Talk:Jacobian matrix and determinant

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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.

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designed to define the jacobian matrix, and i see that that definition is a stub. I have a copy of the page before it was fixed. i'm posting it in the stub for jacobian matrix. I think, then, it would be a good idea to discuss whether we might want to combine the two into one page? I'm for this. I think the ideas neeed to be presented closely together in order for fluent comprehension, and a brief and clear page describing first the jacobian matrix, and then the jacobian, would be simple to construct as well as being a better way to present the topic.

4865:{\displaystyle {\begin{aligned}{\color {Red}y_{2}}&{\color {Red}=4x_{1}^{2}-2\sin(x_{2}x_{3})}\\{\color {Blue}y_{1}}&{\color {Blue}=5x_{2}}\\y_{3}&=x_{2}x_{3}\end{aligned}}\Longrightarrow {\begin{vmatrix}{\color {Red}8x_{1}}&{\color {Red}-2x_{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}}=+8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}={\color {YellowOrange}+40}x_{1}x_{2}.} 4362:{\displaystyle {\begin{aligned}{\color {Blue}y_{1}}&{\color {Blue}=5x_{2}}\\{\color {Red}y_{2}}&{\color {Red}=4x_{1}^{2}-2\sin(x_{2}x_{3})}\\y_{3}&=x_{2}x_{3}\end{aligned}}\Longrightarrow {\begin{vmatrix}{\color {Blue}0}&{\color {Blue}5}&{\color {Blue}0}\\{\color {Red}8x_{1}}&{\color {Red}-2x_{3}\cos(x_{2}x_{3})}&{\color {Red}-2x_{2}\cos(x_{2}x_{3})}\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}={\color {OliveGreen}-40}x_{1}x_{2}.} 4982: 95: 85: 64: 2288: 31: 2990:

that Partial (and not Total) Derivatives are sufficient for it's existence. I thought that for a function to be differentiable then it also had to be continuous at the point in question; i.e. that all derivatives (Partial and Total) would exist. I suppose, for example, that if Z = f(X,Y) and if Y is constant then there would only be dZ/dX and that dZ/dY would be zero and therefore that the total derivative would not exist at the point. Could someone please clarify this for me. Thanks.

4999:. The two vectors define a square (the red part on the left), which get mapped to a distorted rectangle (the red part on the right). The jacobian defines a linear transformation at each point, shown here is its action on two basis vectors. The resulting vectors define a parallelepiped whose area is an approximation for the area of the distorted rectangle. The area of parallelepiped is equal to the jacobian determinant, and as the rectangle gets smaller the error goes to zero. 2578:{\displaystyle {\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}} 3130:=0; if you take spherical and set the polar (zenith) angle to pi/2 you will get an extra term in the radial derivative that is not supposed to be there in 2D. This is part of the reason why we use θ to mean azimuthal angle in 2D Polar but then change θ to the Zenith angle in spherical and let φ be the azimuthal angle. It is fundamentally wrong to think that spherical and polar coordinates are this simply related.) 2014: 1578: 1799: 22: 181: 1185: 1806: 182:

http://books.google.com/books?id=afnlx8sHmQIC&pg=PA193&lpg=PA193&dq=jacobian+riemannian&source=bl&ots=JvH2bAV4c0&sig=RMdv-mJvEn4nWHOaDsqZ7vKFa6k&hl=en&ei=bEO6SvfxD5ux8QbMk92MCg&sa=X&oi=book_result&ct=result&resnum=5#v=onepage&q=jacobian%20riemannian&f=false

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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

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First off I will admit that I only got "C's" and occasionally "B's" in Multivariable Calculus and Vector Calculus classes; so maybe I simply don't know what I'm talking about. But early in the article it states that the Jacobian Matirix can exist even if the function is NOT differentiable at a point;

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let me further add, that i think, thou my memory is shaky here, that a basis is a set of vectors. That is, they can only be something like: {(4,3,0), (5,0,4), (1,3,2)} such that {(4x,3x,0x), (5y,0y,4y), (1z,3z,2z)} are linearly independant. Thus, they depend on a pre-established system of variables,

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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

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Also -- the natural extension of 2D Polar coordinates is NOT spherical with the polar angle (from z-axis) set to pi/2. It is in fact cylindrical coordinates. (If it is not immediately obvious why spherical coordinates would produce the wrong results, consider that the Laplacian must be the same in

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I went ahead an changed the phi theta notation since 71.249.100.27 is correct and while other forms are accepted, nobody seems to have brought a counter argument to his. I also think Wingless might be correct with his claim but I think the current writing may be correct from a conventional point of

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and similarly for the gradient as df/dx and df/dx'. This way everything fits. Note that the orientation of the variable wrt which we differentiate drives the orientation of the output gradient vectors. For the Jacobians, both the differentiated vector function f and the differentiator vector x drive

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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,

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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

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would be defined by way of the basis. That is, one could pick an entirely different system of measurements; entirely unrelated "unit"s of measurement, and have a different 'basis' from which to define 'distances' in a space, which would be equally valid, although (1,0,0) in one system would not be

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It looks to me like the matrix on the right is the transpose of the row matrix on the left. Since I'm not an expert on notation conventions for matrix algebra, I won't try to correct this myself, but could someone else please look at it and decide whether I'm looking at it off-kilter by 90 degrees

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Just checked back here to see if anyone had finally bothered to add the method of computing the scaling factor for CoV when the Jacobian isn't square, now that many sources have been provided, but still nothing, just goes to show that the people running this article don't really give a fuck about

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You introduce the concept of determinant of a non-square matrix. I have never heard of such a concept. It it exists, it is certainly not adopted by mathematics community, and does not deserve to appear in Knowledge. However, if there are sufficiently many articles on this concept, it could be the

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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

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I'd like to insist on this topic. I corrected the first line of my comment, that originally said gradients were defined as row-vectors thus making my comment absurd. Gradient is defined as a column-vector in Knowledge. So either we leave the vectors and matrices orientations out of the definition

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This factor is well-known, it appears when calculating the volume form of a manifold (embedded) in Euclidean space in terms of a chart. See for example (8.1) in Sect. 8.1 in "Functions of several variables" by Wendell Fleming (Springer, 1977, "Undergraduate texts in mathematics"). However, to my

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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'

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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

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I am taking a class on this and Wingless is correct. It is clear from the next two examples and from the text above this section that the example shows a transformation from Cartesian coordinates to spherical coordinates. I confirmed this from a Schaum's Outlines (Advanced Calculus) book. I went

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I do not know what is the "scaling factor of a parameterization" nor what is a CoV formula (no Knowledge article about them). In any case, even if its definition uses a Jacobian matrix, this scaling factor does not belong to this article. Moreover, the term of "scaling factor" suggests that the

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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

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There really should be some explanation of how one takes the determinant of the non-square Jacobian, specifically that the scaling factor for the CoV formula given by |det(J)| is really just a special case of sqrt(det(J^TJ)) which comes from the change in Riemannian metric when you perform the

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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

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Also it being continuous is not a sufficient condition for differentiability. For example, the Weierstrass function is continuous everywhere but differentiable nowhere. Then it also follows from intuition that a function could be differentiable along one direction but not differentiable along

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I originally put in the matrix here, and put in most of the structure. I did make a mistake in terminology, thou, as i see has been corrected. I defined the jacobian matrix, where the "Jacobian" per say, refers to the determinant of that matrix. My point is is that this page was originally

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Nonsense, Jacobian has become an English word with an English pronunciation. All modern English words have a historical origin with another pronunciation; this doesn't mean we should try to pronounce it in the historical way. "Paris" is not pronounced "paree" in English. And "Jacobian" is

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Yes, and it's worse than that: phi is supposed to represent the azimuthal angle ( from z axis) while theta should stay in the x-y plane as it does in polar form. Will someone who's savvy with the formula code please correct the notation, both in the parameterization and in the Jacobian?

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What is "obviously impossible"? The first row of the Jacobian matrix is (df1/dx1 ... df1/dxn), which is the gradient of the first component of the function f. The Jacobian matrix, as defined on this page, is form b) in your list, which seems to be precisely what you want. --

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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.

