Talk:Function of several real variables

Real coordinate space

You may reuse Real coordinate space #The domain of a function of several variables and some other pieces there, as well as make wikilinks back and fourth. Also, do not forget about category: Multivariable calculus. Incnis Mrsi (talk) 14:05, 26 June 2013 (UTC)

Done link and category, I'll continue writing later today. M∧Ŝc²ħεИ_τlk 14:12, 26 June 2013 (UTC)

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I see, the passage about multivariable continuity paradoxes/caveats was once added by later destroyed, and Function of several real variables#Calculus now starts from differentiability. May be it is pedagogically correct, but the article should not downplay the fact that even the continuity of a function is not a (very) simple property. Paradoxes with derivatives (such as of ⁠(x + i y)³/x² + y²⁠ at (0, 0)) are not paradoxes with differentiation itself. They arise from problems with the definition of continuity, where a “naïve” approach fails. Incnis Mrsi (talk) 10:09, 20 November 2013 (UTC)

Please show the diffs of the paragraph added and removed. Maybe it could reinserted somehow. M∧Ŝc²ħεИ_τlk 17:57, 20 November 2013 (UTC)

To do

Thanks for the creation of this article. This is a good thing for WP. However, this article is yet a stub, relatively to the amount of material that should be in it and is yet lacking. I'll use the template {{to do}} to summarize this lacking material. D.Lazard (talk) 04:24, 27 June 2013 (UTC)

A good list, but presumably some things (like the Taylor expansion, partial derivatives, Hessian matrix, real multivariable calculus) are to be mentioned and linked to, not to be covered in too much depth here. M∧Ŝc²ħεИ_τlk 06:09, 27 June 2013 (UTC)

By now, I've at least scraped the surface of a number of entries in the list. Any corrections of my inaccuracies are appreciated in advance.

Maybe some examples could be removed but not sure which ones. The geometric examples are always nice for easy visualization (surfaces of conics and surfaces/volumes of certain 3d solids). Considering the recent posts at Wikipedia talk:WikiProject Mathematics#Linear function, Linear Equation, Linear Inequality full of errors and inconsistencies and "matheese" about "linear equations" and "linear functions", etc. we should at least keep the plane and hyperplane examples... as well as non-linear examples for contrast. The inclusion of a variety of examples in physics/engineering/applied maths is essential. For compactness maybe we could tabulate some examples but it would not be encyclopaedic to do so... Will look at trimming later. M∧Ŝc²ħεИ_τlk 10:08, 28 June 2013 (UTC)

The article is desperate for diagrams, which are coming shortly. M∧Ŝc²ħεИ_τlk 11:12, 29 June 2013 (UTC)

Multiple issues

User:Maschen has done a hard work in writing from scratch such a fundamental article. Nevertheless, in its present state the article has multiple issues. Some of them are easy to solve, but other are more difficult. Here are the main issues:

• Level of the target reader: One may consider that the reader of this article has some knowledge of what is a function and what is a function of one real variable. Otherwise he would not arrive to this article. In fact, when searching for "function", the first answer is Function (mathematics) and when searching for "function of", Function of a real variable appears before the present article. It follows from this remark that the section "Introduction" is off-topic and most of the section "General definition (real-valued functions of several real variables)" must be simplified and rewritten to remove over detailed explanations of the similarity between univariable and multivariable cases and emphasizing on the differences.
• Level of the target reader (2): Although on may suppose that the reader knows something about univariable case, it is essential to suppose that he does not know anything about the multivariate case. Therefore the definition of the continuity and the differentiability must be given explicitly (this is not the case) and illustrated by examples showing that the continuity in each variable does not implies continuity, and that differentiability in each variable does not implies differentiability.
• Confusion in the terminology: The article uses in the definitions many terms that either do not have any formal definition or have a formal definition but only in a different context. Such are: "dependent variable", "independent variable", "continuously varying variable". Moreover, the article is confusing by not clearly distinguishing a function, its graph and the equation of its graph. This confusion is common when teaching kids (I do not know if it is pedagogically a good thing), but must be avoided in a encyclopedia.

There are many other issues. As they are less fundamental, either I'll correct them by editing directly the article or they will be the object of another post. D.Lazard (talk) 14:33, 30 June 2013 (UTC)

Thanks for constructive feedback. In order of the points:

• It seems most you've done most of the objectives, but I'm not sure right now how to simplify. The geometric description of ℝⁿ is important and as simple as I can get it. If it's the set notation, maybe it could be cut out, but there is not masses and masses of unfamiliar notation and it wouldn't be a bad thing to show how set notation (elements, subsets, Cartesian product) is used. About emphasis on the differences, that would be important, but in the first stages I was trying to make the transition from one variable to many variables clearer using the geometric description of ℝⁿ, so it resulted in similarities more.
• Well, I'd better get decent sources (not just Schaum) and use them to sharpen the statements for continuity and differentiability, also for inline citations.
• The terminology will be fixed and explicated.

