Talk:Jacobian matrix and determinant

message: I have serious doubts as to the validity of this entire section. From "Calculus on Manifolds" by Spivak, a function is defined to be differentiable at a point if it has a linear approximation there, and further this linear approximation is unique.
Further, the example used does not give the correct definition for the directional derivative, which is lim_ h->0 |f(a+h)-f(a)|/h. And though the definition given will work, the answer derived is incorrect, because there is confusion about which coordinate system (x.y)/(r,theta) is being used here.
The function is defined in (r,theta) space, but limit is being taken in (x,y) space, tending towards (0,0) along the (1,1) line. This corresponds to traversing the function in (r,theta) space in a line tending towards (0,pi/4), not towards (0,0) as requested.
In (x,y) space, the function is defined as xy/sqrt(x^2+y^2), and so neither it nor its jacobian are defined at (x,y)=(0,0). The function is defined at (r,theta)=(0,0), and so is it's jacobian = (0,0). A taylor series will show this very easily.
For the above reasons, I am removing this section.   ObsessiveMathsFreak (talk) 14:36, 11 February 2008 (UTC)
thread_title:  Differentiable versus partial derivatives 

message: Isn't the Jacobain matrix just the tensor product of the grad operator? That is, is this correct?:
:J_{ij}(\mathbf{f}(\mathbf{x})) = \mathbf{J}(\mathbf{f}(\mathbf{x})) = \mathbf{f}(\mathbf{x})\otimes\nabla = \frac{\partial f_i(\mathbf{x})}{\partial x_j}
—Ben FrantzDale 16:57, 5 May 2006 (UTC)
thread_title:  Tensor Product? 

message: Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p;
Orientation as in Orientability or Orientation (mathematics)? --Abdull 12:56, 25 May 2006 (UTC)
replies: text: As in both of those; they're the same concept. —Keenan Pepper 18:45, 25 May 2006 (UTC)
thread_title:  Orientation 

message: I was looking for information about Vanishing Jacobians on Google, and I saw this article. This article talks nothing about Vanishing. Maybe somebody should add something about Vanishing Jacobians?
James 03:25, 25 June 2007 (UTC)
thread_title:  Vanishing Jacobians 

message: Doesn't the Hessian need to be checked to see whether it's a stationary point vs. an extremely unstable point? (i.e., a maximum). Ashi Starshade 18:17, 5 July 2007 (UTC)
replies: text: There are two closely connected meanings of "stationary point". The meaning intended in the article is that x is a stationary point for the dynamical system x' = f(x) if f(x) = 0. Another meaning is that x is a stationary point for a function f if the derivative of f is zero; this is the meaning that the article stationary point mentions. I don't quite understand your comment, but it seems that you're confusing these two meanings. -- Jitse Niesen (talk) 20:50, 6 July 2007 (UTC)
thread_title:  In dynamical systems, Stationary point

message: Excuse me, but in this section it says 
:for instance, in the (1,1) direction (45°) this equals \sin 45^\circ=\sqrt{2}/2.  
Should it not say something like 
:\sin (2(45^\circ))=1 
since
:f(r,\theta) = r \sin (2\theta) 
?
Dimbulb15 (talk) 23:31, 3 February 2008 (UTC)
thread_title:  Differentiable versus partial derivatives 