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Went ahead and changed this to standard physics notation. Here we use θ as the polar (zenith) angle and φ as the azimuthal angle (x-y plane, angle from x-axis). It is OK to use θ as the azimuthal, but then the notation should be (r,φ,θ) or the map should be changed to

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defines it as a column vector. That's the most common definition, I think, though it often makes more sense to define it as a row vector. Anyway, I changed the text to remove the contradiction; it now says that each row of the Jacobian matrix is the transpose of the

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It would be nice to have a section on generalizations. Given a map f between two Riemannian manifolds (M, g_M) and (N, g_N) a generalization of the determinant of the Jacobian is given (if I'm not mistaken) by the ratio of the determinants of f^*(g_N) and g_M. --gm

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Personally I find form b) as being much more useful, as when we pas to the partial derivatives of matrix expressions we can almost mimic the scalar differentiation rules. Using a) in this context leads to incredibly confusing expressions, full of transposed marks.

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It says something about the inverse function theorem... Is there a way to test if a given region in a mapping's domain is 1:1 instead of a neighborhood around a point? I thought if the jacobian is identically 0 for this region it's 1:1 but it's not (by examples).

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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)

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2D as it must in the natural extension of that coordinate system in 3D projected down to 2D. Working out the results in cylindrical coordinates, it is easy to verify that the Laplacian of polar coordinates is simply the Laplacian of cylindrical with

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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.

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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.

2009:{\displaystyle \partial f/\partial x^{\top }={\begin{bmatrix}\partial f_{1}/\partial x_{1}&\cdots &\partial f_{1}/\partial x_{n}\\\vdots &&\\\partial f_{m}/\partial x_{1}&\cdots &\partial f_{m}/\partial x_{n}\end{bmatrix}}} 4875:

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.

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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.

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I'm not sure how to write this concisely (my main strength is in graphic design) or if the diagram needs more identifying markers, like a variable name for the point or red region. I'm interested in input on this before I put it on the article

1573:{\displaystyle \partial f/\partial x={\begin{bmatrix}\partial f_{1}/\partial x_{1}&\cdots &\partial f_{1}/\partial x_{n}\\\vdots &&\\\partial f_{m}/\partial x_{1}&\cdots &\partial f_{m}/\partial x_{n}\end{bmatrix}}} 783:, ect. Is the system of normalized variables used to measure the space. One could have just as easily (and may find it usefull for other purposes) defined a topologically equavelent space with a different 'basis', orthogonal to this one. 718: 533: 231:

Hello. About notation, I seem to recall the notation DF for the Jacobian of F. Have others seen that notation? -- On a different note, maybe we can mention that if F is a scalar field then the gradient of F is its Jacobian. Happy editing,

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I changed the language. The formula looks good to me (I'm not bothered by using phi or theta, so long as the domains are correct, and they are). If somebody could verify my correction then maybe the 'dubious' label should be removed.

1794:{\displaystyle \partial f^{\top }/\partial x={\begin{bmatrix}\partial f_{1}/\partial x_{1}&\cdots &\partial f_{m}/\partial x_{1}\\\vdots \\\partial f_{1}/\partial x_{n}&\cdots &\partial f_{m}/\partial x_{n}\end{bmatrix}}} 1005: 6755: 4915:

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.

2159:(thus giving this freedom to the user), either we use consistent definitions. I'm for this second alternative but I am not matematician so I can't have a strong position on this. --Note: I've just made up an account so I sign now as 200:

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),

5517:. In any case, the text that you have added to the article (and that I have reverted) does not belongs to the article without a reference to a reliably published article, which defines the determinant of a non-square Jacobian ( 6533:(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 6446:

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

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You wouldn't want to call it a "determinant" though, which simply isn't defined for a non-square matrix in general. Once I read up a bit on differential forms, I'll revisit how to best include things like that in this

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I am not sure if this generalization is widely used. In any case, in the rectangular case, the rank of the Jacobian matrix plays a key role for determining the critical points. I have just added a section about

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I changed the Jacobian entry at Row 3, Column 2 from -r.sin(phi) to -r.sin(theta), an obvious slip by someone but I also agree that x,y,z would be much more widely recognized than x1,x2,x3 in this context.

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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?

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algorithm. In both applications, the corresponding Jacobian term can be non-square matrix. It seems reasonable and acceptable to calculate Jacobian determinant of non-square using the formula in the

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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?

6843: 6688:). 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 ( 4383: 3880: 3792:

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."

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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.

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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". :)

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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.

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Since Jacobi is of German origin I guess his name and its derivatives should be pronounced only in the second way. English language rules can not be applied to words of foreign origin.

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Feel free to add a section about the generalizations that you have cited, but, please, mention their use, if they have some. In fact, if these generalizations are not used in some

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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

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From my understanding of basis, it is the selection of a set of measurements from which one defines a coordinate system to describe a space. Thus, the unit vectors:

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pull-back. I suppose this could go in change of variables formula page, but it's really got to go somewhere as it's maddening to not understand how this works.

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Because we face a terminological problem. Sources give the relevant formula, but hesitate to give a name ("generalized Jacobian" or another) to the coefficient.

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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?

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It could be placed in Section "Jacobian determinant", after ...", and the n-volume of a parallelepiped is the determinant of its edge vectors." A nonlinear map

3058:"The transformation from spherical coordinates to Cartesian coordinates is given by..." Shouldn't this be from cartesian coordinates to spherical coordinates? 6208: 35: 2260:

This should be (∇fᵢ)ᵀ rather than ∇ᵀfᵢ since the matrix product of a row vector and a scalar valued function in this way isn't generally a valid operation

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I noticed that too, but I'm pretty sure there isn't any way around it. Its just how the human mind parses curves. I could change "f" to "f : R -: -->

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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

3224:). If you look at example three and pay attention to the nomenclature used to describe the problem, my edits follows. The Jacobian determinant of ( 6880: 786:

However, this is merely a very fuzzy intuitive interpretation, and I'm not justifying the use. I am explaining what i think was the intention. -

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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?

1358:{\displaystyle \partial f/\partial x={\begin{bmatrix}\partial f/\partial x_{1}&\cdots &\partial f/\partial x_{n}\end{bmatrix}}^{\top }} 117: 6589: 6895: 3267: 3183: 5021:

A nice picture. But, on the first look, I got it as a surface in three dimensions. Alas, I have no idea how to avoid such wrong impression.

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sends a small square to a distorted rectangle close to the parallelepiped, the image of the square under the best linear approximation of

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the distribution of the output matrix elements. This way, we could also define the two following (quite useless I admit) constructions:

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source and the target of the mapping have the same metric, which is a nonsense if the source and the target have different dimensions.

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entry it is said that the rows of the Jacobian matrix are the gradients of each component of f wrt x. This is obvoiusly impossible!

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Furthermore, how does one order the limits if the boundary is, say, a circle. For an area integral the dxdy is a signed quantity (

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The IPA pronunciation guide is totally wrong. I've never heard anybody pronounce it /dʒɨˈkoʊbiən/ and especially not /jɨˈkoʊbiən/

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Jacobian determinant of non-square can appear in application involving transformation of variables or change of variables, such as

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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).

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Sect. 5.3 in "Vector calculus, linear algebra and differential forms" by John Hubbard and Barbara Hubbard (Prentice-Hall, 2002);

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given, but a change from spherical coordinates to Cartesian coordinates would require the functions to be written in terms of (

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regret, he does not call this factor "generalized Jacobian" (or another way; he just denotes and uses it). According to EoM,

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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).

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pronounced like "John" by every mathematician I know. That is why the dictionary correctly lists that pronunciation first.

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and the answer we get when doing a change of variables that involves using the absolute value of the Jacobian determinant?

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Isnt' there an error in the simplification of the determinant given as example? Where is the term x3 cos(x1) gone? --

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I disagree. The Jacobian determinant is fundamental in many areas of mathematics, and, in particular, in the general

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programmer, I appreciate the help that a good compiler gives me with keeping track of the types of my expressions.

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policy, we cannot make up new definitions but have to stick with those already in use. Thanks for your comments. --

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Further, the example used does not give the correct definition for the directional derivative, which is lim_ h-: -->

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Rm then f is totally differentiable at a point p if and only if each component f_i:U-R is also differentiable. -

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15:20, 21 August 2006 (UTC) though I'm the same who started this topic. You can email me if you wish. Cheers. --

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Sect. 4.11c in "Introduction to calculus and analysis" vol 2 by Richard Courant and Fritz John (Springer, 1989);

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I propose to give a double definition, with a clear notation describing the adopted convention, as follows :

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under the spherical coordinates section. Thanks now I have wasted half an hour of studying for finals :-)

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There are two closely connected meanings of "stationary point". The meaning intended in the article is that

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mentions. I don't quite understand your comment, but it seems that you're confusing these two meanings. --

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I agree that this should be made more visible. However, rewriting the lead has been requested by the tag

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For a non-expert, the notation is confusing, so best to annotate variables (and expressions) with types.

