Talk:Leibniz integral rule

General form
The "General form" needs an independent proof as the style of proof in the two previous cases differ.


 * There is now a proof of the general form, but I noticed it is not very rigorous. I just corrected an error in derivation so that it is now completely correct.  Some issues that possibly merit further discussion: the limits of integration (a and b) should be written explicitly to depend on $$ \Delta\alpha $$; the invocation of the MVT on the integral needs further explication (it assumes some continuity results, etc); the statement involving the terms inside the integral (in the limit as $$ \Delta\alpha \rightarrow 0$$) involves the interchanging of two limiting processes (differentiation and integration) and hence is only valid in certain cases. Clayt85 (talk) 22:11, 2 September 2009 (UTC)

The proof in the section "Basic Form" is not a valid proof. The argument is correct up until the step when it is shown that d/dx of the integral is equal to the limit of the integral of difference quotients. But the proof fails to address why we the limit and integration operations commute (this is in fact the only nontrivial part of the proof). The proof mentions something called uniform continuity vaguely, but in fact, uniform continuity is not even relevant. What is needed to show that the limit and integral signs commute is uniform integrability of the family of divided difference quotients. Then the result would follow from Vitali's convergence theorem.

But in fact, there is a simpler approach. The proof can be done fairly simply using an interchange of integrals and Fubini's theorem, as described at http://math.bu.edu/people/rharron/teaching/MAT203/LeibnizRule.pdf. In any case, it is clear that a lot more goes into the proof than what is stated on the Wikipedia page, and the Wikipedia page falsely makes the proof seem easier than it is. particle25 —Preceding unsigned comment added by 68.48.202.216 (talk) 04:00, 7 November 2010 (UTC)

I corrected the proof of "Basic Form" but my proof, while adding in some detail missing from the original, can probably be cleaned up a little bit. particle25 —Preceding unsigned comment added by 68.48.202.216 (talk) 04:50, 7 November 2010 (UTC)

So I actually removed most of my edits, as I think the proof I supplied actually does not work. Instead, I just added a reference to a correct proof and the observation that while the argument displayed is not fully rigorous. —Preceding unsigned comment added by 68.48.202.216 (talk) 05:14, 7 November 2010 (UTC)

Finally, I posted a correct proof for the "Basic Form." Quite simply, pointwise convergence and uniform boundedness on a set of finite measure is sufficient to pass the limit under the integral, as stated in the Bounded Convergence Theorem of Royden on page 78. 68.48.202.216 (talk) 15:46, 7 November 2010 (UTC)particle25

The result for the 3d time dependent case is incorrect if the velocity field isn't constant. That can easily be seen e.g. by considering the integral of a constant vector field normal to a plane circular loop whose radius is increasing linearly with time. The text hints at that: "the proof does not consider the surface distorting as it moves", but seems to imply that the result remains valid even though the proof isn't. The correct formula is even simpler than the quoted one: the second and fourth terms on the RHS of the penultimate equation are dropped. I have not edited the page but minimally the text should be clarified. Joincto (talk) 17:36, 21 June 2011 (UTC)

Merge with Differentiation under the integral sign?
There appears to be a lot of duplication between these articles.Dfeuer (talk) 18:19, 7 October 2012 (UTC)

The "Proof of basic form" is still not correct
I would like to thank everyone who contributed to this page. I have found it very helpful. The "Proof of basic form" is, I believe, not complete correct. However, it can be fixed easily. If we knew apriori that the continuous limit as $$ h \to 0 $$  of the integral existed, the proof shown would work. But showing that the continuous limit of the integral exists is part of what must be proved. The author proves that the limit exists and has the right value for ONE sequence $$ \{ 1/n \} $$. But in order to show that the continuous limit as $$ h \to 0 $$  exists and has the correct value, it must be shown that the sequential limit exists and has the correct value for ALL sequences  $$ \left\{ {a_n } \right\} $$  such that  $$ a_n \to 0 $$. Fortunately, the exact same method of proof will work for ANY sequence $$ \left\{ {a_n } \right\} $$ for which $$ a_n \to 0 $$. Once you prove that the limit exist and has the correct value for all sequences, you can complete the proof by invoking the theorem that says: For any real valued function $$ f $$  on a closed rectangle in  $$ \mathbb{R}^n  $$  (or, more generally, on any Separable Metric Space): $$ \lim_{x \rightarrow a } f(x) $$  exists and equals L      if and only if  $$  \lim_{n \rightarrow \infty } f(a_n) $$  exist and equals L for EVERY sequence  $$ \left\{ {a_n } \right\} $$  such that  $$ a_n \to x $$  as  $$ n \to \infty $$. I hope you find this comment helpful. By the way, I looked at the Fubini proof and, while it's very short and elegant, it's not very intuitive. I prefer this proof because it is more "nuts and bolts". Also, I think using Fubini is really overkill for this simple version of the theorem. Catgod119 (talk) 00:45, 19 January 2013 (UTC)