message: "The transformation from spherical coordinates to Cartesian coordinates is given by..."
Shouldn't this be from cartesian coordinates to spherical coordinates? Wingless (talk) 20:36, 16 June 2008 (UTC)
replies: text: Well, I changed it back, since it is clear from the formulas that F in the first example goes from spherical coordinates to Cartesian coordinates.  --Lambiam 19:18, 12 December 2010 (UTC)
replies: text: You can change it back, but I worked out this type of problem in several homework with solutions, so unless the Jacobian is bidirectional (I am quite possibly wrong about this) then the correction I made makes sense. Functions form rows, variables form columns, which is correct from the function given, but a change from spherical coordinates to Cartesian coordinates would require the functions to be written in terms of  (x, y, z). If you look at example three and pay attention to the nomenclature used to describe the problem, my edits follows. The Jacobian determinant of (x, y, z) to (r, θ, φ) is r2 sin θ. Actually you can confirm this from http://en.wikipedia.org/wiki/Multiple_integral under the spherical coordinates section. Thanks now I have wasted half an hour of studying for finals :-)   —Preceding unsigned comment added by 130.245.246.244 (talk) 22:47, 12 December 2010 (UTC)
replies: text: The text in our article Multiple integral is about domain transformations, which work contravariantly. For the univariate case, when you want to go from a function  to a function , for example because the new domain E is more convenient to work in than the old domain D, you can do that if you have a transformation T between the domains that goes in the opposite direction, from the new domain to the old domain: . Then you can define function g by g(x) = f (T(x)). For use in an integral, using x for the variable in the new domain, which corresponds to variable y in the old domain by the relation y = T(x), we get then f (y)dy = g(x) |JT| dx, where JT is the one-element matrix (dy/dx). This generalizes to the multivariate case.  --Lambiam 08:50, 13 December 2010 (UTC)
thread_title:  Example?! 

message: For a non-expert, the notation is confusing, so it's best to annotate variables and expressions with types.   So, the exposition would be SO much easier to follow if it contained ample phrases like "where p is a column vector of numbers' (or  a column vector of functions, etc).  As a computer programmer, I appreciate the help that a good compiler gives me with keeping track of the types of my expressions.   ThinkerFeeler (talk) 05:20, 4 October 2008 (UTC)
replies: text: All variables should be defined, which includes mentioning its type; that is just proper writing style in mathematics. However, it's possible to overdo it, so this is something that's better discussed with concrete example. Therefore, could you please be more specific, or perhaps edit the article yourself (people will protest if you go to far)? I found only one place where p was introduced and the type was only implicit, and I fixed that. -- Jitse Niesen (talk) 12:04, 4 October 2008 (UTC)
thread_title:  For a non-expert, the notation is confusing, so best to annotate variables (and expressions) with types. 

message: As an example for the post above (annotate variables for non experts), could the variables  x_1, x_2, and  x_3  please be annotated. As an engineer, I assumed these to be the x,y, and z directions in the cartesian system, however this is not clear.
MrLaister  —Preceding unsigned comment added by 129.11.76.230 (talk) 10:13, 23 October 2008 (UTC)
replies: text: I added a bit, saying that  x_1, x_2, x_3  are the Cartesian coordinates and r,\phi,\theta the spherical coordinates. Is that enough? -- Jitse Niesen (talk) 10:58, 23 October 2008 (UTC)
thread_title:  Example 1 - Spherical to Cartesian coordinates, definition of <math> x_1, x_2, x_3 </math> 

message: A more knowledgeable person should write something about the following application of Jacobians.
The jacobian is a key element in the power flow problem. The power flow problem is the problem of having all sorts of different parameters in a power system between generators( at the power station) and loads(homes, businesses, ...) etc. Computers at the power station need to solve huge systems of non linear equations several times per minute, and as it turns out the best way to do it is a guess and check method based on newton's method. Newton's method requires a derivitave. When solving the system of equations for a multivariate context in newtons method, you use the jacobian matrix.
If I had known that this was used for a problem like this in Calculus III, I would have paid more attention to the meaning and conceptual properties of the jacobian matrix.
75.162.43.124 (talk) 23:20, 26 March 2009 (UTC)
thread_title:  Applications 

message: This phrase is clearly vandalism:
"by some people, especially physicists who have trouble with complicated ideas such as the determinant."
I removed it from the section "Jacobian determinant."  —Preceding unsigned comment added by 68.163.47.166 (talk) 20:47, 16 June 2010 (UTC)
replies: text: LOL! just LOL. Gotta hand it to that Vandal. -- 85.210.133.129 (talk) 03:02, 29 April 2011 (UTC)
replies: text:  Nice try, Vandal 206.207.225.66 (talk) 11:13, 3 March 2012 (UTC)
thread_title:  Vandalism 