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Excuse me I submitted with some errors. Reread what I posted now and you will see what I mean. Thanks

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Many thanks for everyone who contributes to the article and talk page. They are really very helpful!

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p;

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I am working on a course in robotics and am pasting material from this page to the following:

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No, the gradient relates to a scalar function and the Jacobian relates to a vector function.

5274: 6805: 6454: 6428: 6420: 6347: 6332: 6282: 6253: 6249: 6229: 6216: 6030: 5870: 5792: 5711: 5685: 5656: 5632: 5621: 5599: 5317: 5231: 5091: 5026: 2742: 2657: 1000:{\displaystyle \left({\frac {f}{y}}\right)_{g}={\frac {\partial {(f,g)}}{\partial {(y,g)}}}} 6750:{\displaystyle \mathbf {R} \to \mathbb {R} ,\mathbf {u} \to {\vec {u}},\mathbf {f} \to f\,} 6001: 5648: 3558:

As an example for the post above (annotate variables for non experts), could the variables

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as a synonym. (At the moment, it's not mentioned at all.) If there are those who consider

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A stationary point is not a fixed point. i removed that part. It was in brackets. Thanks

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I fail to see any notable relation between these two. Why do you think they are related?

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providing the best quality information and are more interested in their mini power trip.

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Sect. 12.4 and 13.2.3 in "Mathematical analysis II" by Vladimir Zorich (Springer, 2004).

4935:

I see absolutely no point in including the matlab-code in this article. Why is it here?

1007:

Yes when i wrote jacobian and determine it, i get 0, zero, null, notting!?! Am i right?

5608:

Another source: Theorem 8.4 in Sect. 8.1 (what a coincidence!), and then Sect. 8.3, in

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via integral in dx dy. Second, in the Area formula and Coarea formula (see EoM, too).

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You mention "the CoV formula" but do not tell us, where did you take this name from.

5544:

So how do you propose to compute the scaling factor for a parametrization R^3 -: -->

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Vanishing means becoming zero. This term is not specifically related to Jacobians.

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First, they appear when calculating area of a surface (or integral over a surface)

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In response to the tag that the lead is too long, I think the tag can be removed.

5545:

R^4? This is how it's done, it's the square root of the determinant of J^T * J.

3202:

in the first example goes from spherical coordinates to Cartesian coordinates. --

247:

I've always heard the initial consonant pronounced as a affricate, as in "John".

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another, wherefore a partial derivative may exist but a total differential won't.

6343: 6328: 6278: 6212: 5707: 5681: 5652: 5617: 5595: 5514: 5313: 5227: 5087: 5022: 1220: 260: 241: 113: 6037:? If it is Lebesgue measure, then the absolute value appears. Note that taking 5354: 3482:

yourself (people will protest if you go to far)? I found only one place where

794: 90: 5594:

author(s) call it this way; but I must admit that this usage is not notable.

6342:

By the way: please sign your messages (on talk pages) by four tildas: ~~~~.

538:(By the way, the Latin phrase "per se" doesn't have an "a" or a "y" in it.) 6419:

This could be a bit cultural, but I would most spontaneously use the term ‘

2283:

Isn't this still a problem? (as it stands on the page as of this writing):

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to illustrate how a local area deforms and how its area deforms by det J.

255:

So have I, and it's listed as the first of two possible pronunciations at

6642:{\displaystyle \left({\frac {\partial {f_{i}}}{\partial {x_{j}}}}\right)} 5706:

Thus we do not have a good reason to place that formula in this article.

4954:

I just added reqdiagram. This could do with a picture of a distortion of

2177: 1245: 6586:, and yet "normal" cursive is used for functions in partial derivatives 5613: 5222:

Jacobian determinant is also defined when the matrix is not square; see

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I don't think that the matrix described there is related to this page.

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a column-stacked vector of all column gradients of the components of f.

1248:

of a scalar function f(x) wrt a vector x is defined as a column vector:

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Is there a similar distinction between the mathematical definition of

256: 6371:

https://en.wikiversity.org/Robotic_Mechanics_and_Modeling/Kinematics

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Sect. 2c, p. 37), but I do not want to see it called "determinant".

5643:

Sure. I regret not seeing it called "generalized Jacobian" (as I do

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was introduced and the type was only implicit, and I fixed that. --

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etc. is cute, but it is not used as far as I know. According to our

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for the variable in the new domain, which corresponds to variable

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Well, I changed it back, since it is clear from the formulas that

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a row-stacked vector of all row gradients of the components of f.

2590:(so easy to do with matrices!) or if it really needs correction? 4977:

I created a diagram that mimics one in one of my math textbook:

3293:. For the univariate case, when you want to go from a function 1011: 6776:

It would be nice to reference this guy in the first paragraph

6292: 5896:" because we are expected to use an unsigned concept of area. 5334:

uses this generalization to specify a Jacobian for a surface.

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Example 1 - Spherical to Cartesian coordinates, definition of

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The variables should be consistent with other Wiki articles (

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if one wishes to use the (r,θ,φ) notation as is standard.

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More formally, if S is an open subset of Rn and f:U-: -->

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for instance, in the (1,1) direction (45°) this equals

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Jacobians and gradients. consistent definitions please!

5521:) and references to articles that mention and use it ( 5116:

Sounds good. I'm putting it in the article for now. --

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What happens if I changed the order of the functions?

4278: 4055: 2385: 2297: 2106:{\displaystyle \partial f^{\top }/\partial x^{\top }=} 1842: 1631: 1406: 1283: 6828: 6808: 6694: 6655: 6592: 6567: 6189: 6168: 6148: 6128: 6101: 6069: 6043: 6004: 5974:{\displaystyle \int \int _{(x,y)\in S}f(x,y)\ dx\ dy} 5905: 5873: 5847: 5827: 5795: 5739: 5277: 4381: 3878: 3709: 3656: 3607: 3564: 3508: 2914: 2854: 2796: 2700: 2660: 2532: 2489: 2432: 2389: 2339: 2301: 2291: 2190: 2068: 2028: 1809: 1598: 1380: 1256: 1038: 920: 818: 733: 557: 372: 306: 212:For the above reasons, I am removing this section. 6797:

https://en.wikipedia.org/Spherical_coordinate_system

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Use of Jacobian in integration - change of variables

