What are non-linear PDEs? And what is topology? And what are either used for?
I know PDEs are partial differential equations, but I haven’t the slightest idea what that actually means.
Since its probably important to the level of any explanation, the highest math I’ve taken is pre-calculus in high school, but that was a couple years ago and I don’t remember much of it.
I’m not an expert on either of these topics, but I’ve taken a class on partial differential equations and I’ve done some reading on topology.
I’d suggest this article in the wikipedia for background on topology.
From the article
Topology is sometimes jokingly referred to as “rubbert sheet geometry” because it is concerned not with size or shape of a geometrical object (2 or 3 or even more dimensions) but about the properties that are independent of precise size (aka “metrics”). So, a donut and a coffee mug (with a handle) are topologically equivalent solids: each one is a three dimensional object with a single hole in it.
What is it used for, well, one branch of topology, graph theory, is used extensively in the design of computer networks. Graph theory studies points (nodes) and segments connecting those nodes. As noted with the Konigsberg bridge problem above, topology looks at the properties that do not depend on the size of the segments connecting the nodes. This is a ridiculously brief introduction but it will, I hope, give you some flavor for what topology is about.
On to partial differential equations. To be honest, I couldn’t tell you much about non-linear differential equations. I did take a class in partial differential equations. Again to vastly oversimplify, the point of partial differential equations is that you can handle several variables at once. So, for example, if a body is moving in three dimensions over time, you would use partial differential equations to describe it. In practice, pde’s are extremely important in physics for subjects such as the exchange of heat and fluid dynamics.
I’m sure someone who really knows what they are talking about will be along shortly, but those are my best stabs at answering your questions.
I’m not familiar with the American school system, so I’m not sure how much you know, but do you know what a derivative is? Differentiating a function?
If so:
A differential equation is an equation involving a function and its derivatives, like
f(x) = (1/3)f’(x)
or
f(x) = xf’(x) + f’’(x)
or
1 = f(x)*f’(x)/e^f’(x)
Solving a differential equation means figuring out what f(x) is, preferably in a simple and explicit way like f(x) = e^(3*x) (that’s one solution to the first equation above), sometimes as something a bit more unweildy, like an infinite series of trigonometric functions, and sometimes you can only solve it numerically.
Of the above, the first two are linear: they just involve the function and its derivatives, multiplied with explicit functions of the variable. The third one isn’t, since it’s got the multiplied together, and in an exponent. You want your DEs to be linear, they’re much nicer that way.
And a PDE is simply a DE with multivariable functions.
As the last post inferred, all first-year calculus courses will include work on simple DEs–wouldn’t be differential calculus without them–, some simple PDEs, but obviously they get much harder later, in the courses that specialize in those areas.
Good description of topology. Speaking as a topologist though, I would consider graph theory to be part of combinatorics, not topology. It seems Wikipedia disagrees, but there you go.
Other uses of topology appear in other branches of science, particularly particle physics, relativity, and astronomy. String theory has more topology than you can shake a rubber sheet at: the idea that particles are not points but instead are infinitesmal loops is very topological. In general relativity, the universe itself becomes a rubber sheet, and gravity is described as a warping of the sheet in response to the presence of mass. And one of the most interesting projects in astronomy right now (to me, anyway) is an attempt to determine the shape of the entire universe by looking for repeating patterns in the cosmic background radiation, like locating the mirrors in a funhouse by comparing different reflections. The math behind that idea is directly descended from topological concepts like the universal cover and the developing map.
I think the best way of defining topology (which Wikipedia doesn’t use) is with the Klein programme, but that immediately assumes a lot of algebra.
For the other part, partial differential equations are equations which relate various partial derivatives of a function to each other. For example, the wave equation in the line, which states u[sub]xx[/sub] = ku[sub]yy[/sub]. Linearity is a property basically coming down to saying each derivative of the function is only multiplied by other (fixed) functions and these are added up. The upshot is that the sum of two solutions is also a solution. The problem is that most of the interesting equations from physics aren’t.
