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A quadratic form Q(z) is said to be positive definite if Q(z)>0 for z!=0. A real quadratic form in n variables is positive definite iff its canonical form is ...

The one-dimensional wave equation is given by (partial^2psi)/(partialx^2)=1/(v^2)(partial^2psi)/(partialt^2). (1) In order to specify a wave, the equation is subject to ...

Calculus I

Critical damping is a special case of damped simple harmonic motion x^..+betax^.+omega_0^2x=0, (1) in which D=beta^2-4omega_0^2=0, (2) where beta is the damping constant. ...

To generate random points over the unit disk, it is incorrect to use two uniformly distributed variables r in [0,1] and theta in [0,2pi) and then take x = rcostheta (1) y = ...

A root-finding algorithm which makes use of a third-order Taylor series f(x)=f(x_n)+f^'(x_n)(x-x_n)+1/2f^('')(x_n)(x-x_n)^2+.... (1) A root of f(x) satisfies f(x)=0, so 0 ...

In elliptic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(sinh^2u+sin^2v), h_z=1, and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving a Stäckel ...

A synonym for analytic function, regular function, differentiable function, complex differentiable function, and holomorphic map (Krantz 1999, p. 16). The word derives from ...

A theorem, also called the iteration theorem, that makes use of the lambda notation introduced by Church. Let phi_x^((k)) denote the recursive function of k variables with ...

The Maxwell (or Maxwell-Boltzmann) distribution gives the distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics. Defining a=sqrt(kT/m), ...