112:, a collaborative effort to improve the coverage of 6834: 6814: 6749: 6680: 6641: 6578: 6201: 6174: 6154: 6134: 6114: 6087: 6055: 6021: 5973: 5888: 5859: 5833: 5813: 5781: 5304: 4864: 4361: 3727: 3695: 3620: 3593: 3547: 3337:direction, from the new domain to the old domain: 3325:is more convenient to work in than the old domain 2959: 2894: 2833: 2721: 2686: 2577: 2217: 2105: 2051: 2008: 1793: 1572: 1357: 1179: 999: 887: 771: 712: 527: 344: 3437:). This generalizes to the multivariate case. -- 6390:The lead section has been improved, not too long 6268:Jacobian and Covariance in Multivariant Function 5616:. (But still, no special name for this object.) 3814:LOL! just LOL. Gotta hand it to that Vandal. -- 1229:As in both of those; they're the same concept. — 6886:Knowledge level-5 vital articles in Mathematics 5567:Please, sign your post with four tildes (~~~~). 3329:, you can do that if you have a transformation 2654:is a stationary point for the dynamical system 1372:is defined as in this page I'm commenting now: 172:Jacobians for maps between Riemannian manifolds 5147:Kjetil Halvorsen 18:02, 28 August 2012 (UTC) 3735:the spherical coordinates. Is that enough? -- 3095:view. I'll let someone else make that change. 2834:{\displaystyle \sin 45^{\circ }={\sqrt {2}}/2} 2741:is zero; this is the meaning that the article 5998:Roughly, it depends on the interpretation of 5513:object of its own article or of a section in 2218:{\displaystyle \partial f^{\top }/\partial x} 779:is a basis, i interpret this as saying that x 8: 6561:Boldface is used for functions (for example 5249:, I doubt that they fulfill the criteria of 3850:What if i change the order of the equations? 2960:{\displaystyle f(r,\theta )=r\sin(2\theta )} 704: 558: 519: 373: 6681:{\displaystyle \nabla ^{\mathrm {T} }f_{i}} 6315:is the Jacobian matrix of a differentiable 5459:Remember when? Pepperidge farms remembers. 5218:Generalizations of the Jacobian determinant 6772:should we mention Carl Gustav Jacob Jacobi 5782:{\displaystyle \int _{0}^{2\pi }\sin(x)dx} 5546: 5491: 5464: 5422: 58: 6827: 6807: 6802:So far in every other script I have seen 6734: 6720: 6719: 6711: 6704: 6703: 6695: 6693: 6672: 6661: 6660: 6654: 6625: 6620: 6608: 6603: 6597: 6591: 6568: 6566: 6188: 6167: 6147: 6127: 6106: 6100: 6079: 6074: 6068: 6063:in the n-dimensional case you do not get 6042: 6003: 5913: 5904: 5872: 5846: 5826: 5794: 5749: 5744: 5738: 5485:Determinant of non-square Jacobian matrix 5276: 4853: 4843: 4829: 4812: 4800: 4776: 4767: 4740: 4728: 4708: 4699: 4690: 4675: 4665: 4646: 4634: 4621: 4611: 4592: 4580: 4570: 4561: 4553: 4540: 4530: 4513: 4497: 4485: 4475: 4469: 4454: 4444: 4419: 4414: 4402: 4392: 4386: 4382: 4380: 4350: 4340: 4326: 4309: 4297: 4273: 4264: 4237: 4225: 4201: 4191: 4172: 4160: 4147: 4137: 4118: 4106: 4096: 4087: 4076: 4067: 4058: 4050: 4037: 4027: 4010: 3991: 3981: 3956: 3951: 3939: 3929: 3923: 3911: 3899: 3889: 3883: 3879: 3877: 3857:The Jacobian determinant of the function 3708: 3687: 3674: 3661: 3655: 3612: 3606: 3582: 3569: 3563: 3539: 3526: 3513: 3507: 3258:http://en.wikipedia.org/Multiple_integral 2913: 2874: 2853: 2823: 2816: 2807: 2795: 2782:Differentiable versus partial derivatives 2699: 2659: 2557: 2542: 2531: 2514: 2499: 2488: 2457: 2442: 2431: 2414: 2399: 2388: 2380: 2359: 2345: 2338: 2321: 2307: 2300: 2292: 2290: 2251:Allright I agree with the solution. Thks 2204: 2198: 2189: 2094: 2082: 2076: 2067: 2035: 2027: 1992: 1980: 1974: 1954: 1942: 1936: 1908: 1896: 1890: 1870: 1858: 1852: 1837: 1828: 1816: 1808: 1777: 1765: 1759: 1739: 1727: 1721: 1697: 1685: 1679: 1659: 1647: 1641: 1626: 1612: 1606: 1597: 1556: 1544: 1538: 1518: 1506: 1500: 1472: 1460: 1454: 1434: 1422: 1416: 1401: 1387: 1379: 1349: 1335: 1323: 1304: 1292: 1278: 1263: 1255: 1168: 1151: 1142: 1132: 1115: 1107: 1093: 1085: 1077: 1063: 1055: 1043: 1037: 977: 955: 949: 940: 926: 919: 888:{\displaystyle J_{f}(x)|J_{f}(x)|^{-1/n}} 875: 868: 863: 847: 838: 823: 817: 760: 741: 732: 556: 371: 333: 314: 305: 188:Differentiable versus partial derivatives 2998:) 21:55, 2 March 2008 (UTC)JeepAssembler 2786:Excuse me, but in this section it says 6876:Knowledge vital articles in Mathematics 6745: 724:the same as (1,0,0) in another system. 702: 655: 605: 561: 517: 470: 420: 376: 60: 19: 6844:2001:9E8:89B4:E600:C081:908B:3709:E723 6448: 5443:The first vowel sound is still wrong. 3854:Let's use the example of the article: 2895:{\displaystyle \sin(2(45^{\circ }))=1} 2636:In dynamical systems, Stationary point 2052:{\displaystyle \partial f/\partial x=} 6891:C-Class vital articles in Mathematics 6293:https://en.wikipedia.org/User:Omermar 4830: 4709: 4700: 4691: 4635: 4581: 4562: 4486: 4470: 4403: 4387: 4327: 4161: 4107: 4088: 4077: 4068: 4059: 3940: 3924: 3900: 3884: 3333:between the domains that goes in the 3321:, for example because the new domain 3256:. Actually you can confirm this from 2733:is a stationary point for a function 7: 6557:Inconsistent notation for functions? 6142:is the interval whose endpoints are 5529:, and is not relevant to Knowledge. 2120:Joan Sola LAAS-CNRS Toulouse France 772:{\displaystyle (x_{1},\dots ,x_{n})} 345:{\displaystyle (x_{1},\dots ,x_{n})} 106:This article is within the scope of 6300:The question is, covariance matrix 1021:Isn't the Jacobain matrix just the 49:It is of interest to the following 6662: 6657: 6617: 6600: 5610:"Manifolds and differential forms" 4895:the magnitude of the determinant. 3703:are the Cartesian coordinates and 3383:in the old domain by the relation 3375:)). For use in an integral, using 2845:Should it not say something like 2550: 2535: 2507: 2492: 2450: 2435: 2407: 2392: 2352: 2342: 2314: 2304: 2209: 2199: 2191: 2095: 2087: 2077: 2069: 2040: 2029: 1985: 1967: 1947: 1929: 1901: 1883: 1863: 1845: 1829: 1821: 1810: 1770: 1752: 1732: 1714: 1690: 1672: 1652: 1634: 1617: 1607: 1599: 1549: 1531: 1511: 1493: 1465: 1447: 1427: 1409: 1392: 1381: 1350: 1328: 1317: 1297: 1286: 1268: 1257: 1161: 1135: 1126: 974: 952: 180:Ok, here is a reference for this: 14: 6901:Mid-priority mathematics articles 6649:or for the rows of the Jacobian ( 3696:{\displaystyle x_{1},x_{2},x_{3}} 3548:{\displaystyle x_{1},x_{2},x_{3}} 126:Knowledge:WikiProject Mathematics 6871:Knowledge level-5 vital articles 6735: 6712: 6696: 6569: 6319:at a point. How to "conflate" a 5172:Jacobi Matrix external reference 4980: 2346: 2308: 1152: 1116: 1108: 1094: 1086: 1078: 1064: 1056: 259:. I've added it to the article. 