I’m not so sure of that: Klein’s programme was largely concerned with geometry, which is only one aspect of topology (albeit the most significant one these days, in terms of current research). Also, it wasn’t intended to be a definition of the subject as much as a research direction or a point of view. Your mileage may very, I suppose.
There’s an interesting set of lecture notes here here anyone who wants to know what we’re babbling about, although they get technical very quickly.
It’s tough to explain what nonlinear partial differential equations are to someone who’s never studied any calculus. A first course in differential equations would typically have three semesters of calculus as a prerequisite, and that first course wouldn’t get much into non-linear partial DE’s. But I’ll take my best shot, at the risk of greatly oversimplifying:
In the kind of equations you study in high school algebra, the unknowns stand for numbers, and solving equations means figuring out what numbers the variables could stand for to make the equation true. In differential equations, the unknowns are functions, and to solve a differential equation, you have to figure out what the function could be so that it, and its derivatives. satisfy the equation. The derivative of a function is another function which relates to the first function’s rate of change—this is one of the main topics that calculus is all about.
Just as in high school algebra equations can be classified in different ways (like according to whether they’re linear, quadratic, or higher degree; whether they involve one or two variables; etc.), differential equations can be classified according to whether they’re ordinary or partial differential equations. The functions in partial DE’s are functions of more than one variable—that is, their values depend on more than one other quantity. PDE’s are typically tougher to work with and solve than ODE’s. Another way of classifying DE’s is according to whether they’re linear or non-linear. Basically, non-linear DE’s can involve those unknown functions and their derivatives in more complicated kinds of ways than linear DE’s do.
The Klein programme ultimately says that geometry and topology are essentially studying what’s preserved under certain classes of transformations. In topology, the transformations are continuous maps, which have homeomorphisms as the anlogue of isomorphism. In particular, consider the group of homeomorphisms from a topological space to itself and study the invariants of that group action.
I’m currently taking a 3rd year course in PDEs - and I can safely say I still don’t really know what they are. More precisely, I don’t have any sort of intuitive sense of what a PDE is. Sure, I can do the mechanical steps they tell us to do, but I don’t understand what we’re trying to accomplish.
Specifically, there are mathematical objects called topolgies tht can be imposed on sets. If you have some set X, then a collection of subsets T is a topology if it meets three conditions.
1.) T contains the empty set, and contains X.
2.) T is closed under finite intersections.
3.) T is closed under all unions, finite or infinite.
The visualization of topology as rubber sheet geometry follows from this definition, although explaining why would take about two semesters of graduate level mathematics. But what topologoy really allows us to do is to classify sets. For instance, we know that a line contains an infinite number of points, and a plane also contains an infinite number of points. So how do we know that a line and a plane aren’t really the same thing? It seems like an obvious difference, but topology provides the official answer.
Differential equations are basically just equations with derivatives in them.
Partial differential equations are equations with partial derivatives in them.
For an example of a partial derivative: the Heat Index depends on heat and humidity, but heat is not a function of humidity, and humidity is not a function of heat.
So if you take the derivative (rate of change) of the Heat Index with respect to humidity, you have a partial derivative - you didn’t take temperature into account.
You would use differential equations in various problems that involve rates of change.
For example, if the rate of growth of a population is proportional to the size of the population and the number of people that could be added before it hits its carrying capacity, you could use a differential equation to predict the growth of the population over time.
This could look like dP/dt = P(a - bP) which is a non-linear DE, since you have the dependent variable P multiplied by itself.
If you solved that DE, you would get a function P(t) of population with respect to time.
That should be “air temperature” and humidity, not heat and humidity.
This is a vast oversimplification. We know from set theory that they’re different because one has only one element and one has uncountably many (assuming we’re talking about points and lines in some Euclidean space, and also assuming the subspace topology). Topology tells us that they’re not homeomorphic: that there cannot exist a bijective continuous function with continuous inverse from one onto the other, with continuity being defined by the topologies assigned to each one.
This definition of a topology is a generalization of a similar notion of open sets in a metric space.
