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surface integration with respect to area

surface integral
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Does anybody know what is the right preposition for the verb 'integrate', i.e. which of the following phrases is right:

function f(x) is integrated <over> a sphere
function f(x) is integrated <on> a sphere

or may be they both are correct?

Intereseting question. While they may not be of earth-shattering importance (and certainly are mathematically irrelevant), linguistic questions like this can be fun and interesting diversions, so I decided to have a look. After the business with the tuplets, I already had the dictionary open and ready for business.

I decided to start by browsing through a few math books to see what usages I might find. Unfortunately, this can be tricky because, as often as not, someone simply writes an expression like \int_S and doesn't bother describe what they are doing using words.

Generally speaking, I saw the phrase "intgration over" used most frequently. However, other prepositions also showed up. For instance, Whittaker and Watson use the phrase "integration along" in their discussion of Cauchy's theorem.

My guess would be that both usages are acceptable. From the syntactic point of view, "on" and "over" are prepositions, so both usages parse correctly. The question is one of semantics. However, given a well-behaved function and a sphere, there's a unique number which can be rightly called the integral of that function, so no matter which phrase one uses, it will be understood to refer to the same quantity. I suppose that if one where dealing with pathological functions for which one could come up with different anwers by using different procedures, then one might decide to define "integration on" to refer to one procedure and "integration over" to refer to the other. As far as I know, noone has done this.

Personally, I prefer the term "over" and would usually say something like "given a function f on the sphere, we integrate this function f over the sphere with respect to the measure m on the sphere". I guess that this is because, for me, the word "on" in mathematics suggests a structure (like a function or a measure) that just sits on the sphere and looks at you waiting for something to happen while "over" suggests a process that is happening and moves from one part of the sphere to another. This is kind of like what happens in everyday language where one speaks of "standing ON a rock" as opposed to "walking OVER a rock".

Maybe the difference is one of connotation --- "integrate over the sphere" suggests that to me that the integral is a process that involves the sphere whilst "integrate on" suggests that the sphere is a passive bystander. Thinking this way, the choice of word would be a reflection of one's intuitive picture of integration.

Since we have an active Finnish-speaking member on this site, it might be interesting to compare this with what happens in Finnish, since that language belongs to a different family.

P.S. For the heck of it, I had a look at the Oxford English dictionary and found that they allowed -tuple as a noun, so I guess both usages are OK. In this case, listing both as synonyms should solve the problem.

in spanish both on and over can be described with the same word "sobre" and thus we say
"la integral sobre una esfera|region, etc"

I don't think "along" is used other than over curves, in spanish we use "a lo largo de" for along but it's also heard
"integral sobre la curva"
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

Indeed the Finnish language is not Indo-European but a Uralic (or Finno-Ugrian) language, the predecessor of which was spoken as a lingua franca under the last Ice Age widely in the northern Central and Eastern Europe (in the Periglacial Zone); nowadays the Uralic languages form only separate insels (as Estonian, Hungarian, Mordwinian, Cheremis, Nenetsian etc.). Many scientists say that e.g. the Germans have changed their original Finno-Ugric language to an Indo-European one, when the Indo-Europeans came from the south bringing the agriculture to those mammouth-hunters (in the German one can find some Finno-Ugric substrate features).

As for integration over some domain, we use in Finnish the comparable preposition or postposition "yli" -- I think this usage has come to us from other European languages (as from French "sur", German "über" or Swedish "över"). In the line integrals we use the pre- or postposition "pitkin" which corresponds the English word "along".

While we're on the subject, it might be worth mentioning that in Polish, the usual expression for "intgrate over a sphere" is "calkowac po kule", although Banach prefers "calkowac w kule" --- the preposition "po" in this context (this word tends to be used with many different meanings) can be translated as "over" (or as "on") and "w" as "in".

What I find interesting about linguistics and mathematics is how the meanings of words get stretched from their usual meanings in order to describe abstract concepts which can be very far removed from the everyday experiences which language was designed for. A good example would be the phrase "closed set" --- clearly the sense of the word "closed" has changed from the sense in a phrase like "closed door".

This discussion again brings up the suggestion that has been made on more than one occasion in the past of adding some sort of lexicon to Planet Math. I'm wondering how hard or easy it would be to add a place in the entries for foreign terms. There's already a place for pronounciation; I assume that this could be something similar.

Many thanks for replies! The main reason for the question was that there is to my opinion inconsistency in the following names:

"example of integration with respect to surface area on a helicoid"


"example of integration with respect to surface area of the paraboloid"

so taking into account all discussions they both should be changed to:

"example of integration with respect to surface area OVER a helicoid"
"example of integration with respect to surface area OVER the paraboloid"

also what seems strange is once usage of indefenite article and other times usage of definite one. Probably one should only one, but which?


