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Understanding Mathematics: Functions of more than one variable

As you know, the main goal of Brilliant is to provide you with fun and challenging problems in maths and sciences. There are, however, many questions related to interesting mathematical results that cannot be turned into problems with a well defined "correct answer." Arguably, such questions are the most powerful driving force behind new mathematical discoveries. Similarly to "Thinking like a theorist", the goal of the "Understanding Mathematics" series is to raise some of these questions and give each of you an opportunity to express your opinion.

David and I will be hosting these discussions, which will alternate between Mathematics and Physics. We plan to devote two weeks to each discussion topic: a question will be posted every other Saturday, and you will have a week and a half to ponder it and present your thoughts. If there is sufficient interest, then on the following Wednesday (approximately 10 days later) we will present an answer which touches on the important aspects presented, and entertain further comments. Before going on, let me introduce myself. My name is John Smith, and I am the new mathematical discussions leader at Brilliant. I look forward to exploring the depths of mathematics with you.

Today our topic of discussion is as follows. Are there interesting/important functions of more than one variable? Note that different interpretations of this question are possible, which could lead to completely different answers. (Also, see the update below.)

There are many one-variable functions that are so important and ubiquitous that they have their own names. The most common examples are the trigonometric functions (such as sine and cosine) and the exponential and logarithm functions (\(e^x\) and \(\log(x)\)), but there are many others (such as the Gamma function \(\Gamma(x)\)). Yet we don't often encounter ``named'' functions of two or more (real) variables. Why is this the case? Do you know of examples of functions of several (real) variables that have proper names? Why are these functions important? This is one way to interpret the topic of discussion.

A different approach is to ask whether there are examples of functions of two or more variables that have interesting behavior, which is very different from anything that you can observe for one-variable functions. You are also welcome to suggest your own interpretations of the discussion topic; please explain your viewpoint in the comments.

Update. As can be seen from the comments you posted, examples of useful functions of several variables abound. Therefore I'd like to make the first viewpoint on the topic of our discussion more precise. Are there important genuine functions of two or more variables? By "genuine" I mean that the function cannot be expressed in terms of one-variable functions using arithmetic operations and compositions. Here are some functions that are not genuinely functions of several variables. We can write \(y^x=e^{x\cdot\log(y)}\), which represents the two-variable function \(f(x,y)=y^x\) in terms of the exponential function, the logarithm function, and the multiplication operation. Similarly, the function \(g(x,y)=\sqrt{x^2+y^2}\) can be expressed in terms of the squaring function, the square root function, and the addition operation.

Note by John Smith
4 years, 1 month ago

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A lot of simple operations can be thought of as binary functions, i.e. \(2 + 5 = \mathrm{Add}(2,5)\), etc. Functions like \(e^x\) and \(\log x\) can be thought of special cases of more general two-variable functions like \(f(x,y)=y^x \) and \(f(x,y)=\log_y{x}\). Ricky Escobar · 4 years, 1 month ago

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@Ricky Escobar Very true and interesting. But note that for each of these "general" two variable functions, we usually only use one set value of one of the arguments, and take the other as the true variable. Bob Krueger · 4 years, 1 month ago

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Dividing mathematics into functions of one variable, and 'genuine' functions of more than one variable is like dividing the entire physical universe into bananas and non-bananas.

That's a paraphrased joke from, perhaps, Von Neumann. The point is that functions of one real number variable are simple enough that we can study them, complicated enough that we don't get quickly bored doing so, and generalizable enough that by studying them we gain valuable (but sometimes flawed) intuition for functions that do not have just some subset of real numbers for the domain and range.

Your so called non-genuine functions of multiple variables are ones that we can seek to understand by seeing the effect of each variable individually. But even in this non-genuine realm we can see some behavior that is not accounted for by our single variable inspired intuition.

For example, there are functions of two variables such that if we restrict the domain to any line in the plane we have a continuous functions (defined by a single variable parameterized along the line of domain) but considered as a function of two variables it is discontinuous. E.g. \(f(x,y)=\frac{xy^2}{x^2+y^4}\) if \((x,y)\neq (0,0)\) and \(f(0,0)=0\)

Even though the function is defined in familiar terms there is a new type of behavior because locally, the input of the function can vary along two dimension instead of just one.

Going deeper, we will often want to consider functions for which we cannot even describe elements the domain with a list of numbers. If you want to know how long it will take a ball to slide to the bottom of some ramp you need to define a function whose input is a curve which defines the ramp, and the output is the time required. In principle it takes an uncountable list of coordinates to define the points of the curve, but we can do it more succinctly by define the curve as the graph of a function.

Here it seems that there should be infinitely many dimensions along which we can make a small change to a given curve in the domain. Yet still, in attempts to understand the properties of this function, we make efforts to isolate the possible variations into one 'dimension' at a time, and analyze change in that dimension modeled by a single variable function. Colin Hinde · 4 years, 1 month ago

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@Colin Hinde I think you raised some very important arguments.

Infinite variable functions indeed appear everywhere in nature, like in your ball-ramp problem. However, the solution of such problems will most likely contain functions like \­( e^x \­), and other named univariable functions, while the "ramp-function" is specific to the ramp-problem, and so is not very interesting from a purely theoretical viewpoint. Marcell Simkó · 4 years, 1 month ago

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@Colin Hinde I think that your comments address the second interpretation of the topic of discussion. Indeed, you are right that by combining one-variable functions using arithmetic operations and compositions, one can obtain functions of several variables that behave in ways, which one-variable intuition cannot explain. I agree with you that this phenomenon is very important.

