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Level Curves of a Two Variable Function

At some point, you have probably seen the equation: \[ y = x^2 \] You probably have a good idea of what this looks like when graphed. When it comes to functions of one variable, we have a pretty good grasp of the basic shape any given function will make when graphed. When it comes to functions of 2 variables, this intuition is not as sharp. For instance, imagine what this function may look like: \[ \frac{1}{z} = \frac{1}{x} + \frac{1}{y} \] It is not immediately obvious what this looks like when graphed. The tried and true method of plotting points could be used to generate some idea of what it may look like, but this would be quite tedious. Instead of plotting one point at a time, let's graph entire functions. If z is set to some constant, then the expression can be written as a function of 1 variable. This function can be graphed at the respective z value to give one level of the two variable function. If this is done enough times, a clear shape becomes visible and gives insight into what the function looks like.

Paratmerizations of Surfaces

The function mentioned earlier: \[ \frac{1}{z} = \frac{1}{x} + \frac{1}{y} \] expresses the relationship for the total resistance of an electrical cicruit that contains 2 resistors that are in parallel. It makes sense the function is increasing as both x and y increase, but graphing the function and finding the level curves will produce more insight to how this function behaves. The function has many different ways to be written that result in the same graph. One of these ways is to write it is a function of x and y: \[ f(x,y) = (\frac{1}{x} + \frac{1}{y})^{-1} \] This can also be written as a parameterization of two independent variables: \[ f(s,t) = (s,\frac{st}{s-t},t) \] This third representation defines the function in a parameterized form rather than an explicit form. This was achieved by setting the values of x and z to independent variables s and t and expressing y as a function of x and z in terms of the independent variables. In this form, a surface level can be created for any combination of s and t. For the scope of this problem s and t will always be greater than or equal to 1.

Creating the Level Curves

At this point, we have a parameterization for the two function variable. Let's break down what exactly each component of the parameterization does. The parameterization defines the function as any point that is of the form (s,\frac{st}{s-t},t) for some input s,t. Say for instance s=2 and t=1, then the point (2,2,1) is a point that lies on the surface of the function: \[ \frac{1}{z} = \frac{1}{x} + \frac{1}{y} \]. This also means that if there are two resistors in parallel, both with resistance of 2 ohms, then the equivalent resistance of the cicruit is 1. Instead of using just a single point, t can be set to a constant and s left as a variable. This would allow for the y component of the parameterization to be graphed as function of s since t is defined as a number. This is where the level curves come from. Let's once again say that t = 1, but leave s as a variable: \[ f(s,1) = (s,\frac{s}{s-1},1) \] This allows for a curve to be graphed on the plane z = 1. The parameter t can be changed to any number, but within the scope of this example it must be greater than 1, to generate a level curve. Once enough of these curves have been created, the general shape of the function can be observed.

The Curves in All Their Glory

Using OpenScad, I was able to created the level curves with respect to the z axis. The domain of these curves was [1,10]X[1,10] and 10 level curves in total were created. These curves were taken at every .5 values for z = [1,5]. Here is a picture of the level curves by themselves:
The largest value of z depicted here is 5 which happens to only include the single point (10,10,5) as a result of the chosen domain. These level curves indicate that for each z value, a curve is shown that indicates what value are a part of the function. This outline provides a great deal of insight as to what the shape of the function will look like. The following picture shows the level curves imposed on the function:
The grooves represent the level curves and there are 10 in total.

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