Talk:Partial derivative

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[edit] Thanks!!!

Thanks to all the writers of this page! Saved my behind (-:

Thanks again, Sam Krupa

[edit] Partial derivative of Area of a circle

Since there is only one variable in the formula for the area of a circle, as the article mentions, isn't the example using it a bad one? (Unsigned comment by 68.78.139.53 on 17 May 2006)

Although it's been many, many years since I've done any maths, I would agree - this example does seem pointless. --A bit iffy 09:54, 11 June 2006 (UTC)
I agree. I've removed it. --Spoon! 02:22, 11 September 2006 (UTC)

[edit] Disagree with recent changes

The third and the fourth paragraph, that is the discussion about the total derivative and the Jacobian are way out of place. Those things are inserted in the middle of the discussion about the partial derivatives, and that is inappropriate.

All that stuff needs to go in the last section, where the gradient is discussed. Other opinions? Oleg Alexandrov 02:27, 23 Mar 2005 (UTC)

[edit] Comparison with "d" notation

I'd like to see a section addressing the semantic difference between ∂ and d. For example, if y is only a function of x, then what is the difference between ∂y/∂x and dy/dx. Also, in sloppy engineering notation I've seen dy appear alone; I think I've seen ∂y alone as well. This may be rigerous use of Infinitesimals or it may be engineering shorthand; I'm not sure. —BenFrantzDale 02:00, 5 December 2005 (UTC)

I would agree with such a section, but preferably at the bottom of the article; otherwise I would think that it may confuse more than illuminate. Some connections with the Leibniz notation may be made. Oleg Alexandrov (talk) 02:59, 5 December 2005 (UTC)
Now I see, the page I needed to look at was total derivative. —BenFrantzDale 17:40, 15 December 2005 (UTC)

[edit] Vote for new external link

Here is my site with partial derivative example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Seperation_of_variables

[edit] Ordering of the sections

I am a bit perplexed as to why the formal definition is at the end and the example is at the beginning, I think some reordering is in order here! Retardo 22:13, 5 May 2006 (UTC)

[edit] Surfaces?

In computational mechanics, it is common to denote the surface of a volume as S=∂V. My sense is that this has some underlying meaning in differential geometry. Is it just a notational convenience or is there some rigerous extension of partial differentiation onto volumes? Also, how does this relate to the dV you find in a volume integral? (There dV basically means "one tiny point in V" whereas ∂V basically means "one thin sliver of the surface"; does this relate to the difference between ∂ and d?) —Ben FrantzDale 23:29, 18 May 2006 (UTC)

In topology, ∂S denotes the boundary of a surface S. As far as I know, this is unrelated to the symbol's use to denote a partial differential (I've only just started seeing this formally, so I may be wrong). I also read this on an article here on Wikipedia which was about the different uses of the symbol; an article that I can't seem to locate now. If anyone can find it, I'd like to know. Anyway, I think this makes sense since the boundary of a volume is a surface, as the boundary of an area is a path (at least intuitively). Hope this helps. Commander Nemet 23:25, 31 May 2006 (UTC)

[edit] Integration equivalent of partial derivative

Since F'(P)=\frac{dF(P)}{dP}\,\! and (with respect to P) F'(P,Q)=\frac{\partial F(P,Q)}{\partial P}\,\!, doesn't it follow that F(P_d,Q)-F(P_b,Q)=\int_{P_b}^{P_d}F(P,Q)\partial P\,\!?  ~Kaimbridge~18:44, 20 May 2006 (UTC)

Yes, that is true, assuming that by the integrand you really meant FP(P,Q) instead of F itself. Also, I believe that the differential here would be just dP, not \partial P, though I can't give you a good reason why. What you would get after this integration is a function of Q, which, if you were doing a double integral, would be integrated in the next step. I always thought of the double (or triple, etc.) integral as the "equivalent" of the partial derivative. Commander Nemet 03:09, 1 June 2006 (UTC)
Actually, I meant F'(P,Q)\,\!. In terms of full notation, should it be
F(P_d,Q)-F(P_b,Q)=\int_{P_b}^{P_d}F'(P,Q)\partial P=\int_{P_b}^{P_d}G_{p}(P,Q)\partial P\,\!, or should F'(P,Q)\,\! be F'_{p}(P,Q)\,\!, with a subscript, too? As someone asked above in "Comparison with "d" notation", if you have "dP", you know it's regarding P, not Q, so what does \partial P\,\! add? You could just as easily say
E'(P)=F'(P,Q)=\sin(P)\cos(Q)\,\!, in which case
E'(P)dP=F'(P,Q)\partial P\,\!, right? In terms of it being a "double integral", a double integral integrates one variable, then the other. In the example I give above, only "'P'" is integrated.  ~Kaimbridge~19:28, 4 June 2006 (UTC)
In regards to your first question, the notation that I learned for partial derivatives includes, in this case, \begin{matrix} \frac{\partial F(P,Q)}{\partial P} \end{matrix}, FP(P,Q), and \partial_PF(P,Q). F'(P,Q) is not used because it does not indicate which variable the derivative is taken with respect to.
In regards to your second question, I'm not really sure what you're asking, but I can tell you that, as far as I know, the partial differential symbol is used only in derivatives, so in the integral, for example, it would be just dP. I think the reason for this is that a differential is only "partial" in the context of a derivative. To me, at least, having a differential be "partial" otherwise would be meaningless. Commander Nemet 22:49, 5 June 2006 (UTC)

[edit] Greek or Cyrillic?