A metric space is a set with a “metric”–a function d which assigns distances between pairs of points in the set. Specifically, for any x, y, z in the set:
-
d(x,y) is always a nonnegative real number (the numbers we generally use to represent distances),
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d(x,x)=0 (the distance from a point to itself is zero),
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d(x,y)=d(y,x) (the distance from x to y is the same as the distance from y to x)
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d(x,y) <= d(x,z) + d(z,y) (triangle inequality, or, intuitively, the shortest distance between two points is a “straight line”).
In a metric space, a set O is open if and only if, for any point in the set, you can find a ball centered at that point which lies entirely in O (a “ball” being the set of points within some fixed distance from the center).
The open set structure given in the definition provided by ITR champion is a generalization of this.
From here, you get a nice generalization of continuous functions, which was a big catalyst for the development of topology. A function from one topological space to another is continuous if and only the preimage of every open set is itself open. This definition is equivalent to the definition commonly used in metric spaces.
However, not all topological spaces are “metrizable” (i.e., there are topological spaces whose topology is not equivalent (not “homeomorphic”) to any metric space topology). For such spaces, I’m not sure that “rubber sheet geometry” is a very apt description. To me, “rubber sheet geometry” connotates “nice” spaces, like lines, planes, spheres, Klein bottles, Moebius strips,…that sort of thing (but maybe that’s just me). I just wanted to mention there are really bizarre topological spaces out there, completely unlike these “nice” examples I just mentioned (like the Stone-Cech compactifications of various spaces such as the line or plane or integers). Spaces that really can’t be pictured geometrically, even if you are capable of picturing 4th, 5th, or 6th dimensionsal objects in your head.
Even if you are thinking of such spaces there are examples of “bizarre” spaces. A lot of the time, “niceness” comes down to a topological space being “Hausdorff”. That is, given any two points, there is an open set around each one such that the two open sets are disjoint. In algebraic geometry, almost no spaces considered are Hausdorff, since open sets are the complements of solutions to sets of polynomial equations.
The technique of identifying a sublattice of subsets as a topology on a set is really a dual notion (and usually a more tractable one) to an identified definition of convergence. Take a partially ordered set and map it into the space X. If you want, you can restrict the poset to be N with the usual (total) order, but this misses some cases. Now the question is if this “net” (“sequence”, for N) converges to a point p in X. A function from one topological space to another is continuous if it sends convergent nets to convergent nets, and sends the limit of a convergent net to the limit of its image. A homeomorphism is continuous with a continuous inverse. Note the parallel with the definition of morphisms of algebraic structures: the preservation of an operation. The open set definition of a topology defined continuous functions by reflecting a property, which always feels like somewhat of a hack to me.
Of course (and those who have been watching me in other threads might expect this) there’s a much nicer and more general definition stemming from the notion of a topos.
Is it the line or the plane that has only one element?
Ah. I misread “plane” as “point”.
Actually, this makes my point easier. Consider an appropriate étale topology on the plane, wrap the plane into a cylinder and the line into a circle and the two are homeomorphic.
Like, you couldn’t have asked a more difficult question!?! I am an engineer who found this one of the most difficult concepts in my upper calculus courses to understand. You just want to start at the top and work your way down! :eek:
I bet you sleep on a bed of nails, too, right? And swallow swords? Sheesh!
- Jinx
[geeky math hijack]
(a) If we’re asking whether or not two spaces are homeomorphic, we don’t get to choose whatever topology we like after the question is asked. The line and the plane are both uncountable, so giving both the discrete topology makes them homeomorphic, but that doesn’t mean they’re homeomorphic under the metric topology or whatever topology the questioner had in mind.
In short, I still don’t see how your objection makes ITR Champion’s post an “oversimplification”. Continuity on the reals can be and is defined well before the notion of topology in general, so when someone asks if the line is homeomorphic to the plane…i.e., if R is homeomorphic to R[sup]2[/sup]…it makes sense to say that topology provides the answer to that question, as ITR Champion did.
(b) Since when are a circle and a cylinder homeomorphic?
[/geeky math hijack]