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 6521:, and in the statements of the 6304:is meant. Generally, it is the 5417:ipa pronunciation of "jacobian" 3728:{\displaystyle r,\phi ,\theta } 3351:. Then you can define function 146:This article has been rated as 6881:C-Class level-5 vital articles 6842:, describing the inclination. 6791:Notation Spherical Coordinates 6767:00:48, 24 September 2022 (UTC) 6739: 6725: 6716: 6700: 6573: 6363:Material Copied to Wikiversity 6262:20:31, 22 September 2017 (UTC) 5952: 5940: 5926: 5914: 5808: 5802: 5770: 5764: 5716:07:20, 23 September 2017 (UTC) 5479:19:16, 20 September 2015 (UTC) 5299: 5287: 5044:R" to make it more explicit -- 4681: 4658: 4627: 4604: 4550: 4460: 4437: 4207: 4184: 4153: 4130: 4047: 3997: 3974: 2954: 2945: 2930: 2918: 2883: 2880: 2867: 2861: 2710: 2704: 2681: 2675: 2176:Indeed, the Knowledge article 1156: 1148: 1120: 1112: 1101: 1098: 1090: 1082: 1071: 1068: 1060: 1052: 990: 978: 968: 956: 864: 859: 853: 839: 835: 829: 766: 734: 699: 657: 643: 607: 599: 563: 514: 472: 458: 422: 414: 378: 363:would be something like this: 339: 307: 1: 6852:02:54, 16 December 2023 (UTC) 6579:{\displaystyle \mathbf {f} )} 6385:15:48, 20 February 2020 (UTC) 6238:18:42, 27 February 2017 (UTC) 6221:16:22, 27 February 2017 (UTC) 6088:{\displaystyle \int _{a}^{b}} 5993:15:45, 27 February 2017 (UTC) 5453:20:44, 27 November 2017 (UTC) 5192:18:07, 11 November 2012 (UTC) 3760:14:52, 23 December 2009 (UTC) 3442:08:50, 13 December 2010 (UTC) 3276:22:47, 12 December 2010 (UTC) 3207:19:18, 12 December 2010 (UTC) 3192:19:50, 11 December 2010 (UTC) 3140:06:58, 21 December 2009 (UTC) 2600:18:48, 2 September 2019 (UTC) 1029:? That is, is this correct?: 297:Why in the world do you call 222:14:36, 11 February 2008 (UTC) 120:and see a list of open tasks. 6896:C-Class mathematics articles 5690:13:25, 24 January 2016 (UTC) 5661:19:24, 22 January 2016 (UTC) 5639:18:52, 22 January 2016 (UTC) 5626:18:15, 22 January 2016 (UTC) 5604:18:02, 22 January 2016 (UTC) 5581:17:41, 22 January 2016 (UTC) 5561:17:11, 22 January 2016 (UTC) 5539:09:00, 22 January 2016 (UTC) 5506:05:02, 22 January 2016 (UTC) 5437:00:12, 8 February 2015 (UTC) 4972:14:14, 18 October 2011 (UTC) 3745:10:58, 23 October 2008 (UTC) 3644:10:13, 23 October 2008 (UTC) 3594:{\displaystyle x_{1},x_{2},} 3289:transformations, which work 3105:04:25, 4 December 2009 (UTC) 3089:02:16, 2 December 2009 (UTC) 2980:23:31, 3 February 2008 (UTC) 205:towards (0,0) as requested. 6551:14:46, 21 August 2022 (UTC) 6512:13:50, 21 August 2022 (UTC) 6415:Total derivative as synonym 6398:length is appropriate IMO. 5411:03:35, 19 August 2014 (UTC) 5396:11:01, 18 August 2014 (UTC) 5375:16:53, 7 January 2015 (UTC) 3650:I added a bit, saying that 3496:12:04, 4 October 2008 (UTC) 3476:05:20, 4 October 2008 (UTC) 3433:is the one-element matrix ( 3166:21:36, 2 January 2010 (UTC) 2631:09:49, 8 October 2009 (UTC) 2270:19:48, 17 August 2022 (UTC) 2256:12:56, 23 August 2006 (UTC) 2239:07:58, 22 August 2006 (UTC) 2168:15:20, 21 August 2006 (UTC) 2152:11:39, 17 August 2006 (UTC) 2141:11:07, 17 August 2006 (UTC) 2125:11:37, 17 August 2006 (UTC) 804:Would it be correct to say 727:Thus, when it is said that 282:10:32, 8 October 2009 (UTC) 6917: 6352:09:42, 8 August 2018 (UTC) 6337:09:42, 8 August 2018 (UTC) 6296:09:14, 8 August 2018 (UTC) 6287:07:54, 8 August 2018 (UTC) 6182:; think what happens when 5213:14:27, 2 August 2013 (UTC) 5126:07:09, 22 April 2015 (UTC) 5096:06:59, 22 April 2015 (UTC) 5054:06:39, 22 April 2015 (UTC) 5031:06:23, 22 April 2015 (UTC) 5013:05:56, 22 April 2015 (UTC) 4926:11:05, 22 March 2011 (UTC) 4905:10:04, 22 March 2011 (UTC) 4887:23:38, 10 March 2011 (UTC) 3824:03:02, 29 April 2011 (UTC) 3782:23:20, 26 March 2009 (UTC) 3048:03:14, 29 April 2011 (UTC) 3027:11:00, 22 April 2008 (UTC) 2729:. Another meaning is that 236:21:28, 23 Mar 2004 (UTC) 6527:implicit function theorem 6410:21:32, 28 June 2021 (UTC) 6115:{\displaystyle \int _{S}} 5525:). Without this, this is 4987:Basically we have a map 3842:11:13, 3 March 2012 (UTC) 3808:20:47, 16 June 2010 (UTC) 3068:20:36, 16 June 2008 (UTC) 3008:21:55, 2 March 2008 (UTC) 2776:16:09, 4 April 2012 (UTC) 2617:03:25, 25 June 2007 (UTC) 1217:Orientation (mathematics) 793:and are based off of the 542:20:36 Mar 26, 2003 (UTC) 264:03:36, 5 March 2006 (UTC) 251:01:23 Mar 24, 2003 (UTC) 244:00:32 Mar 24, 2003 (UTC) 145: 78: 57: 6523:inverse function theorem 6491:13:01, 17 May 2022 (UTC) 6441:12:11, 17 May 2022 (UTC) 6327:vector-valued function? 5344:14:56, 2 July 2014 (UTC) 5322:16:48, 16 May 2014 (UTC) 5305:{\displaystyle z=f(x,y)} 5263:15:05, 16 May 2014 (UTC) 5236:11:53, 16 May 2014 (UTC) 4945:08:40, 19 May 2011 (UTC) 3281:The text in our article 3173:ahead and changed this. 2755:20:50, 6 July 2007 (UTC) 2645:18:17, 5 July 2007 (UTC) 1234:18:45, 25 May 2006 (UTC) 1224:12:56, 25 May 2006 (UTC) 910: 907:23:49, 4 Aug 2004 (UTC) 152:project's priority scale 6815:{\displaystyle \theta } 6786:05:25, 4 May 2023 (UTC) 5889:{\displaystyle x=2\pi } 5814:{\displaystyle \sin(x)} 2687:{\displaystyle x'=f(x)} 1194:16:57, 5 May 2006 (UTC) 227:Notations, Determinants 109:WikiProject Mathematics 6866:C-Class vital articles 6836: 6816: 6757:). Does anyone agree? 