Cramer's Rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique soltution. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750. The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(n³) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian Elimination. 

Graphs of Polar Equations

There are many different types of polar graphs such as limacons, rose curves, circles and lemniscates. The four different types of Limacons are: limacons with inner loop, cardioid (heart-shaped), dimpled limacons and convex limacons. There are also four different types of rose curves: two with r=acosnfeta and two with r=asinnfeta. There are also two different types of circles: r=acosfeta and r=asinfeta. Lastly there are two different types of lemniscates: r squared= a squared sin2feta and r squared=a squared cos2feta. For limacons a must be greater than zero and b must be greater than zero. For rose curves: the number of petals if the number is odd, and two times the number of petals if the number is even, the number of petals is equal or greater than two. 

Polar Coordinares

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radical coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth. When feta is bigger than zero than it's counterclockwise and if feta is less than zero than it's clockwise. Polar coordinates are written as (r, feta).