message: Let's use the example of the article:
The Jacobian determinant of the function F : R3 → R3 with components
:\begin{align}
  {\color{Blue}y_1} &{\color{Blue}= 5x_2} \\
  {\color{Red}y_2} &{\color{Red}= 4x_1^2 - 2 \sin (x_2x_3)} \\
  y_3 &= x_2 x_3
\end{align} \Longrightarrow 
\begin{vmatrix}
  {\color{Blue}0} & {\color{Blue}5} & {\color{Blue}0} \\
  {\color{Red}8 x_1} & {\color{Red}-2 x_3 \cos(x_2 x_3)} & {\color{Red}-2x_2\cos(x_2 x_3)} \\
  0 & x_3 & x_2
\end{vmatrix} = -8 x_1 \cdot \begin{vmatrix}
  5 & 0 \\
  x_3 & x_2
\end{vmatrix} = {\color{OliveGreen}-40} x_1 x_2.
What happens if I changed the order of the functions?
:\begin{align}
  {\color{Red}y_2} &{\color{Red}= 4x_1^2 - 2 \sin (x_2x_3)} \\
  {\color{Blue}y_1} &{\color{Blue}= 5x_2}\\
  y_3 &= x_2 x_3
\end{align} \Longrightarrow 
\begin{vmatrix}
  {\color{Red}8 x_1} & {\color{Red}-2 x_3 \cos(x_2 x_3)} & {\color{Red}-2x_2\cos(x_2 x_3)} \\
  {\color{Blue}0} & {\color{Blue}5} & {\color{Blue}0} \\
  0 & x_3 & x_2
\end{vmatrix} = +8 x_1 \cdot \begin{vmatrix}
  5 & 0 \\
  x_3 & x_2
\end{vmatrix} = {\color{YellowOrange}+40} x_1 x_2.
That is, if i change the order of the functions, the determinant changes its signal. This is consistent with determinant properties, I know, but my point is: which is the "right" order of the equations? Apparently the functions are NOT ordered. Luizabpr  —Preceding undated comment added 23:38, 10 March 2011 (UTC)
replies: text: If you change the order of the functions you change the orientation of the axes, so a 'right-handed' coordinate system becomes a 'left-handed' one. This becomes a sign change in the determinant. This is mentioned in the article, though not very clearly. When you are doing an integral you need to take the magnitude of the determinant. Paul Matthews (talk) 10:04, 22 March 2011 (UTC)
thread_title: What if i change the order of the equations?

message: I re-arranged the order of things, putting all the examples towards the end. 
Previously it didn't make much sense because the examples used ideas like change of variable in integrals that did not come up until later in the article. Paul Matthews (talk) 11:05, 22 March 2011 (UTC)
thread_title: Order of material

message: I see absolutely no point in including the matlab-code in this article. Why is it here? Lapasotka (talk) 08:40, 19 May 2011 (UTC)
thread_title:  Why matlab-code? 

message: I want to add a (long) list of Jacobians, especially of matrix functions. Should that be in this article, or should I make a separate article?
Kjetil Halvorsen 18:02, 28 August 2012 (UTC)
thread_title:  List of Jacobians 

message: I don't think that the matrix described there is related to this page.  — Preceding unsigned comment added by 79.183.148.124 (talk) 18:07, 11 November 2012 (UTC)
thread_title:  Jacobi Matrix external reference 

message: To the contributors: thanks for making this page so useful and comprehensible to non-specialists. It even contains pseudocode! A bit more on applications of the Jacobian would make it better, but a very good article - stellar by the standards of the average math article. Much better than 'Start' class. Doug (talk) 14:27, 2 August 2013 (UTC)
thread_title:  Good job 

message: The last-but-one sentence of chapter Jacobian Matrix ends on
"the former the first derivative of a scalar function of several variables, the latter the first derivative of a vector function of several variables."
I think you got "former" and "latter" the wrong way round.
Puddington (talk) 11:01, 18 August 2014 (UTC)
replies: text: No, the gradient relates to a scalar function and the Jacobian relates to a vector function. Mark Hurd (talk) 03:35, 19 August 2014 (UTC)
thread_title:  Typo? 

