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To find the minimum distance between a point in the plane (x_0,y_0) and a quadratic plane curve y=a_0+a_1x+a_2x^2, (1) note that the square of the distance is r^2 = ...

There are two different definitions of the polar angle. In the plane, the polar angle theta is the counterclockwise angle from the x-axis at which a point in the xy-plane ...

An integer which is expressible in more than one way in the form x^2+Dy^2 or x^2-Dy^2 where x^2 is relatively prime to Dy^2. If the integer is expressible in only one way, it ...

The Pontryagin number is defined in terms of the Pontryagin class of a manifold as follows. For any collection of Pontryagin classes such that their cup product has the same ...

The term used in physics and engineering for a harmonic function. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a ...

The general bivariate quadratic curve can be written ax^2+2bxy+cy^2+2dx+2fy+g=0. (1) Define the following quantities: Delta = |a b d; b c f; d f g| (2) J = |a b; b c| (3) I = ...

To compute an integral of the form int(dx)/(a+bx+cx^2), (1) complete the square in the denominator to obtain int(dx)/(a+bx+cx^2)=1/cint(dx)/((x+b/(2c))^2+(a/c-(b^2)/(4c^2))). ...

A statistical distribution in which the variates occur with probabilities asymptotically matching their "true" underlying statistical distribution is said to be random.

The sum of the reciprocals of roots of an equation equals the negative coefficient of the linear term in the Maclaurin series.

The period for a quasiperiodic trajectory to pass through the same point in a surface of section. If the rotation number is irrational, the trajectory will densely fill out a ...