From what I can tell, the traditional way to teach Lagrange multipliers is to start with a function $f(x,y,z)$ and to look for extrema of $f$ subject to $g(x,y,z)=k$.
That is, we restrict $(x,y,z)$ to be on the level curve $g(x,y,z)=k$.
We then look at the level curves of $f$ and find the one(s) tangent to the level curve $g(x,y,z)=k$.
An example of this can be found here: http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx
I came across what seems to me to be a different approach here: https://sites.lafayette.edu/thompsmc/files/2014/01/Section_14_8.pdf
In this pdf, the constraint $g(x,y,z)=k$ is not referred to as a level curve. Rather it's shown as a cylinder intersecting the shape in question [that is, the graph of $f(x,y,z)$]. We're looking for extrema along the curve of intersection of the 2 shapes.
We then look at level curves of both $f$ and $g$. And we find that the pair of level curves that are tangent correspond to an extrema of $f$. Unlike the traditional approach, here we have multiple level curves of $g$.
I can't seem to reconcile these two views and am wondering if there's a sense in which one of the 2 approaches outlined is a more generalized version of the other.
Can someone help guide me on this?
Thanks!
Best Answer
Preliminary note. In the first tutorial the objective is a quadratic function and the constraint is a sphere. While in the second tutorial the objective is a sphere and the constraint a cylinder (or a quadratic function).
The main difference is that in the first tutorial only the variables $(x,y)$ are considered, and both $f$ and $g$ go from $\mathbb{R}^2$ to $\mathbb{R}$. In contrast, the second document depicts functions with three variables $(x,y,z)$ and represents the constraint $g(x,y)=z$ for different levels of $z$, hence the different level curves for the constraint on the last figure of page 4. If you fix $z$ to and arbitrary number, and eliminate $z$ from the choice variables, you end up with a problem which is comparable to the one considered in your first reference. Both views are consistent, but represent slightly different problems.