Parametric Equations

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. are parametric equations for the unit circle, where t is the parameter. 

Together, these equations are called a parametric representation of the curve. A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter. The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.)

Partial Fraction Decomposition 

Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x(x + 1). Then you write the fractions with one of the factors for each of the denominators. Of course, you don't know what the numerators are yet, so you assign variables (usually capital letters) for these unknown values: A/x + B/(x + 1) Then you set this sum equal to the simplified result: (3x + 2)/[x(x + 1)] = A/x + B/(x + 1) Multiply through by the common denominator of x(x + 1) gets rid of all of the denominators: multiply each term by [x(x + 1)] / 1 3x + 2 = A(x + 1) + B(x)