What also seems quite strange to me is that googling the expression

"integral with respect to surface area"/"integrals with respect to surface area"

brings very few results, whereas

"surface integral"/"surface integrals"

seem to be much more popular. Thus may be the better name for the entry

"integration with respect to surface area"

would be just

"surface integration"

which is much shorter and easier readable, or? At least to make it in 'other name' field won't hurt. But for the names of examples I would choose "surface integration" since it would be defenitely much more readable

"example of surface integration over a helicoid"

(it seems that defenite article is more appropriate) then current

"example of integration with respect to surface area on a helicoid".


To be consistent, I will change all ocurences of any other preposition to "over" and revise the names of the titles. I am still thinking about your suggestion to consolidate some of these entries. I am also thinking of other ideas such as attaching a short table of formulas for surface integrals over several popular surfaces (e.g. spheres, cones, cylinders, ellopsiods, ...) for as a handy refernce. In this case maybe I would turn some of these examples into derivations of the formulas in that table. All in all, this entry is in flux as I experiment with different ways of presenting this material and organizing it.

I deliberately chose the title "integration with respect to surface area" rather than "surface integrals" for a specific technical reason. Whilst the term "surface integration" denotes any kind of integral over a surface, the phrase "with respect to surface area" specifies that one is using a specific measure to perform the integral. Since the focus of this article is on computing integrals with respect to this measure, I thought this title would be more appropriate. Perhaps a better choice of title would be "surface integrals with respect to area measure", but I feel reluctant to use the term "measure" because this entry is meant to be accessible to calculus students and one usually doesn't hear about measures until one get to analysis class. As a compromise, I'll try the title "surface integrals with respect to area". If that's not too good either, let me know and I'll try again.

However, adding "surface integration" as a related topic or a keyword is an excellent idea, so I will do that.

In closing, I would like to thank Serg and Robert for their suggestions with respect to organizing this entry. In many ways, writing a Planet Math entry differs from more conventional mathematical writng, such as writing a book or an article. One runs into issues like using hypertext effectively and keeping terminology and notation consistent with other entries. One must keep in mind that an entry might be accessed by readers with different goals and levels of mathematical sophistication and that the links should be understandable to sthe reader. While it is relatively straightforward to write short entries, writing, editing and organizing a longer entry like this one can be something of a learning experience. Your advice and suggestions are most helpful and have already led to an improved entry.


Dear Ray

First of all many thanks for your kind words:

> In closing, I would like to thank Serg and Robert
> for their suggestions with respect to organizing this entry.
> Your advice and suggestions are most helpful
> and have already led to an improved entry.

What you wrote about difficulties in writing encyclopedia entries is completely true (at least I feel that way), but you seem succesfully overcome them, when looking on the amount of materials you are producing ;). It is really amazing how fast you can write the texts ;).

Now back to our discussion:

> Whilst the term "surface integration"
> denotes any kind of integral over a surface

In fact, I havn't seen any other integrations over the surface except the one you are taling about in the entry. But it is probably possible to have some disastrous measures on the surface and thus having some strange integrals. Thus I would probably keep your old title "..." (you have already changed it so I can't type it :). The reason I find your new title:

"surface integration with respect to area"

suspicuous, is that if you google it, then you'll see nothing (probably after some time it will give your entry), which means that such term is extremely rare in use. By the way the same was with your old title, and that's why I proposed "surface integration" as more frequently used term. Thus, I would choose some name used in most books on calculus.

Anyway, as you said this entry is in flux, thus we'll keep on talking about improvements.


Interesting question indeed. I think "over" is preferable -- i guess i think of integration as a process that sweeps across the surface, so to say. Both can be defended, of course.

Come to think of it, "on" is wider in its applicability. If you draw a curve on the sphere, and do a line integral along the curve, you're still integrating "on the sphere" (as opposed to a line integral on a plane say). Integrating "over" the sphere only feels right for \int something d^2 A, but it too is still an instance of doing your integration "on" the sphere.

On a related note: i'm never sure whether to say a graph is embedded "on" or "in" a sphere. OT1H the "em-" in "embedded" really seems to want "in". OTOH i find myself using "on" to emphasise it's a surface (maybe only to avoid some readers getting the wrong idea and thinking i'm talking about the interior of a solid ball).

Oh, before i forget: in the (perhaps aptly named ;) section "A common mistake": some typos


--regards, marijke

Thanks for putting a note on the old theme! ;)

> s/exmaple/example/
> s/Sweeden/Sweden/

For the above typos it is better to submit a correction to the entry ;)


knowledge can become a science
only with a help of mathematics

Don't bother with the corrction in this case, because I read the message and made the changes. However, as a general policy, a corection is an appropriate place to point out mispellings.

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