6751: 6682: 6643: 6580: 6317:vector-valued function 6204: 6176: 6156: 6136: 6116: 6089: 6057: 6023: 6022:{\displaystyle dx\ dy} 5975: 5890: 5861: 5835: 5815: 5783: 5306: 4866: 4363: 3729: 3697: 3622: 3595: 3549: 2961: 2896: 2835: 2723: 2722:{\displaystyle f(x)=0} 2688: 2579: 2219: 2107: 2053: 2010: 1795: 1574: 1359: 1181: 1001: 889: 773: 714: 529: 346: 6837: 6817: 6752: 6683: 6644: 6581: 6463:at every point where 6205: 6202:{\displaystyle a: --> 6177: 6157: 6137: 6117: 6090: 6058: 6024: 5976: 5891: 5862: 5836: 5816: 5784: 5307: 4867: 4364: 3730: 3698: 3623: 3621:{\displaystyle x_{3}} 3596: 3550: 2962: 2897: 2836: 2737:if the derivative of 2724: 2689: 2580: 2220: 2108: 2054: 2011: 1796: 1575: 1360: 1182: 1002: 890: 774: 715: 530: 347: 36:level-5 vital article 6835:{\displaystyle \pi } 6826: 6806: 6692: 6653: 6590: 6565: 6539:Jacobian determinant 6449:The Jacobian matrix 6187: 6166: 6146: 6126: 6099: 6067: 6041: 6002: 5903: 5871: 5845: 5825: 5793: 5737: 5527:WP:Original research 5359:Reversible Jump MCMC 5275: 4379: 3876: 3707: 3654: 3605: 3562: 3506: 2912: 2852: 2794: 2698: 2658: 2289: 2227:no-original-research 2188: 2066: 2026: 1807: 1596: 1378: 1254: 1036: 918: 816: 731: 555: 370: 304: 132:mathematics articles 6531:Jacobian conjecture 6084: 6056:{\displaystyle n=1} 5860:{\displaystyle x=0} 5757: 5523:WP:secondary source 4424: 3961: 3118:× [0,2π) × [0,π) → 3019:ObsessiveMathsFreak 2606:Vanishing Jacobians 2184:Your proposal with 903:is the dimension)? 214:ObsessiveMathsFreak 6832: 6812: 6747: 6746: 6678: 6639: 6576: 6504:Hooman Mallahzadeh 6469:is differentiable. 6309:of a random vector 6199: 6172: 6152: 6132: 6112: 6085: 6070: 6053: 6019: 5971: 5886: 5857: 5831: 5811: 5779: 5740: 5302: 4862: 4837: 4820: 4748: 4713: 4704: 4695: 4684: 4630: 4576: 4548: 4503: 4481: 4463: 4410: 4398: 4359: 4334: 4317: 4245: 4210: 4156: 4102: 4081: 4072: 4063: 4045: 4000: 3947: 3935: 3917: 3895: 3725: 3693: 3618: 3591: 3545: 3015:heaviside function 2957: 2892: 2831: 2719: 2684: 2575: 2569: 2565: 2522: 2465: 2422: 2371: 2367: 2329: 2215: 2103: 2049: 2006: 2000: 1791: 1785: 1582:Later in the same 1570: 1564: 1355: 1343: 1177: 997: 885: 769: 710: 703: 656: 606: 562: 525: 518: 471: 421: 377: 342: 234:Wile E. Heresiarch 101:Mathematics portal 45:content assessment 6728: 6633: 6519:substitution rule 6354: 6306:covariance matrix 6246:differential form 6175:{\displaystyle b} 6155:{\displaystyle a} 6135:{\displaystyle S} 6035:differential form 6012: 5964: 5956: 5834:{\displaystyle x} 5705: 5669:And more sources: 5646: 5563: 5551:comment added by 5519:WP:primary source 5508: 5496:comment added by 5481: 5469:comment added by 5439: 5427:comment added by 5247:secondary sources 5182:comment added by 5168: 5154:comment added by 5140:List of Jacobians 4911:Order of material 3832:Nice try, Vandal 3798:comment added by 3634:comment added by 3283:Multiple integral 3266:comment added by 3182:comment added by 3169: 3152:comment added by 3079:comment added by 2821: 2766:comment added by 2564: 2521: 2464: 2421: 2366: 2328: 1244:In this wiki the 1175: 995: 934: 797:of that system. - 166: 165: 162: 161: 158: 157: 6908: 6841: 6839: 6838: 6833: 6821: 6819: 6818: 6813: 6756: 6754: 6753: 6748: 6738: 6730: 6729: 6721: 6715: 6707: 6699: 6687: 6685: 6684: 6679: 6677: 6676: 6667: 6666: 6665: 6648: 6646: 6645: 6640: 6638: 6634: 6632: 6631: 6630: 6629: 6615: 6614: 6613: 6612: 6598: 6585: 6583: 6582: 6577: 6572: 6497:Title of article 6480: 6474: 6468: 6462: 6429:total derivative 6425:total derivative 6421:total derivative 6402:Fantasticawesome 6341: 6210: 6207: 6206: 6200: 6181: 6179: 6178: 6173: 6161: 6159: 6158: 6153: 6141: 6139: 6138: 6133: 6121: 6119: 6118: 6113: 6111: 6110: 6094: 6092: 6091: 6086: 6083: 6078: 6062: 6060: 6059: 6054: 6031:Lebesgue measure 6028: 6026: 6025: 6020: 6011: 5980: 5978: 5977: 5972: 5963: 5955: 5936: 5935: 5895: 5893: 5892: 5887: 5866: 5864: 5863: 5858: 5840: 5838: 5837: 5832: 5820: 5818: 5817: 5812: 5788: 5786: 5785: 5780: 5756: 5748: 5703: 5644: 5311: 5309: 5308: 5303: 5194: 5167: 5148: 5086:near the point. 4984: 4931:Why matlab-code? 4890: 4871: 4869: 4868: 4863: 4858: 4857: 4848: 4847: 4838: 4825: 4824: 4817: 4816: 4805: 4804: 4772: 4771: 4753: 4752: 4745: 4744: 4733: 4732: 4714: 4705: 4696: 4685: 4680: 4679: 4670: 4669: 4651: 4650: 4631: 4626: 4625: 4616: 4615: 4597: 4596: 4577: 4575: 4574: 4549: 4545: 4544: 4535: 4534: 4518: 4517: 4504: 4502: 4501: 4482: 4480: 4479: 4464: 4459: 4458: 4449: 4448: 4423: 4418: 4399: 4397: 4396: 4368: 4366: 4365: 4360: 4355: 4354: 4345: 4344: 4335: 4322: 4321: 4314: 4313: 4302: 4301: 4269: 4268: 4250: 4249: 4242: 4241: 4230: 4229: 4211: 4206: 4205: 4196: 4195: 4177: 4176: 4157: 4152: 4151: 4142: 4141: 4123: 4122: 4103: 4101: 4100: 4082: 4073: 4064: 4046: 4042: 4041: 4032: 4031: 4015: 4014: 4001: 3996: 3995: 3986: 3985: 3960: 3955: 3936: 3934: 3933: 3918: 3916: 3915: 3896: 3894: 3893: 3869:with components 3810: 3734: 3732: 3731: 3726: 3702: 3700: 3699: 3694: 3692: 3691: 3679: 3678: 3666: 3665: 3646: 3627: 3625: 3624: 3619: 3617: 3616: 3600: 3598: 3597: 3592: 3587: 3586: 3574: 3573: 3554: 3552: 3551: 3546: 3544: 3543: 3531: 3530: 3518: 3517: 3350: 3320: 3306: 3278: 3194: 3168: 3146: 3091: 2966: 2964: 2963: 2958: 2901: 2899: 2898: 2893: 2879: 2878: 2840: 2838: 2837: 2832: 2827: 2822: 2817: 2812: 2811: 2778: 2743:stationary point 2728: 2726: 2725: 2720: 2693: 2691: 2690: 2685: 2668: 2584: 2582: 2581: 2576: 2574: 2573: 2566: 2563: 2562: 2561: 2548: 2547: 2546: 2533: 2523: 2520: 2519: 2518: 2505: 2504: 2503: 2490: 2466: 2463: 2462: 2461: 2448: 2447: 2446: 2433: 2423: 2420: 2419: 2418: 2405: 2404: 2403: 2390: 2376: 2375: 2368: 2365: 2364: 2363: 2350: 2349: 2340: 2330: 2327: 2326: 2325: 2312: 2311: 2302: 2224: 2222: 2221: 2216: 2208: 2203: 2202: 2112: 2110: 2109: 2104: 2099: 2098: 2086: 2081: 2080: 2058: 2056: 2055: 2050: 2039: 2015: 2013: 2012: 2007: 2005: 2004: 1997: 1996: 1984: 1979: 1978: 1959: 1958: 1946: 1941: 1940: 1913: 1912: 1900: 1895: 1894: 1875: 1874: 1862: 1857: 1856: 1833: 1832: 1820: 1800: 1798: 1797: 1792: 1790: 1789: 1782: 1781: 1769: 1764: 1763: 1744: 1743: 1731: 1726: 1725: 1702: 1701: 1689: 1684: 1683: 1664: 1663: 1651: 1646: 1645: 1616: 1611: 1610: 1579: 