Solving Systems of Equations and Inequalities
Figure 2. Functions for solving systems of equations and inequalities.
Here is the cylindrical-solution-form representation of the unit ball.
Please note the nested structure of the answer. The cylindrical parts described in the introduction can be obtained by expanding
Here is a necessary and sufficient condition for positiveness of the quadratic .
This tells when the cubic has three real roots and the largest root is positive.
In many practical applications we may want to solve systems containing only polynomial (or rational function) inequalities, and it may be sufficient to find the set of solutions up to a set of measure zero. This is the case if we want to integrate over a volume described by inequalities, or if we want to plot a set of solutions of inequalities (e.g., the
For the unit ball the set A is the interior of the ball, and the set B is the unit sphere.
This computes the volume of the unit ball.
We can now plot the solution of the inequality
Here is the "generic" description of the solution set of
This shows how to use the description to numerically compute the area of the solution set.
Converted by Mathematica April 24, 2000