message: The IPA pronunciation guide is totally wrong. I've never heard anybody pronounce it /dʒɨˈkoʊbiən/ and especially not /jɨˈkoʊbiən/  — Preceding unsigned comment added by 71.227.186.180 (talk) 00:12, 8 February 2015 (UTC)
replies: text: The first vowel sound is still wrong. 92.19.54.161 (talk) 20:44, 27 November 2017 (UTC)
thread_title:  ipa pronunciation of "jacobian" 

message: Remember when these articles were supposed to be in English and the crap contain should at least be readable to someone without a terminal degree in WTFology?  — Preceding unsigned comment added by 24.206.154.127 (talk) 19:16, 20 September 2015 (UTC)
thread_title:  Remember when?  Pepperidge farms remembers. 

message: It would be useful to have examples of how the Jacobian is used when changing variables in integration.  In particular, it would be useful to clarify when one uses the Jacobian determinant versus when one uses the absolute value of that determinant.
A commenter in a previous section says that we always use the absolute value of the Jacobian determinant when changing variables in an integration.  Is that true?  Is it a theorem or is it tradition? 
For example,  \int_0^{2\pi} \sin(x) dx  has an unambiguous mathematical definition that makes its value zero and this definition is different than the traditional answer to the problem "Find the area between graph of  \sin(x)  and the x  axis between  x = 0  and  x = 2\pi " because we are expected to use an unsigned concept of area.
Is there a similar distinction between the mathematical definition of \int \int_{(x,y)\in S} f(x,y)\  dx\ dy and the answer we get when doing a change of variables that involves using the absolute value of the Jacobian determinant?
Tashiro~enwiki (talk) 15:45, 27 February 2017 (UTC)
replies: text: Roughly, it depends on the interpretation of  dx\ dy : is it Lebesgue measure or a differential form? If it is Lebesgue measure, then the absolute value appears. Note that taking n=1 in the n-dimensional case you do not get \int_a^b but rather \int_S where S is the interval whose endpoints are a and b; think what happens when a>b. Boris Tsirelson (talk) 16:22, 27 February 2017 (UTC)

text: This issue is pretty technical and depends on how the integration limits are defined.  In a one dimensional integration the limits are ordered and if a change of variable changes the sign of dx then the limits also change appropriately.  In a multidimensional definite integral the limits are a boundary and "changing the order of the limits" becomes somewhat problematic and technical. Easier to just take the absolute value, see what the new boundary is and "get the right answer". :)  Cloudswrest (talk) 18:42, 27 February 2017 (UTC)

text: Furthermore, how does one order the limits if the boundary is, say, a circle.  For an area integral the dxdy is a signed quantity (differential form), but the "order of the limits" is the way you go around the circle.  This line of reasoning leads to Green's theorem. Cloudswrest (talk) 20:31, 22 September 2017 (UTC)
thread_title: Use of Jacobian in integration - change of variables 

message: Hello,
Could you please add information about the relation between the Jacobian matrix and the Covariance matrix, particularly in a function that has many parameters? (I am requesting this becuase I do not know this topic).
omermar 11:22, 6 August 2018 (UTC)
replies: text: I fail to see any notable relation between these two. Why do you think they are related? Boris Tsirelson (talk) 07:54, 8 August 2018 (UTC)
replies: text: Becuase I've read that there is a relation. I am not sure what type. I think that cov=JT*J ? (JT is transpose of Jacobian). https://en.wikipedia.org/wiki/User:Omermar 09:14, 8 August 2018 (UTC)
replies: text: The question is, covariance matrix of what is meant. Generally, it is the covariance matrix of a random vector. While Jacobian matrix is the Jacobian matrix of a differentiable vector-valued function at a point. How to "conflate" a random vector and a (non-random) vector-valued function? Boris Tsirelson (talk) 09:42, 8 August 2018 (UTC)