1577: 1576: 1571: 1569: 1568: 1561: 1560: 1548: 1543: 1542: 1523: 1522: 1510: 1505: 1504: 1477: 1476: 1464: 1459: 1458: 1439: 1438: 1426: 1421: 1420: 1391: 1364: 1362: 1361: 1356: 1354: 1353: 1348: 1347: 1340: 1339: 1327: 1309: 1308: 1296: 1267: 1186: 1184: 1183: 1178: 1176: 1174: 1173: 1172: 1159: 1155: 1147: 1146: 1133: 1119: 1111: 1097: 1089: 1081: 1067: 1059: 1051: 1050: 1006: 1004: 1003: 998: 996: 994: 993: 972: 971: 950: 945: 944: 939: 935: 927: 894: 892: 891: 886: 884: 883: 879: 867: 852: 851: 842: 828: 827: 778: 776: 775: 770: 765: 764: 746: 745: 719: 717: 716: 711: 534: 532: 531: 526: 355:a "basis" of an 351: 349: 348: 343: 338: 337: 319: 318: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 6916: 6915: 6911: 6910: 6909: 6907: 6906: 6905: 6856: 6855: 6824: 6823: 6822:goes from 0 to 6804: 6803: 6793: 6774: 6690: 6689: 6668: 6656: 6651: 6650: 6621: 6616: 6604: 6599: 6593: 6588: 6587: 6563: 6562: 6559: 6535:Jacobian matrix 6499: 6478: 6472: 6464: 6458: 6417: 6392: 6365: 6344:Boris Tsirelson 6329:Boris Tsirelson 6313:Jacobian matrix 6279:Boris Tsirelson 6270: 6250:Green's theorem 6213:Boris Tsirelson 6184: 6183: 6164: 6163: 6144: 6143: 6124: 6123: 6102: 6097: 6096: 6065: 6064: 6039: 6038: 6000: 5999: 5909: 5901: 5900: 5869: 5868: 5843: 5842: 5823: 5822: 5791: 5790: 5735: 5734: 5724: 5708:Boris Tsirelson 5682:Boris Tsirelson 5653:Boris Tsirelson 5618:Boris Tsirelson 5596:Boris Tsirelson 5487: 5461: 5419: 5383: 5332:Enneper surface 5314:Boris Tsirelson 5273: 5272: 5228:Boris Tsirelson 5220: 5200: 5177: 5174: 5149: 5142: 5088:Boris Tsirelson 5023:Boris Tsirelson 4964:—Ben FrantzDale 4952: 4950:Diagram request 4933: 4913: 4880: 4849: 4839: 4819: 4818: 4808: 4806: 4796: 4793: 4792: 4787: 4777: 4763: 4747: 4746: 4736: 4734: 4724: 4722: 4716: 4715: 4706: 4697: 4687: 4686: 4671: 4661: 4642: 4632: 4617: 4607: 4588: 4578: 4566: 4554: 4547: 4546: 4536: 4526: 4519: 4509: 4506: 4505: 4493: 4483: 4471: 4466: 4465: 4450: 4440: 4400: 4388: 4377: 4376: 4346: 4336: 4316: 4315: 4305: 4303: 4293: 4290: 4289: 4284: 4274: 4260: 4244: 4243: 4233: 4231: 4221: 4219: 4213: 4212: 4197: 4187: 4168: 4158: 4143: 4133: 4114: 4104: 4092: 4084: 4083: 4074: 4065: 4051: 4044: 4043: 4033: 4023: 4016: 4006: 4003: 4002: 3987: 3977: 3937: 3925: 3920: 3919: 3907: 3897: 3885: 3874: 3873: 3852: 3793: 3790: 3767: 3705: 3704: 3683: 3670: 3657: 3652: 3651: 3629: 3608: 3603: 3602: 3578: 3565: 3560: 3559: 3556: 3535: 3522: 3509: 3504: 3503: 3463: 3454: 3431: 3420: 3395:), we get then 3338: 3308: 3294: 3291:contravariantly 3268:130.245.246.244 3261: 3184:130.245.246.244 3177: 3175:User: Anonymous 3147: 3074: 3056: 2987: 2985:Is It Possible? 2910: 2909: 2870: 2850: 2849: 2803: 2792: 2791: 2784: 2761: 2696: 2695: 2661: 2656: 2655: 2638: 2608: 2568: 2567: 2553: 2549: 2538: 2534: 2529: 2524: 2510: 2506: 2495: 2491: 2485: 2484: 2479: 2474: 2468: 2467: 2453: 2449: 2438: 2434: 2429: 2424: 2410: 2406: 2395: 2391: 2381: 2370: 2369: 2355: 2351: 2341: 2336: 2331: 2317: 2313: 2303: 2293: 2287: 2286: 2194: 2186: 2185: 2090: 2072: 2064: 2063: 2024: 2023: 1999: 1998: 1988: 1970: 1965: 1960: 1950: 1932: 1926: 1925: 1923: 1921: 1915: 1914: 1904: 1886: 1881: 1876: 1866: 1848: 1838: 1824: 1805: 1804: 1784: 1783: 1773: 1755: 1750: 1745: 1735: 1717: 1711: 1710: 1704: 1703: 1693: 1675: 1670: 1665: 1655: 1637: 1627: 1602: 1594: 1593: 1584:Jacobian matrix 1563: 1562: 1552: 1534: 1529: 1524: 1514: 1496: 1490: 1489: 1487: 1485: 1479: 1478: 1468: 1450: 1445: 1440: 1430: 1412: 1402: 1376: 1375: 1370:Jacobian matrix 1342: 1341: 1331: 1315: 1310: 1300: 1279: 1277: 1252: 1251: 1242: 1201: 1191:—Ben FrantzDale 1164: 1160: 1138: 1134: 1039: 1034: 1033: 1019: 1017:Tensor Product? 973: 951: 922: 921: 916: 915: 913: 905:142.177.126.230 862: 843: 819: 814: 813: 782: 756: 737: 729: 728: 553: 552: 368: 367: 329: 310: 302: 301: 257:Merriam Webster 229: 190: 174: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 6914: 6912: 6904: 6903: 6898: 6893: 6888: 6883: 6878: 6873: 6868: 6858: 6857: 6831: 6811: 6792: 6789: 6773: 6770: 6744: 6741: 6737: 6733: 6727: 6724: 6718: 6714: 6710: 6706: 6702: 6698: 6675: 6671: 6664: 6659: 6637: 6628: 6624: 6619: 6611: 6607: 6602: 6596: 6575: 6571: 6558: 6555: 6554: 6553: 6498: 6495: 6494: 6493: 6433:130.243.94.123 6416: 6413: 6391: 6388: 6364: 6361: 6360: 6359: 6358: 6357: 6356: 6355: 6339: 6269: 6266: 6265: 6264: 6241: 6240: 6224: 6223: 6198: 6195: 6192: 6171: 6151: 6131: 6109: 6105: 6082: 6077: 6073: 6052: 6049: 6046: 6018: 6015: 6010: 6007: 5985:Tashiro~enwiki 5970: 5967: 5962: 5959: 5954: 5951: 5948: 5945: 5942: 5939: 5934: 5931: 5928: 5925: 5922: 5919: 5916: 5912: 5908: 5885: 5882: 5879: 5876: 5856: 5853: 5850: 5830: 5810: 5807: 5804: 5801: 5798: 5778: 5775: 5772: 5769: 5766: 5763: 5760: 5755: 5752: 5747: 5743: 5723: 5720: 5719: 5718: 5695: 5694: 5693: 5692: 5679: 5676: 5673: 5670: 5667: 5666: 5665: 5664: 5663: 5606: 5584: 5583: 5568: 5542: 5541: 5486: 5483: 5471:24.206.154.127 5460: 5457: 5456: 5455: 5429:71.227.186.180 5418: 5415: 5414: 5413: 5382: 5379: 5378: 5377: 5351: 5347: 5346: 5327: 5326: 5325: 5324: 5301: 5298: 5295: 5292: 5289: 5286: 5283: 5280: 5266: 5265: 5243: 5219: 5216: 5199: 5196: 5184:79.183.148.124 5173: 5170: 5141: 5138: 5137: 5136: 5135: 5134: 5133: 5132: 5131: 5130: 5129: 5128: 5105: 5104: 5103: 5102: 5101: 5100: 5099: 5098: 5061: 5060: 5059: 5058: 5057: 5056: 5036: 5035: 5034: 5033: 5016: 5015: 5000: 4985: 4978: 4951: 4948: 4932: 4929: 4912: 4909: 4908: 4907: 4885:comment added 4873: 4872: 4861: 4856: 4852: 4846: 4842: 4836: 4833: 4828: 4823: 4815: 4811: 4807: 4803: 4799: 4795: 4794: 4791: 4788: 4786: 4783: 4782: 4780: 4775: 4770: 4766: 4762: 4759: 4756: 4751: 4743: 4739: 4735: 4731: 4727: 4723: 4721: 4718: 4717: 4712: 4707: 4703: 4698: 4694: 4689: 4688: 4683: 4678: 4674: 4668: 4664: 4660: 4657: 4654: 4649: 4645: 4641: 4638: 4633: 4629: 4624: 4620: 4614: 4610: 4606: 4603: 4600: 4595: 4591: 4587: 4584: 4579: 4573: 4569: 4565: 4560: 4559: 4557: 4552: 4543: 4539: 4533: 4529: 