Notation
The current description makes heavy use of subscript notation for derivatives. Subscript/superscript notation is highly ambiguous, having different meanings and conventions in different fields. Chemistry uses subscripts to represent number of occurances of an atom in a molecule, linear algebra uses subscripts to index elements of vectors and matrices, and here we are introducing a 3rd notation for partial derivatives. Readers of this article will often be highschool students and freshman. Some of them will be like me and find this notation makes the formulas too hard to understand. I recommend standardizing on the Liebnitz notation. — Preceding unsigned comment added by 68.232.116.79 (talk) 19:26, 6 April 2013 (UTC)

The problem is more general to mathematics on wikipedia, as there are no notational standards between articles. It is not much of a problem, but then it requires precise defintions beforehand. I am making an effort to point out whether it is a physics, IEEE, or pure mathematics/real analysis notation when I see ambiguity in articles Limit-theorem (talk) 08:30, 10 June 2014 (UTC)

There is a parsing error
Proof of basic form Let: Failed to parse (lexing error): So that, using difference quotients ——— Substitute equation (1) into equation (2), combine the integrals (since the difference of two integrals equals the integral of the difference) and use the fact that 1/h is a constant: :$$\begin{align} u'(x) &= \lim_{h \rightarrow 0} \frac{\int_{y_0}^{y_1}f(x + h, y)\,\mathrm{d}y - \int_{y_0}^{y_1}f(x, y)\,\mathrm{d}y}{h} \\ &= \lim_{h \rightarrow 0} \frac{\int_{y_0}^{y_1}\left( f(x + h, y) - f(x,y) \right)\,\mathrm{d}y}{h} \\ &= \lim_{h \rightarrow 0} \int_{y_0}^{y_1} \frac{f(x + h, y) - f(x, y)}{h} \,\mathrm{d}y \end{align} $$ — Preceding unsigned comment added by 181.177.248.131 (talk) 16:35, 27 April 2015 (UTC)

And then there is some error, a subsequent edition has just erased the first equation after Let. Please correct. — Preceding unsigned comment added by 181.177.248.131 (talk) 16:34, 27 April 2015 (UTC)

"Alternative Proof of General Form with Variable Limits, using the Chain Rule" is incorrect
In the last paragraph, the justification for the continuity of the function

$$ \dfrac{\partial F}{\partial x}(x, y) = \int_{t_1}^{y} \dfrac{\partial f}{\partial x}(x,t) dt $$

seems insufficient. It is certainly continuous in variable y, but it is a function of two variables, x and y.

121.176.253.42 (talk) 23:42, 23 July 2019 (UTC)

Splitting proposal
Shall we split this article into "Leibniz integral rule" and "Integration by differentiating under the integral sign"? Nerd271 (talk) 16:13, 5 April 2020 (UTC)

What is difference between feynman trick integral and Leibniz rule??
I think they both are same. One eye Triangle (talk) 10:58, 3 May 2020 (UTC)


 * Feynman's technique for evaluating integrals uses the Leibniz rule for differentiating under the integral sign. It is a technique that uses the theorem, but is not the theorem itself. As an analogy, think of solving a first-order linear ordinary differential equation using an integration factor. (Hope you have gotten this far in calculus.) The integration factor makes use of the Product Rule, but is not the Product Rule itself. Nerd271 (talk) 15:11, 3 May 2020 (UTC)

Whither Example 3?
In the current version of the page, there are enumerated Examples 1, 2, 4, 5, and 6. Should we renumber these to include an Example 3, restore a (possibly) previously-deleted Example 3, or add a new example so the numeration makes sense? 75.181.102.130 (talk) 15:22, 12 June 2021 (UTC)