text: By the way: please sign your messages (on talk pages) by four tildas: ~~~~. Boris Tsirelson (talk) 09:42, 8 August 2018 (UTC)
thread_title:  Jacobian and Covariance in Multivariant Function 

message: I am working on a course in robotics and am pasting material from this page to the following:
https://en.wikiversity.org/wiki/Robotic_Mechanics_and_Modeling/Kinematics
Thank you. Admazzeo (talk) 15:48, 20 February 2020 (UTC)
thread_title: Material Copied to Wikiversity 

message: In response to the tag that the lead is too long, I think the tag can be removed.
The lead seems great! I don't think it's too long, considering the variety of ways the Jacobian is presented and talked about in math texts. I noticed an improvement from when I relied on the article last Fall for my differential geometry course, though perhaps I'm just imagining it. Either way, its length is appropriate IMO.
Fantasticawesome (talk) 21:32, 28 June 2021 (UTC)
thread_title:  The lead section has been improved, not too long 

message: This could be a bit cultural, but I would most spontaneously use the term ‘total derivative’ for this matrix, and ‘Jacobian’ only for the determinant. At the very least, this article should mention total derivative as a synonym. (At the moment, it's not mentioned at all.) If there are those who consider total derivative as a distinct concept, then it becomes even more important to explain the difference. It could be that the two articles should be merged, but at the moment I won't go as far as proposing that. 130.243.94.123 (talk) 12:11, 17 May 2022 (UTC)
replies: text: The total derivative is a linear map; the Jacobian matrix is a matrix. They could be considered as synonyms only if a "linear map" and a matrix would be the same thing. The relationship is clearly stated in the lead as  I agree that this should be made more visible. However, rewriting the lead has been requested by the tag . D.Lazard (talk) 13:01, 17 May 2022 (UTC)
thread_title:  Total derivative as synonym 

message: Hi, "Jacobian matrix" is the main topic of this article and its determinant is its subtitle. In my opinion, the part "determinant" should be removed from the title, because determinant is a property of every type of matrix. Therefore, the title of this article can only be "Jacobian matrix". Thanks, Hooman Mallahzadeh (talk) 13:50, 21 August 2022 (UTC)
replies: text: I disagree. The Jacobian determinant is fundamental in many areas of mathematics, and, in particular, in the general substitution rule, and in the statements of the inverse function theorem, the implicit function theorem and the Jacobian conjecture (this list is certainly incomplete). So, we need definitively an article about the Jacobian determinant. As the Jacobian determinant requires the Jacobian matrix to be defined, it is silly to try to split this article into Jacobian matrix and Jacobian determinant. D.Lazard (talk) 14:46, 21 August 2022 (UTC)
thread_title:  Title of article 

message: Boldface is used for functions (for example \mathbf{f}), and yet "normal" cursive is used for functions in partial derivatives \left(
\frac{\partial{f_i}}{\partial{x_j}}\right) or for the rows of the Jacobian (\nabla^\mathrm{T}f_i). I know it's not really important but the inconsistency is jarring, especially since different notation is used for the same function. I propose changing the whole thing to "normal" notation (\mathbf{R}\to\mathbb{R}, \mathbf{u}\to\vec{u}, \mathbf{f}\to f\,). Does anyone agree? Barçaforlife (talk) 00:48, 24 September 2022 (UTC)
thread_title:  Inconsistent notation for functions? 

message: It would be nice to reference this guy in the first paragraph 97.113.155.11 (talk) 05:25, 4 May 2023 (UTC)
thread_title:  should we mention Carl Gustav Jacob Jacobi 

message: The variables should be consistent with other Wiki articles (https://en.wikipedia.org/wiki/Spherical_coordinate_system).
So far in every other script I have seen \theta goes from 0 to \pi, describing the inclination. 2001:9E8:89B4:E600:C081:908B:3709:E723 (talk) 02:54, 16 December 2023 (UTC)
thread_title:  Notation Spherical Coordinates