# Meaning of '$z_0$ is a constant' in the definition of a level curve.

I was looking at the definition of a level curve for a two argument function and came across this definition:

A level curve is can be given by:

$$f(x,y)=z_0$$

for some constant $$z_0$$.

I am confused by the use of the word in this case, a constant is a well-defined number, and can also come up in a function where $$f(x)=ux$$ where $$u is a constant and it's clear, that for a given function f the number represented by u is constant will be the same number. However, there is no argument here, we could easily change $$z_0$$ to get a new 'level curve', what is the use of the term here? A function's value is more tangible, but an equation can mean literally anything so how does 'constant' apply here, any number I put here will give an equation of a level curve. • The correct definition of a level curve should be: a level curve is a set of the following form$$\{ (x,y) \in \Bbb R^2 : f(x,y)=z_0 \}$$where z_0 \in \Bbb R and f: \Bbb R^2 \to \Bbb R. For example$$\{ (x,y) \in \Bbb R^2 : x^2 + y^2 =16 \} is a level curve. Commented Sep 23, 2022 at 17:13

In your example $$f(x)=ux$$, that equation represents a defining equation for the function $$f$$, meaning that for all values $$x$$ in the domain of the function $$f$$, the value of $$f(x)$$ is defined to equal $$ux$$.

The equation $$f(x,y)=z_0$$ does not represent a defining equation. It is assumed that $$f(x,y)$$ has already been defined (perhaps by a formula occurring in some defining equation that is recorded elsewhere). Instead, that equation represents an equation to be solved. The level set of $$f$$ corresponding to the constant $$z_0$$ is simply the solution set of the equation $$f(x,y)=z_0$$, which in set builder notation can be expressed as $$\{(x,y) \in \mathbb R^2 \mid f(x,y) = z_0\}$$.

And then you are absolutely right: different values of the constant give different level sets. For instance, if one were given the defining equation $$f(x,y)=x+y$$, one could produce a lot of different lines in the Euclidean plane by assigning the constant value $$z_0$$ in the equation $$f(x,y)=z_0$$: $$z_0=0$$ gives the line $$x+y=0$$, $$z_0=1$$ gives the line $$x+y=1$$; and so on...

• So it's 'constant' in that, have a number that is not itself a variable and it will give an equation to be solved, so if we have $f(x,y)=z_0$ if $z_0$ is replaced with a specific number we get the specific equation for a specific level curve, is this where 'constant' comes into it? Is that If I want the equation of a specific level curve I need to choose a value for that curve and keep it? Commented Sep 23, 2022 at 18:58
• Yes, that's basically correct. But keep this in mind: you may, if you desire, think about two level curves with one mental thought by choosing two constants. And you may think about all level curves simultaneously by simultaneously choosing all possible constants. Commented Sep 23, 2022 at 19:03
• Sorry,that last bit has confused me a bit, do you mean introducing to symbols $z_0$ and $z_1$ or leaving $z_0$ blank so allowing it to 'change' but doing this is sort of jumping from scenario to scenario? Similar to parameterize a function but were doing it to a set so we say $z_0$ is independent? Commented Sep 23, 2022 at 19:21
• Yes, something like that. Look up contour diagrams. Commented Sep 23, 2022 at 19:30
• Oh, you meant visually, makes sense, contour diagrams are a drawing of all level sets thanks. Commented Sep 23, 2022 at 19:34

Did you ever take tours in mountains using a map. In the map you have "level curves. they give you the lines of constant hight, you can call the hight h or z0. so z=f(xy) gives you the shape of a mountain, and for any fixed z you have the hight od the mountain at point(x,y) Or consider a weather chart, there z may be th pressure and p0=f(x,y) gives you the points of fixes pressure p0