Today I ran across this text which I don’t understand but would like to. According to this blog post that links to it, it describes a “mathematical model for trail formation” similar to the urban legend about college campuses where sidewalks are put in only after students have worn pathways through the grass.
I’m not asking exactly what the equations mean in this instance (though I’ll thank anyone who feels like explaining that). What I’m after is this: Which subdisciplines of mathematics would I need to know about, or which university curriculum would I need to follow, in order to be able to understand this “mathematical model” on my own?
Sidenote: It occurs to me that maybe the example I’ve linked here is total nonsense, the author is a boob, and the math is totally inappropriate for the subject matter. If so, I’d still like to know what kind of math I’d need to know so I could have figured that out on my own.
I hope my question makes sense. My liberal-arts education was a fine thing, but it’s ill prepared me for topics like this.
I have no idea about the math, but putting down sidewalks on paths created by students or other walkers on office or government campuses is definitely real and a very good idea.
Without reading your post and only glancing at the link, it appears to be multivariable Calculus. That would be Calc 3 (or Calc 2 depending on how many classes it’s divided up into) in most Undergrad Colleges…It was one of my favorite classes.
I would hazard a guess that to understand this (which even though I’m a math major I’ve forgotten most of what I learned) you would need a good grasp of Calculus and Differential Equations.
I concur with Joey P; multivariable calculus (if only to understand the basic notion of gradient). Though if all you want is to understand this one particular work, you can, of course, get there much quicker by just learning the things you need to know for it (perhaps even reformulating the way it’s presented), rather than going through the full slog of calculus in a typical course sequence.
It’s also worth noting that what’s been linked to is clearly a mere excerpt from a larger article. And there’s not very much to be gained from reading the excerpt alone. All that’s being presented are some definitions; their motivations, applications, surrounding theory, etc., have not been shown.
Multivariate Calculus, usually taught freshman or sophmore year to basically any hard science major in US colleges.
If you just want an feel for what the equations mean instead of needing to manipulate them or prove anything, you can probably find a math major who remembers their stuff to give you an explanation of the symbols in an hour so without spending a year in college. Calculus is pretty open to an intuitive explanations of its symbols.
For example, the e(r,v,t) function in the page you linked to is just an arrow of constant length that points from any point in the space to the travelers desired destination.
Thanks to all for your helpful answers. It’s not just this particular passage I want to understand; rather, I’d like to be the kind of person who doesn’t have to skip over every mathematical section I encounter in an article. If it takes 3 calculus courses to get savvier, then I may just make that my New Year’s Resolution. (I enjoyed Calc 1 as a college freshman, but that was 25 years ago.)
I suppose a more general question would be about a “math for scientists or social scientists” curriculum. I work with biomedical research scientists and I have a layman’s interest in urban planning; these are very different disciplines, of course, but I suspect that experts in these fields and many others have, or should have, a certain math background in common – some understanding of multivariable calculus, for example, and some statistics. Is this true?
ETA: cross-posted with Simplicio, whose note that multivariate calculus is universal in the hard-sciences curriculum is welcome news.
Strongly seconded. Depending on your particular goals, doing something like this for a variety of basic calculus concepts may be preferable to bothering with a full-on calculus course. [And once you’ve acquired some intuitive understanding of the various concepts [figuring out the language or vocabulary, so to speak], you can always flesh it out with more technical details and calculating proficiency later]
My FIL is chief of maintenance* at a hospital. Any time they add a new building, new landscaping, new entrance, new parking lot, new anything really, one of the questions that pops up is where to put the new sidewalks (or mulch pathways depending on where it’s going). His answer is always the same…Instead of guessing, just put down sod everywhere and we’ll come back in a year or two and pave the areas where the grass is trampled down. It’s easier then putting the sidewalk or path in the aesthetically pleasing area just to have people cutting across a different way and then dealing with the trampled grass and mud.
*Maintenance being responsible for things like repairs to all things minor and major, landscaping, snow removal, contracting builders for new roofs/parking lots/etc…not maintenance as in mopping the floors and cleaning up puke.
My son, the Quantum Computational Physical Chemist, tells me that the first part isn’t involving even calculus, but definition of functions. It isn’t until you get to the final steps (starting with Step #16) where actual calculus is involved.
I have no idea what that stuff in the paper means. To me it’s all Star Trek science (i.e. they could be pulling the stuff out of their dark nether regions for all I know). However, my son merely glanced at the paper and had no problems understanding it. So, maybe to understand this paper, you need about as much education as my son.
And, how much education does my son have? He’s been going to college for five years. He has a double major in Chemistry and Physics with a minor in Math. (Actually, he has enough Math to be a triple major, but his degree requires him to have a minor, so he declared Math to be his minor). He’ll be graduating this year with about 150 hours of credit. Then, off he goes to graduate school to procure a Ph.D.
Our only hope is that my son uses his powers for good and not evil.
The math curriculum for the hard sciences and the social sciences will look pretty different, but three semesters of calculus is pretty standard. I’ve posted before about my take on an undergraduate math curriculum, and I think that’s a good place to start looking, although I would recommend differential equations to people who are looking more at applications than theory.
There should be a listing for complex analysis in there too. I must’ve just overlooked that the first time around. I used Saff & Snider as an undergrad, and it was a fine book. Ahlfors is the classic graduate level introduction, but if you’re more interested in applications, it’s probably not the right place to start.
