r|^(-3)r^^, (1) where mu is a positive constant. There are three solution ...

For the hyperbolic partial differential equation u_(xy) = F(x,y,u,p,q) (1) p = u_x (2) q = u_y (3) on a domain Omega, Goursat's problem asks to find a solution u(x,y) of (3) ...

Half-angle formulas and formulas expressing trigonometric functions of an angle x/2 in terms of functions of an angle x. For real x, sin(1/2x) = ...

Refer to the above figures. Let X be the point of intersection, with X^' ahead of X on one manifold and X^('') ahead of X of the other. The mapping of each of these points ...