Unfortunately, I have badly organized the sections and will try and fix that also.

One question - in the list above you included "Tangent hyperplane to the graph of the function, expressed in term of the gradient" - why was that deleted? And I didn't think the subsection it was in ("Curved hypersurfaces") was completely off-topic as it would offer more geometric insight to multivariable functions. No matter. M∧Ŝc²ħεИ_τlk 09:08, 1 July 2013 (UTC)

I have considered the section "Tangent hyperplane to the graph of the function, expressed in term of the gradient" as off-topic because of the emphasis on geometry and equations, which does not belong to the subject of the article. Moreover, it supposes that the reader knows about high dimensional geometry and hypersurfaces. However a section on the best approximation of a differentiable function by a linear function would have almost the same mathematical content and fit exactly the subject of the article.

About the generalities, I'll propose soon a new presentation. D.Lazard (talk) 14:14, 1 July 2013 (UTC)

Periodic functions

Unable to find a ref for now (should be able to in time), but this would be good to add, my rough formulation would be:

A function of the form:

$f\,:\,X\rightarrow \mathbb {R}$

$f({\boldsymbol {x}}+{\boldsymbol {T}})=f({\boldsymbol {x}})$

where T = (T₁, T₂, ..., T_n), and using interval notation:

$X=\{{\boldsymbol {x}}\in \mathbb {R} ^{n}\,:\,x_{i}\in [a_{i},a_{i}+T_{i}]\}$

where i can be any or all of 1, 2, ..., n, is a periodic function of several real variables, with periods T_i. Not all of the T_i have to be nonzero.

For example:

$f\,:\,X\rightarrow \mathbb {R}$

$f(x,y,z)=z\sin(ax)\cos(by)$

with domain:

$X=\{(x,y,z)\in \mathbb {R} ^{3}\,:\,x\in [0,2\pi /a]\,,y\in [-\pi /b,\pi /b]\,,z\in (-\infty ,\infty )\}$

is periodic in x and y, but not z:

$f(x,y,z)=f(x+2\pi /a,y+2\pi /b,z+0)$

A physical example would be the Bloch wave. M∧Ŝc²ħεИ_τlk 13:33, 8 July 2013 (UTC)

"Supposed supposed to contain an open subset of ℝⁿ"?

The end of the lead section says that the domain of a function of several real variables is "supposed to contain an open subset of ℝⁿ". I understand why the domain of some kind of sensible functions might contain an open subset, but the following is a perfectly valid function of several real variables: $f:\{(0,0)\}\to \{\emptyset \}$ where $f(0,0)=\emptyset$. Could someone add a citation or clarify the language? Thanks, and sorry if I'm misunderstanding! siddharthist (talk) 21:14, 23 November 2017 (UTC)

Here "supposed" means that a function whose domain is finite of does not contain any open set of ℝⁿ, is a well defined function, which, generally, is not considered as a function of n real variables. This is rarely explicitly said, but becomes implicit as soon as one consider continuity and differentiability, which are defined only for points in the interior of the domain. D.Lazard (talk) 22:09, 23 November 2017 (UTC)

Section on the domain

The section Domain is somewhat questionable... It claims that it might be difficult to specify the domain of a function one is defining, which seems backwards. I don't believe that one can define a function without specifying its domain. It is also unreferenced. Can anyone support its claims, or should we remove it? siddharthist (talk) 21:25, 23 November 2017 (UTC)

Take any function f that has the real line or the complex plane as its domain. Knowing the domain of 1/f amounts to know the zeros of f, which may be a difficult task. In the case where f is Riemann zeta function, to specify the domain of 1/f is equivalent with proving or disproving Riemann hypothesis. Not an easy task! D.Lazard (talk) 21:59, 23 November 2017 (UTC)

In grade school, an extremely common abuse of language is to ask to "find the domain of" a given function, which indeed is putting the cart before the horse. What is really meant is "What is the maximal subset of $\mathbb {R} ^{n}$ on which we can use the given expression to define a function?", or even more succinctly, "On what subset of $\mathbb {R} ^{n}$ is this expression defined?". This is probably what is being meant by the statement in question.--Jasper Deng (talk) 22:05, 23 November 2017 (UTC)

Thank both of you for helping to clarify this point. It seems to me that the expression 1/f is not a function per se, but might be if one specified its domain, codomain, and arguments correctly. Could someone who has a clearer picture of what this is trying to say rewrite it to actually be correct (i.e. not abuse language by talking about "functions", but rather "expressions")? Or maybe this is totally clear to most readers and I'm just pedantic :) siddharthist (talk) 05:47, 24 November 2017 (UTC)