Also, just to be clear, I never suggested that mathematics should be "divided" into functions of one variable and functions that are genuinely of more than one variable. Indeed, most of the several-variable functions that are interesting and important are not "genuine" in the sense of the discussion. This is precisely the point that I wanted to make. Why is it the case that most of the several variable functions that one encounters in practice are not "genuine" in this sense? Is it just a coincidence, or is there something deeper going on? John Smith Staff · 4 years, 1 month ago

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\(^nC_r\): the number of ways of selecting r objects out of n objects. It can also be written as C(n,r). I don't know the name of this function but I have used it a lot of times.

Greatest Common Divisor (GCD) and Lowest Common Multiple (LCM): These functions take atleast 2 arguments. Eklavya Sharma · 4 years, 1 month ago

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@Eklavya Sharma Good -- several valid points have been brought up so far. Let's start with arithmetic operations, mentioned by Ricky here. Indeed, there is no way to represent the addition map \(Add(x,y)=x+y\) using only one variable functions. In fact, a moment's thought will convince you that if you want to "disassemble" a function of several variables into one-variable parts, you must still allow yourself to use at least one function of two variables.

Now suppose that we allow arithmetic operations as part of our toolbox. Can you think of ways to represent some of the other two-variable functions, mentioned in the comments, using only one-variable functions and arithmetic operations? For example, what about the binomial coefficient \(C(n,r)\) mentioned by Eklavya here, or the functions \(f(x,y)=y^x\) and \(g(x,y)=\log_y x\) mentioned by Ricky? John Smith Staff · 4 years, 1 month ago

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@Eklavya Sharma You are right about the GCD and LCM functions: they are "genuine" functions of two (or more) variables that are important and have names attached to them. When I originally formulated my question, I mostly had in mind functions of a real variable, but as I mentioned, it is good to generalize these questions beyond the initial context. On a side note, the first function you mentioned -- the binomial coefficient \(C(n,r)\), can in fact be extended to accept real numbers as arguments; but as far as I know, there is no useful way to extend the GCD function to arbitrary real numbers. John Smith Staff · 4 years, 1 month ago

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We can really think of a function as a sort of mapping. Our normal univariable functions are just maps from the real number line to the real number line. We can conveniently express these in two dimensions by putting the number lines at right angles. For multi variable functions, we can think of mappings from one Euclidean space into another, not necessarily of the same dimensions. Ahmed posted about the distance function, which is really a mapping from an n dimensional Euclidean space to the real number line. For example, there exists a mapping to stretch the plane, say f(x,y)=(2x,y), as a simple example. Note that restricting the domain could give more interesting results. You can find the equations of much more complex mappings or compositions thereof, say making a square a cube, or making the unit cube a sphere. Other than this function notation, I believe these mappings should be express able in the form of matrices, if you are familiar with that subject. I came across all these ideas whilst meddling in introductory topology. There probably exists specific well known names for some of these function, but I think the reason most of those don't have names is because they aren't used as often as univariable functions. Bob Krueger · 4 years, 1 month ago

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We can do with f(x)=squareroot(x²+y²) as a function that relates the A(x,y) in the plane with its distance from the center. Ahmed Taha · 4 years, 1 month ago

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@Ahmed Taha Right. Or similarly any point in \(R^n\).

However, this could be thought of as a one variable function of a vector, i.e. the norm. Bob Krueger · 4 years, 1 month ago

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@Bob Krueger This is an interesting point. If we generalize the meaning of the word "variable," we can think of pretty much any function as a function of one variable. In some cases this viewpoint is extremely useful. For instance, in linear algebra, whenever you can work with functions operating on vectors by treating a vector as a single variable, instead of writing it out in terms of its "components," you typically get much simpler formulas, and a much cleaner and more understandable theory. John Smith Staff · 4 years, 1 month ago

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@John Smith And if we consider a vector as a single variable, then most of these multi variable functions could be expressed as the function of a vector, right? Bob Krueger · 4 years, 1 month ago

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@Bob Krueger In fact, all of them can! For instance, suppose I have a function \(f(x_1,x_2,x_3,x_4,x_5)\) of five real variables. I can introduce the notation \(v\) for the 5-tuple \((x_1,x_2,x_3,x_4,x_5)\) and write \(f(v)\) everywhere. Thus, at least from the viewpoint of notation, \(f\) looks like a function of one variable.

One should be careful when applying this trick, though: it is useful when it can help you simplify your formulas and clarify the logic of the math that you are doing. However, there are also situations where using this trick would only confuse things. One example that comes to mind is multidimensional integration: for certain calculations involving integrals of functions of two (for instance) variables, it is important to remember that your function really does depend on two real variables, and treating them as a single variable with values in \(\mathbb{R}^2\) would not help you. John Smith Staff · 4 years, 1 month ago

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I know at least one such function which was defined by Srinivas Ramanujan. It is called mock theta function and is defined as follows: \(f(x,y)=\sum_{i=-\infty}^{i=\infty}x^{\frac{i(i+1)}{2}}y^{\frac{i(i-1)}{2}}\) Snehal Shekatkar · 4 years, 1 month ago

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