Whither Figure 1?
In the section "Three–Dimensions Time–Dependent Form" reference is made to a Figure 1. There is no such figure. It used to exist as I distinctly remember reading this page a few years back.....On investigation it was removed on May31st 2021 by, I think, an unknown user. Can it be restored? Spmundi (talk) 01:07, 22 December 2021 (UTC)


 * Yes, it can be restored, and I think it should all be restored. As you have explained, a lot of text was removed without explanation. That action should have been reverted or at least challenged as soon as it happened. It was followed closely by a Bot making some routine edit so it wouldn’t have shown up on the Watchlists of interested Users.
 * It can be easily restored by anyone with a mouse attached to their computer. For the coming week I will have little alternative to an iPad without a mouse but I will be able to do the restore after that. Unless someone else does it first.
 * Thanks for finding this large-scale deletion and drawing it to our attention. Dolphin ( t ) 09:57, 22 December 2021 (UTC)


 * I have now restored all the text removed in 4 edits on 31st May 2021. See the diff. Dolphin ( t ) 09:39, 3 January 2022 (UTC)

Continuity of the derivative required?
According to https://planetmath.org/differentiationundertheintegralsign, only (almost everywhere) existence of the derivative is required, not continuity. 2601:547:500:E930:8D0A:C216:B0C4:2C6B (talk) — Preceding undated comment added 17:14, 16 April 2022 (UTC)

It appears that the continuity condition was added by. 2601:547:500:E930:8D0A:C216:B0C4:2C6B (talk) 17:54, 16 April 2022 (UTC)

Requirements on Limits of integration
In the opening lines of the wikipedia page, it is stated that the limits of the integration need to satisfy the requirements $$-\infty < a(x), b(x) < \infty$$.

However, in the general statement of the theorem further down the page, these requirements are dropped.

Are these requirements missing? Or were they never required to begin with? If missing, then they need to be added to the general theorem. If they are not required, they shouldn't be stated as required in the first place. Whole Oats (talk) 01:44, 18 April 2023 (UTC)

Structural solution for presentation problem requested: parts of equation invisible in Safari browser on iPad
The information block to the right fully blocks (is that why it is called a block ;-)) the last term in the equation:

“$$\begin{align} & \frac{d}{dx} \left (\int_{a(x)}^{b(x)} f(x,t)\,dt \right ) \\ &= f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt \end{align}$$

where the partial derivative $$\tfrac{\partial}{\partial x}$$ indicates that inside the integral, only the variation of $$f(x, t)$$ with $$x$$ is considered in taking the derivative.“

I only discovered the blocking because the remark about the partial derivative made no sense to me, since no partial derivative was visible to me.

That must be something similar to what another user tried to mend. That user wrote:

”22:55, 21 February 2023‎ Dolphin51 talk contribs‎ 53,046 bytes+38‎   →‎top: In my browser the Info box obscured part of this long equation. I have solved the problem by presenting the equation over two lines. If someone knows of a better way to do it - feel free to go ahead.”

While for Dolphin51 the problem may have been solved, for me it is still there.

I am afraid Wikipedia could do with a more structural solution that prevents visual parts of a page from blocking other parts from view. Or is this a browser problem?

Who would know how to get this problem solved? I have a screen photograph ready that illustrates the problem, but cannot see a way to upload it here.Redav (talk) 00:13, 28 January 2024 (UTC)


 * Safari does not show scroll bars unless "active" by default. Set in your system preferences in appearance to always show them.  This is not a wiki problem, but a safari problem.  There are scroll bars there, safari just does not show them since they are not pretty. AManWithNoPlan (talk) 00:38, 28 January 2024 (UTC)
 * ooops. You are rigtht, that setting is only on desktop.  You can scroll it, but there is no visial clue. AManWithNoPlan (talk) 12:06, 28 January 2024 (UTC)
 * https://en.wikipedia.org/w/index.php?title=Wikipedia%3AVillage_pump_%28technical%29&diff=1199981416&oldid=1199966707 AManWithNoPlan (talk) 12:11, 28 January 2024 (UTC)
 * There does not seem to be any solution https://stackoverflow.com/questions/73496949/how-to-make-scrollbar-always-visible AManWithNoPlan (talk) 15:19, 31 January 2024 (UTC)

Errors
I’m not sure, but apparently some of the calculations have problems. I could just be dumb however  48JCL Talk  18:43, 30 April 2024 (UTC)