I may have been too sketchy in my preceding post. In any case, this is not a problem of expressions, this is purely a problem of functions: If f is a non-constant continuous function (in the sense of this article, that is, its domain contains an open set), then, this is a theorem that ${\boldsymbol {x}}\to 1/f({\boldsymbol {x}})$ is also a function whose domain may be difficult to specify, even if f and its domain are well defined. I have edited the article section for clarifying things (I hope). D.Lazard (talk) 11:06, 24 November 2017 (UTC)

I'll remove the old example of the section for the following reasons. This example is confusing, as it is unclear whether the difficult problem is to find some ball or the largest ball. In the latter case, the example duplicates essentially the new example (reciprocal function), while being more WP:TECHNICAL. If the problem is to find some ball, it is generally not difficult, at least if one knows an upper bound of the first derivatives in a neighborhood of the center of the ball, which is the case when f is a polynomial. D.Lazard (talk) 14:57, 24 November 2017 (UTC)

Minor edit: Section on continuity and limit

The statement

   If a is in the interior of the domain, the limit exists if and only if the function is continuous at a.

is technically incorrect, as continuity is a stronger condition than the existence of a limit. Specifically, f is continous at a if and only if the limit exists AND agrees with the value f(a). — Preceding unsigned comment added by Hojahs (talk • contribs) 05:30, 25 November 2020 (UTC)

The statement is not only correct, but it remains true if "the interior of the domain" is replaced by "in the domain". In fact, the definition of a limit is "For every positive real number ε > 0, there is a positive real number δ > 0 such that $|f({\boldsymbol {x}})-L|<\varepsilon$ for all x in the domain such that $d({\boldsymbol {x}},{\boldsymbol {a}})<\delta .$" If one takes x = a, one has $d({\boldsymbol {x}},{\boldsymbol {a}})<\delta .$ Thus the existence of a limit implies that $|f({\boldsymbol {a}})-L|<\varepsilon$ for every ε > 0. This implies $f({\boldsymbol {a}})=L.$ D.Lazard (talk) 10:46, 25 November 2020 (UTC)

@D.Lazard: in my experience this is not the definition of limit commonly used: almost always one takes the hypothesis $0<d({\boldsymbol {x}},{\boldsymbol {a}})<\delta$, specifically removing a from the ball and allowing the possibility of removable discontinuities. (The article limit of a function agrees that this is the more common usage.) --JBL (talk) 16:05, 25 November 2020 (UTC)

I have never seen before the definition with $0<d({\boldsymbol {x}},{\boldsymbol {a}})<\delta ,$ but I must aknowledge that it is used in English Wikipedia. However, in French Wikipedia ${\boldsymbol {x}}={\boldsymbol {a}}$ is not excluded. In German Wikipedia, the Engish definition is given first and is called the "punctured definition"; the French definition is given later in the article, it is presented as "newer" and called "unpunctured". I would guess that the definition used in English Wikipedia is commonly used in US pedagogical mathematics, while the other definition is more commonly used in advanced mathematics. This is only a guess, and would require a verification. As several articles are concerned by this, I'll open a discussion at WT:WPM. D.Lazard (talk) 17:52, 25 November 2020 (UTC)

Restriction of domain to real variables

I am concerned by the redirection of two other pages, relating to multivariate functions and multivariable functions, to this page.

I have no problem or argument with the main content of this page per se (it matches the page name, and I am not qualified to comment on its accuracy or completeness), but the redirection, in my view, equates this important special case, of a "function over several real variables", with the general concept of "a function over several variables".

The general definition of a function over multiple variables does not restrict the function's domain to a set of n-tuples of real numbers, nor the range to the reals.

In discrete mathematics it is quite common to define a function over a domain which is a Cartesian product: e.g. A X B X C, where the sets composed by the product set (A, B and C in my example) are simply collections of objects of any kind that makes sense for the problem at hand.

For example, an edge-labelled graph can be defined as follows:

A simple graph G is a pair (V, E) where V is a set of vertices, and the set of edges E = V X V. We can say E(G) to denote the set E in G.

An edge-labelled graph LG is a tuple (G, L, λ) where L is a set of labels, and there is a function λ : E(G) → L.

The function λ takes as it input a pair of vertices (u ∈ V, v ∈ V), and its value is a label l ∈ L.

This is a multivariate function, a function over/of multiple variables, whose value or result is an arbitrary object called a label. There are no real numbers in sight :-).

Such functions commonly appear in computer science as functions over variables of different types with different names, which is just a restatement of the set-theoretic concept of a function whose domain is a Cartesian product, since types are sets of values, and we name sets to distinguish them. Once this is made clear, the connection of this page to the page on variadic functions also becomes clearer: they both derive from a more general concept of a function whose domain is a product set.

If this is an acceptable framework then I would want to insert some variant of the content just given, with links to this page and to the page on variadic functions, into the page on multivariable functions and redirect the multivariate function page to it (or vice versa).

Apologies for not formatting the mathematical definitions pleasantly or using proper links for the other related pages -- I'm writing in a rush.

Wikimodoo (talk) 07:07, 19 January 2022 (UTC)