Well, hey. I don't see my site show up here that often. If anyone's got comments, questions, or suggestions, I'll try to stop by and respond at some point.
Thanks for writing that tutorial, I've referred to it a few times over the years and as far as I'm concerned it's the definitive Lagrange multiplier tutorial. I was able to accurately picture it in my mind's eye before clicking the link to confirm. You've helped a lot of people!
It's genuinely good to hear that it's been appreciated (let alone so much). I owe the milkmaid example to my Calc 3 professor, many years ago, though I later realized that I'd probably altered it a little from his version.
Just want to thank you for the resource! The MS Paint (?) drawing of the river and accompanying explanation got through to me more than all the textbooks that tried to describe the same thing. I hope you find a way to keep it online for many more generations of math learners!
I appreciate the kind words! And yeah, that could have even been MS Paint: it's been a long time, and I was still using Windows a lot back then. (I made the first version of this page for my wife when the topic came up in one of her graduate math courses: I'd had a much better exposure to Lagrange multipliers in my Calc 3 class than she'd had, and since she was studying in another state at the time I wrote it up in HTML instead of on paper.) I've occasionally thought about going back and redrawing those first few images in some better form, but hey: they still get the idea across perfectly well.
I have every intention of keeping it online indefinitely, as long as my friends doing the hosting don't get sick of me mooching off their web space.
I remember needing to relearn Lagrange multipliers several times throughout undergrad and grad school. Each time I needed a refresher, I ended up back at this "cow and the river" example. For some reason it always seems to map nicely back to whatever the current problem I am focused on is.
> For some reason it always seems to map nicely back to whatever the current problem I am focused on is.
This is the case with many combinatorics and optimization problems. The specifics of the problems may differ, but the underlying math tends to be very similar.
One application of Lagrange Multipliers includes deriving the micro, standard, and grand canonical ensembles from statistical physics [0].
These can be used to simulate physical systems like magnets and particle fields.
The Lagrange Constraints are used to constrain things like probability, energy, and particle numbers. What is constrained accounts for the difference between the ensembles listed above.
The derivation of the Support Vector Machine also involves Lagrange Constraints as well!
Lagrange’s own intuition into the multipliers was based on a purely formal construction of a differential equation that would incorporate the data about a point at equilibrium subject to constraints as well as about the constraints themselves. (I find the idea of “the virtual motion” in statics, where there is no motion, fascinating.) Here’s the story:
