Talk:Constant of integration

joke

It's a subject of one of the great Math jokes... See http://www.netfunny.com/rhf/jokes/old90/constant.html AnonMoos 14:29, 26 September 2006 (UTC)

untitled

Isn't a bit of overkill to use FTC to prove F' identically zero implies F identically constant? It follows immediately from the MVT... 207.62.177.227 (talk) 23:55, 18 November 2009 (UTC)

It is, and it does. The downside to that approach, I think, is that the mean value theorem isn't usually given much emphasis in calculus courses, so if we used it here, then I think we'd be more likely to confuse the reader. (I could be wrong, though.) It's too bad, really, because the Fundamental Theorem of Calculus is really just a souped-up mean value theorem. Ozob (talk) 00:25, 19 November 2009 (UTC)

Upon further reflection, I think the problem may be considerably worse than I originally imagined. Let's say there are the 2 basic versions of FTC we could be using: (1) the one that assumes continuity of f, or (2) the one that does not. In the former case, we're using the corollary of the fact that the accumulation function is an antiderivative, and the proof of this corollary requires using the fact that two functions with the same derivative differ by a constant. In short, the former case is a circular proof. In the latter case, if you work through the proof (say, baby Rudin proof for Reimann-integrability) for the case when f = 0, you'll see that at the pivotal point of the proof, it reduces to nothing more than the MVT (take the trivial partition). In short, the latter case is a rip-off of MVT.

So, choose your poison: either the proof given is circular, or a rip-off. In any case, the conclusion has absolutely nothing to do with integration. If that's true that MVT isn't taught in calc now (maybe this is true just in high school), it's a shame. 76.20.109.233 (talk) 10:22, 20 November 2009 (UTC)

"... where C is an arbitrary constant"

is the choice of C really arbitrary?

theoretically speaking, let us say we have a three dimensional euclidean space where each point in this space is independent of the other. if we were to integrate or differentiate with respect to time, would one not want a unique constant C that essentially facilitate calculation using a two-dimensional approach (a la Lebesgue integration)?

let us say that we have some discretisation LxWxH (resulting in LWH independent points)

if we had LWH primes, would these primes not also act as a sort of "unique key" (C) that would make it easier to integrate?

i admit this but one obscure scenario, but it seems we would want the constant (ideally, systematically) selected such that it satisfies some rigorous conditions, no? probably gonna get mad crickets so w/e. thought i'd throw it out there. — Preceding unsigned comment added by 174.3.155.181 (talk) 03:22, 6 July 2016 (UTC)

It's not a question of convenience or of being systematic. Constants are forced on us by the nature of antidifferentiation, but the only thing forced on us is that they're there. Everything else about them is indeterminate. I suggest looking again at the paragraph in the article that interprets d/dx as a linear operator on a vector space of functions; to me, that is the clearest statement of why the constant has no structure. Ozob (talk) 13:13, 6 July 2016 (UTC)

hi @Ozob:. i agree with what you're saying, but your explanation wouldn't support the claim of arbitrary constants. using a dictionary definition, arbitrary is defined as subject to personal will or judgement without restriction, which clearly conflicts with how the constant is chosen.

i agree that there is no structure in how the constant is chosen/assigned, but it is not arbitrary. i took this to talk because i want to remove that specific word. i do not disagree with what's given in the content sections, just the word used in the lede because it is misleading (there is no "whim" involved for choosing the constants). 174.3.155.181 (talk) 17:32, 6 July 2016 (UTC)

You're making some distinction that I don't understand. What is the distinction between a constant having no structure and a constant being arbitrary? To me these are synonyms. Ozob (talk) 23:40, 6 July 2016 (UTC)

hey @Ozob:, your unfamiliarity with how a subtle term like arbitrary can greatly affect the interpretation of a mathematical statement is totally understandable. it's a slight distinction; arbitrary here entails [[subjective|subjectivity], whereas your description objective.

• that is, choosing constant such that it best satisfies whatever (assumed rigourous) constraints were imposed prior to its evaluation is objective, and not arbitrary.
• choosing constant C arbitrarily implies we can set it to "whatever we want", which is obviously not true but is an admissible "corner case" (remember, arbitrary entails subjectivity, and subjectivity is not the modus operandi of mathematics)

is that clearer? hopefully we can remove that word? that's all i want to do. i hope i gave you a clearer understanding about the subtle difference. 174.3.155.181 (talk) 01:29, 7 July 2016 (UTC)

No, we can indeed set C to "whatever we want". Usually we write a variable C because we intend to express all antiderivatives of the given function, but we may, at our option, set C. We may even set it arbitrarily (in your sense, meaning subjectively). If we do so, then we have a single antiderivative instead of the collection of all antiderivatives. However, if we are only interested in a single antiderivative (our favorite, perhaps), then this is mathematically admissible. Ozob (talk) 03:13, 7 July 2016 (UTC)

@Ozob: sorry, i am not following entirely (edit: after dumping the below thoughts, i do follow what you're saying but make a slight distinction which i hope helps you understand where i'm coming from). let us use the figure from antiderivative (seems to be taken from slope field), so both of us are on the same page:

The slope field of dy/dx=x²-x-2, with the blue, red, and turquoise lines being (x³/3)-(x²/2)-2x+4, (x³/3)-(x²/2)-2x, and (x³/3)-(x²/2)-2x-4, respectively.

• ok, so C is a constant that can be interpreted (for the case in the figure, at least) as a translation of the function. regardless of what constant is chosen, you can see that the function (differential form?) itself is unchanged (as it should be). here, i accept it is 'arbitrary'.
• what bugs me about this figure is that we are implicitly assuming we are given some sort differential form(?) that is primed for integration. in the real world, it is simply unrealistic to assume we have a differential form(?) because most sensors collect noisy measurements.

• in the cases where we cannot assume we are given a differential form, we have to determine it from a noisy function f that presumably "contains" it.

• referring to the aforementioned LxWxH setting, even if you were to determine the differential form(?) from a single sensor (that is placed at any point given by a three tuple (x,y,z)), it will have a C that is a function of its position in space.

• i do not dispute we could "add" some number to this (already existing, but not known) C and recover the correct differential form(?), assuming the analyses on the function is appropriate (say, Lagrange multipliers).
• what i am saying is that if we assume there exists a sufficiently-sophisticated sensor capable of collecting measurements that capture higher-order properties at point(s) in three-dimensional space, then these measurements should already capture a natural C corresponding to the constant of integration for that respective point in space.

• that is, i agree the constants are arbitrary, but only after we do the "hard part" of recovering the local differentiable properties (differential form(?)) of the respective function.
• i see your point of why you call it arbitrary, and i guess i'm arguing that it shouldn't be called arbitrary because natural measurements (which we assume contain a noisy function from which we want to recover the differentiable properties, given by the slope field or whatever) should implicitly have pre-existing constants of integration. even if you disagree, i hope you can see my line of thinking. thank you for this conversation, it has also helped me refine the argument i was making (been a while since that happened). can't thank you enough. gawddddd philosophy of maths is the best man xD 174.3.155.181 (talk) 05:05, 7 July 2016 (UTC)

References

I'm not sure you're using differential form correctly here. Properly speaking, antidifferentiation is the process of finding a preimage of a differential form under the exterior derivative: The derivative starts with a function and yields a differential form, and antiderivatives are just the inverse. The constant corresponds to the kernel of the exterior derivative. These are the locally constant functions, and on a connected domain such as R these are just constants.

The given example shows solutions to the differential equation $dy/dx=x^{2}-x-2$ with several initial values. One may convert the differential equation to an equality of differential forms, specifically, to $dy=(x^{2}-x-2)\,dx$. Solving the differential equation amounts to finding a function $y=f(x)$ such that $dy=(x^{2}-x-2)dx$. Solving a given initial value problem means finding a function y which has the correct differential and takes a prescribed value at a certain point.

As a corollary to this, there is one constant for the solution of the whole differential equation (assuming a connected domain). The constant cannot vary from point to point.

In a multi-dimensional situation such as your box example, when we apply Fubini's theorem to compute a volume integral as an iterated integral, we may select one constant for each point. However in that situation, we are computing a definite integral, so the constant will ultimately cancel. If we want to compute the preimage of a differential form under the exterior derivative, then we are computing the preimage of a 3-form, which is a 2-form, and there is one constant per basis vector; in R³ that means there are three constants, one each for $dx\wedge dy,dx\wedge dz,dy\wedge dz$. (Roughly speaking, you can say that there is a constant in each of the z, y, and x directions.)

The question of measurement is not mathematical. In addition, most differential equations that arise in practice come from physical models with a small number of parameters. These parameters are often universal constants. When they are not, they are frequently space-independent (like the viscosity of a homogeneous fluid). There are exceptions to this rule, but in most physical situations you can think of the differential equation as known. Whether that's accurate is an entirely different question: These are models, not reality, but in most applications there's no detectable difference between the model and reality (e.g., if I'm modeling ocean waves I can't detect quantum effects). Ozob (talk) 13:01, 7 July 2016 (UTC)

when speaking of solutions to the global differential equation (assuming every local domain is connected), you are right that we can view it as a single constant because we already have the local differentials.

it really seems that we could benefit a model-free methodology that would employ the fundamental theorem of calculus using a collection of concepts like, say, the Hasse principle and Hasse-Minkowski theorem (& others, as use of modular arithmetic mandates we revisit work of The Master, Gauss, such as Theorema egregium) for of course, this is way off topic and (even more, sorry!) original research, so i think i'll hold off on talking about this more.

i am not saying improving upon the current approach to the Fundamental theorem of calculus is easy; rather, i just think we all stand to benefit if we could develop a methodology that embodies things like the global-local principle, as these concepts sort of entail Riemannian geometry (calculate differentials locally, integrate globally). a

• again, not easy, but such a methodology may help us in situations such as the one you've described: when modelling ocean waves, we can't "detect" quantum effects. however, if we had a methodology faithful to the above concepts, and assume the sensor is capable of detecting effects at the nanometre scale, then the macroscopic model (ocean waves) should contain these quantum effects at a finer scale; of course, studying/extracting such quantum effects would require new analyses on the data output from the aforementioned "unicorn" methodology that deploys the fundamental theorem in riemannian geometry.

again, way off topic and i apologise, but it's hard not to get excited when discussing this stuff.

in closing, thank you so much for helping me understand my line of thinking better, @Ozob:. i guess my big mistake was assuming even the most rudimentary philosophical questions re: measurement were relevant to mathematics. upon reading your extremely informative (i can't thank you enough), i cannot disagree with anything because your wise explanation suggests you have much more experience than i do. really, thank you so much.

It's no problem! I don't mind explaining things to people who are interested.

Here are some links that you might find interesting. Microlocal analysis is a way of approaching the study of differential equations that explicitly includes information about phase space (the space where differentials live). The adeles are a number-theoretic approach to including local information, where "local" here means relative to all places of a number field or function field (so that in particular it includes not only the real and complex numbers but also p-adic numbers). A multiresolution analysis is a technique based on wavelets that makes it possible to separate the behavior of a function into big, coarse features and small, fine ones. Finally, since the question of measurement has come up, I love to point out that the uncertainty principle isn't the same as the observer effect; the fact that we can't simultaneously measure position and momentum to extremely high degrees of precision is a mathematical fact, not a physical one. The exact same principle says that an electromagnetic signal (like radio, Wi-Fi, cell phones, etc.) doesn't have a well-defined frequency on very short time scales.

Anyway, I hope you find some of this fun and interesting. Take care, Ozob (talk) 00:19, 8 July 2016 (UTC)

I've been summoned to this RfC, but I don't think I've grasped what the debate is about. If I integrate a function $F:\mathbb {R} \rightarrow \mathbb {R}$, I'll write a $k$ in the result, to mean "any real number". If the function is $F:\mathbb {C} \rightarrow \mathbb {C}$, my $k$ will I suppose be "any complex number". I think this is what is meant when a mathematician uses the word "arbitrary". Maproom (talk) 11:19, 20 July 2016 (UTC)

Who summoned you, and why? I thought this discussion was over. Ozob (talk) 12:44, 20 July 2016 (UTC)

The feedback request service summoned me, by a message on my talk page dated 04:25, 20 July 2016. Maproom (talk) 19:56, 20 July 2016 (UTC)

But why? The discussion is long over. I've never heard of the feedback request service before. I'm confused. Maybe I should stop commenting. Ozob (talk) 00:57, 21 July 2016 (UTC)

I see no evidence that this discussion is over. The {{rfc|sci|rfcid=3A0BE72}} tag at the top implies that it is still open. But I have noticed that the feedback request service, which I assume is triggered by the placement of that tag, often summons me to discussions long after they have started. I shall try to take this up with someone responsible for its operation, if I can find them. Maproom (talk) 06:46, 21 July 2016 (UTC)

Well, it does not look the discussion is following the right pattern. Do I have to remind everyone this is not a forum? C is an arbitrary constant of integration, if in the relevant sources it is considered so. Our opinions, for qualified they can be is irrelevant. And for what I can see C is an arbitrary constant of integration. This is exactly the wording I could find on my books at home, and in the sources I could check on the web. Silvio1973 (talk) 10:16, 26 July 2016 (UTC)

Necessity - wrong calculation?

The last line looks very suspcious:

$\int 2\sin(x)\cos(x)\,dx=-{\frac {1}{2}}\cos(2x)+C=\sin ^{2}(x)+C=-\cos ^{2}(x)+C$

Let's take the inenr part: $-{\frac {1}{2}}\cos(2x)+C=\sin ^{2}(x)+C$

The half cosine of double frequency has the same "look" as squared sine, but they are for sure not equal. Just put in a 0 and calculate the values. $\sin(0)+C=C$ but $-{\frac {1}{2}}\cos(0)+C=C-0.5$. So the article claims that $0=0.5$. The euqal sign can just work out if you take to DIFFERENT constants. But this shows that C is arbitrary and can be chosen to be 0. --2003:DE:F17:C400:C85D:EC77:702A:EADE (talk) 15:38, 9 June 2022 (UTC)

Mathematical jibberish

What on earth is this statement supposed to mean?

"So setting C to zero can still leave a constant."

Can someone please precisely explain what it means mathematically to "leave a constant"?? The original explanation provided back in 2004 was much more intelligible. — Preceding unsigned comment added by 2601:205:4401:1070:A4BB:444D:98FC:FDED (talk) 01:07, 10 February 2024 (UTC)