Search Results for ""

"Poincaré transformation" is the name sometimes (e.g., Misner et al. 1973, p. 68) given to what other authors (e.g., Weinberg 1972, p. 26) term an inhomogeneous Lorentz ...

Solutions to holomorphic differential equations are themselves holomorphic functions of time, initial conditions, and parameters.

Poincaré's lemma says that on a contractible manifold, all closed forms are exact. While d^2=0 implies that all exact forms are closed, it is not always true that all closed ...

If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...

Poinsot's spirals are the two polar curves with equations r = acsch(ntheta) (1) r = asech(ntheta). (2)

The equation of a line ax+by+c=0 in slope-intercept form is given by y=-a/bx-c/b, (1) so the line has slope -a/b. Now consider the distance from a point (x_0,y_0) to the ...

Let a line in three dimensions be specified by two points x_1=(x_1,y_1,z_1) and x_2=(x_2,y_2,z_2) lying on it, so a vector along the line is given by v=[x_1+(x_2-x_1)t; ...

Given a plane ax+by+cz+d=0 (1) and a point x_0=(x_0,y_0,z_0), the normal vector to the plane is given by v=[a; b; c], (2) and a vector from the plane to the point is given by ...

In the case of a general surface, the distance between two points measured along the surface is known as a geodesic. For example, the shortest distance between two points on ...

In Euclidean space R^3, the curve that minimizes the distance between two points is clearly a straight line segment. This can be shown mathematically as follows using ...