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A topological space having a countable dense subset. An example is the Euclidean space R^n with the Euclidean topology, since it has the rational lattice Q^n as a countable ...

Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory ...

The shoelace formula, also known as Gauss's area formula, the shoelace algorithm, shoelace method, or surveyor's formula, is a name sometimes given to the polygon area ...

Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

A surface in 3-space can be parameterized by two variables (or coordinates) u and v such that x = x(u,v) (1) y = y(u,v) (2) z = z(u,v). (3) If a surface is parameterized as ...

A pair (M,omega), where M is a manifold and omega is a symplectic form on M. The phase space R^(2n)=R^n×R^n is a symplectic manifold. Near every point on a symplectic ...

The set of all points x that can be put into one-to-one correspondence with sets of essentially distinct values of four homogeneous coordinates x_0:x_1:x_2:x_3, not all ...

A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus. ...

By the definition of the functions of trigonometry, the sine of pi is equal to the y-coordinate of the point with polar coordinates (r,theta)=(1,pi), giving sinpi=0. ...

By the definition of the functions of trigonometry, the sine of pi/2 is equal to the y-coordinate of the point with polar coordinates (r,theta)=(1,pi/2), giving sin(pi/2)=1. ...