It seems that 68.103.26.177 has changed the introduction to say the symbol is from the Greek instead of Cyrillic alphabet. I'm trying not to be hasty and call it vandalism, but unless there is some deeper meaning I am unaware of, the partial differential symbol looks a heck of a lot more like the cursive de (Cyrillic) than a Greek delta. Furthermore, the intro now purports to say that the Greek letter in question is actually a "d"—something that doesn't exist in that alphabet. Commander Nemet 23:25, 31 May 2006 (UTC)

  • I agree with you that this is just wrong, if not vandalism. I therefore reverted it to the correct version. PanchoS 00:29, 29 June 2006 (UTC)
  • I think the firs person to use that notation was leibniz
According to Cajori's History of Methematical Notations v. 2, p. 220 Leibniz used a 'δ' for the partial derivative. Thomasmeeks 17:31, 11 December 2006 (UTC)

[edit] Can we get a computerized notation set up?

Many other mathematics-related articles have an implementation of the topic in various programming languages. Will anybody set up this page with such an example? -- kanzure 16:41, 19 July 2006 (UTC)

[edit] Is modern origin of ∂ Cyrillic?

If so, please cite here. I'm doubtful based on # of hits for Google of:

  • "partial derivative" Greek
  • "partial derivative" Cyrillic

Earlier notation used Greek small delta δ, which looks like ∂ than the Cyrillic Б. If no one can give an authoritative cite for Cyrillic, my vote would be for deleting sentence so suggesting it. Greek origin looks more plausible. Thomasmeeks 02:25, 28 November 2006 (UTC)

The answer to the question in this section title seems to be in the following classic article:
Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46
accessible through JSTOR from many colleges and universities. Thomasmeeks 01:08, 29 November 2006 (UTC)
Should the reference to the Cyrillic letter in the lead be deleted? For now, I believe so. Florian Cajori's A History of Mathematical Notations (1928-29), still widely regarded as definitive for the period through its writing, makes no reference to the cursive Cyrillic letter de in the section on partial derivatives (v. 2 pp. 220-42). He refers only to the "rounded letter" after referencing earlier partial notations, including the 'ό' and 'd'. No other scholarly sources in the math reference section or the relevant sections of book collections on math history at a good university library make the Cyrillic connection. Nor does a search of Google Scholar give any primary source for: Cyrillic "partial derivative." So, I don't believe that the Cyrillic reference meets the Wikipedia: verifiability requirement. Thomasmeeks 18:50, 11 December 2006 (UTC) ('rounded' for 'curved' edit Thomasmeeks 20:16, 11 December 2006 (UTC))Thomasmeeks 20:18, 12 December 2006 (UTC)

[edit] Number of Google 'hits' inadmissable

Is there a WP policy that discourages arguments based on the number of Google 'hits'? If not, can someone please propose one? Here for example: From searching Google today:

  • "partial derivative" American ~54000
  • "partial derivative" English ~36400
  • "partial derivative" French ~24000
  • "partial derivative" Greek ~12300
  • "partial derivative" Cyrillic ~519

On this basis we'd conclude that: Americans invented partial derivatives, and they probably spoke (a form of) English; their invention was aided by the French and quickly adopted by the Greek; and Russians generally know nothing about partial derivatives.
But let's also search >"partial derivative" Amrica<. It returns ~35800 'hits'. At least, Google claims that number of hits. It can't be verified. Disagree? Okay, then tell me what hit number 10000 is.
And what about other search engines???
—DIV (128.250.204.118 01:34, 15 November 2007 (UTC))

[edit] Deletion of 2nd sentence of 2nd paragraph

Deletion from the previous Edit of the article of:

'∂' also corresponds to the small Greek delta 'ό' which was also used before the 20th century as partial derivative notation.

The problem is that that sentence follows a related parenthetical comment. In retrospect (I introduced the second sentence), if the sentence is going to be included at all, there is a case that both should be parenthetical (to avoid breaking the substanive exposition) or neither (possibly as a footnote or short separate section on the origin of the notation). Thomasmeeks 14:56, 29 November 2006 (UTC)

[edit] a step backwards

"a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary"

the total derivative is, in fact, the sum of the partial derivatives. i seem to remember explaining this before. but no matter. keep removing other people's contributions if you dont understand them —Preceding unsigned comment added by 212.159.75.167 (talk) 20:26, 13 October 2007 (UTC)

[edit] numericalization

We should also discuss the numericalization in this article. At least illustrate how to numericalize the 2nd derivative. anyone agree? Jackzhp (talk) 15:22, 3 May 2008 (UTC)