Multivariable Calculus Concept Heegyung Kim November 8, 2021 George Mason University MATH 401: Mathematics Through 3D Printing

The level set is the set of points in the domain of a function such that the function is constant. The level curve is the curve of points (x,y) where f(x,y) is some constant value. A horizontal slice is taken from a graph of z = f(x,y) at a constant value z = c, and can be also written as c = f(x,y). In other words, the level curve can show the graph of a function at height c. It lies in the xy plane.

For this assignment, my first thought was to illustrate the concept of level sets by creating the graph of a function f(x,y) and its level curves. Then, after I printed the object, I realized that the object didn’t represent the level curves (or it could say it shows the level curves in the direction of the x and y-axis). Rather, it showed the multiple lines of x = constant and y = constant on a plane and combined with the graph of a function 4x^2+ (1/3)x^3-x^4-y^4. Also, instead of showing z = f(x,y), it showed f(c,y) and f(x,d) where c, d is a fixed number. Fortunately, this 3D printed graph showed relatively maxima which can be another calculus concept.

For printing my object, the graph, I used Mathematica and converted it to an SLT file. Also, I created a stand so that the bottom part of my object can be displayed. For creating a stand, I used OpenSCAD because it was less complicated than in Mathematica. Also, to be able to hold the object correctly, I created the stand according to the original dimension of the object. My original dimension of the object was really small, so I had to scale up to 842.6% as well as the stand. For the printer setting, the support density was 20% with a support overhand angle of 60°. The brim was used for plate adhesion type. The dimension for the object was 36.2 x 19.3 x 75 mm and for the stand was 50.6 x 25.3 x 62.7 mm. The Monoprice Mini 3D Printer was used and it took about 5 hours to print the object.

Sources: https://mathinsight.org/level_sets https://web.ma.utexas.edu/users/m408m/Display14-1-3.shtml