4525: 4522: 4520: 4516: 4512: 4508: 4507: 4500: 4496: 4492: 4489: 4484: 4478: 4474: 4468: 4467: 4462: 4457: 4453: 4447: 4443: 4439: 4436: 4433: 4430: 4427: 4422: 4417: 4413: 4409: 4406: 4401: 4395: 4391: 4385: 4384: 4370: 4369: 4358: 4353: 4349: 4343: 4339: 4333: 4330: 4325: 4320: 4312: 4308: 4304: 4300: 4296: 4292: 4291: 4288: 4285: 4283: 4280: 4279: 4277: 4272: 4267: 4263: 4259: 4256: 4253: 4248: 4240: 4236: 4232: 4228: 4224: 4220: 4218: 4215: 4214: 4209: 4204: 4200: 4194: 4190: 4186: 4183: 4180: 4175: 4171: 4167: 4164: 4159: 4155: 4150: 4146: 4140: 4136: 4132: 4129: 4126: 4121: 4117: 4113: 4110: 4105: 4099: 4095: 4091: 4086: 4085: 4080: 4075: 4071: 4066: 4062: 4057: 4056: 4054: 4049: 4040: 4036: 4030: 4026: 4022: 4019: 4017: 4013: 4009: 4005: 4004: 3999: 3994: 3990: 3984: 3980: 3976: 3973: 3970: 3967: 3964: 3959: 3954: 3950: 3946: 3943: 3938: 3932: 3928: 3922: 3921: 3914: 3910: 3906: 3903: 3898: 3892: 3888: 3882: 3881: 3851: 3848: 3847: 3846: 3845: 3844: 3834:206.207.225.66 3827: 3826: 3816:85.210.133.129 3789: 3786: 3766: 3763: 3748: 3747: 3724: 3721: 3718: 3715: 3712: 3690: 3686: 3682: 3677: 3673: 3669: 3664: 3660: 3615: 3611: 3590: 3585: 3581: 3577: 3572: 3568: 3555: 3542: 3538: 3534: 3529: 3525: 3521: 3516: 3512: 3500: 3499: 3498: 3462: 3459: 3453: 3450: 3449: 3448: 3447: 3446: 3445: 3444: 3429: 3418: 3307:to a function 3132:24.218.141.183 3055: 3052: 3051: 3050: 3040:85.210.133.129 3035: 3030: 3029: 2986: 2983: 2968: 2967: 2956: 2953: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2926: 2923: 2920: 2917: 2903: 2902: 2891: 2888: 2885: 2882: 2877: 2873: 2869: 2866: 2863: 2860: 2857: 2843: 2842: 2830: 2826: 2820: 2815: 2810: 2806: 2802: 2799: 2783: 2780: 2768:128.12.100.219 2758: 2757: 2718: 2715: 2712: 2709: 2706: 2703: 2683: 2680: 2677: 2674: 2671: 2667: 2664: 2642:Ashi Starshade 2637: 2634: 2607: 2604: 2603: 2602: 2587: 2586: 2585: 2572: 2560: 2556: 2552: 2545: 2541: 2537: 2530: 2528: 2525: 2517: 2513: 2509: 2502: 2498: 2494: 2487: 2486: 2483: 2480: 2478: 2475: 2473: 2470: 2469: 2460: 2456: 2452: 2445: 2441: 2437: 2430: 2428: 2425: 2417: 2413: 2409: 2402: 2398: 2394: 2387: 2386: 2384: 2379: 2374: 2362: 2358: 2354: 2348: 2344: 2337: 2335: 2332: 2324: 2320: 2316: 2310: 2306: 2299: 2298: 2296: 2279: 2277: 2276: 2275: 2274: 2273: 2272: 2258: 2244: 2243: 2242: 2241: 2214: 2211: 2207: 2201: 2197: 2193: 2182: 2171: 2170: 2155: 2154: 2144: 2143: 2102: 2097: 2093: 2089: 2085: 2079: 2075: 2071: 2048: 2045: 2042: 2038: 2034: 2031: 2003: 1995: 1991: 1987: 1983: 1977: 1973: 1969: 1966: 1964: 1961: 1957: 1953: 1949: 1945: 1939: 1935: 1931: 1928: 1927: 1924: 1922: 1920: 1917: 1916: 1911: 1907: 1903: 1899: 1893: 1889: 1885: 1882: 1880: 1877: 1873: 1869: 1865: 1861: 1855: 1851: 1847: 1844: 1843: 1841: 1836: 1831: 1827: 1823: 1819: 1815: 1812: 1788: 1780: 1776: 1772: 1768: 1762: 1758: 1754: 1751: 1749: 1746: 1742: 1738: 1734: 1730: 1724: 1720: 1716: 1713: 1712: 1709: 1706: 1705: 1700: 1696: 1692: 1688: 1682: 1678: 1674: 1671: 1669: 1666: 1662: 1658: 1654: 1650: 1644: 1640: 1636: 1633: 1632: 1630: 1625: 1622: 1619: 1615: 1609: 1605: 1601: 1567: 1559: 1555: 1551: 1547: 1541: 1537: 1533: 1530: 1528: 1525: 1521: 1517: 1513: 1509: 1503: 1499: 1495: 1492: 1491: 1488: 1486: 1484: 1481: 1480: 1475: 1471: 1467: 1463: 1457: 1453: 1449: 1446: 1444: 1441: 1437: 1433: 1429: 1425: 1419: 1415: 1411: 1408: 1407: 1405: 1400: 1397: 1394: 1390: 1386: 1383: 1352: 1346: 1338: 1334: 1330: 1326: 1322: 1319: 1316: 1314: 1311: 1307: 1303: 1299: 1295: 1291: 1288: 1285: 1284: 1282: 1276: 1273: 1270: 1266: 1262: 1259: 1241: 1238: 1237: 1236: 1200: 1197: 1188: 1187: 1171: 1167: 1163: 1158: 1154: 1150: 1145: 1141: 1137: 1131: 1128: 1125: 1122: 1118: 1114: 1110: 1106: 1103: 1100: 1096: 1092: 1088: 1084: 1080: 1076: 1073: 1070: 1066: 1062: 1058: 1054: 1049: 1046: 1042: 1023:tensor product 1018: 1015: 1010: 1008: 992: 989: 986: 983: 980: 976: 970: 967: 964: 961: 958: 954: 948: 943: 938: 933: 930: 925: 912: 909: 882: 878: 874: 871: 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5910: 5906: 5897: 5883: 5880: 5877: 5874: 5854: 5851: 5848: 5841:axis between 5828: 5805: 5799: 5796: 5776: 5773: 5767: 5761: 5758: 5753: 5750: 5745: 5741: 5733:For example, 5731: 5728: 5721: 5717: 5713: 5709: 5701: 5700: 5699: 5691: 5687: 5683: 5680: 5677: 5674: 5671: 5668: 5662: 5658: 5654: 5650: 5642: 5641: 5640: 5637: 5634: 5629: 5628: 5627: 5623: 5619: 5615: 5614:Reyer Sjamaar 5611: 5607: 5605: 5601: 5597: 5593: 5588: 5587: 5586: 5585: 5582: 5578: 5574: 5569: 5566: 5565: 5564: 5562: 5558: 5554: 5550: 5540: 5536: 5532: 5528: 5524: 5520: 5516: 5511: 5510: 5509: 5507: 5503: 5499: 5495: 5484: 5482: 5480: 5476: 5472: 5468: 5458: 5454: 5450: 5446: 5442: 5441: 5440: 5438: 5434: 5430: 5426: 5416: 5412: 5408: 5404: 5400: 5399: 5398: 5397: 5393: 5389: 5380: 5376: 5372: 5368: 5364: 5360: 5356: 5355:Robot Control 5352: 5349: 5348: 5345: 5341: 5337: 5333: 5329: 5328: 5323: 5319: 5315: 5296: 5293: 5290: 5284: 5281: 5278: 5270: 5269: 5268: 5267: 5264: 5260: 5256: 5252: 5251:WP:notability 5248: 5244: 5240: 5239: 5238: 5237: 5233: 5229: 5225: 5217: 5215: 5214: 5210: 5206: 5197: 5195: 5193: 5189: 5185: 5181: 5171: 5169: 5165: 5161: 5157: 5153: 5145: 5139: 5127: 5123: 5119: 5115: 5114: 5113: 5112: 5111: 5110: 5109: 5108: 5107: 5106: 5097: 5093: 5089: 5085: 5081: 5077: 5073: 5069: 5068: 5067: 5066: 5065: 5064: 5063: 5062: 5055: 5051: 5047: 5042: 5041: 5040: 5039: 5038: 5037: 5032: 5028: 5024: 5020: 5019: 5018: 5017: 5014: 5010: 5006: 5001: 4998: 4994: 4990: 4986: 4983: 4979: 4976: 4975: 4974: 4973: 4969: 4965: 4961: 4957: 4949: 4947: 4946: 4942: 4938: 4930: 4928: 4927: 4923: 4919: 4918:Paul Matthews 4910: 4906: 4902: 4898: 4897:Paul Matthews 4893: 4892: 4891: 4888: 4884: 4879: 4859: 4854: 4850: 4844: 4840: 4834: 4831: 4826: 4821: 4813: 4809: 4801: 4797: 4789: 4784: 4778: 4773: 4768: 4764: 4760: 4757: 4754: 4749: 4741: 4737: 4729: 4725: 4719: 4710: 4701: 4692: 4676: 4672: 4666: 4662: 4655: 4652: 4647: 4643: 4639: 4636: 4622: 4618: 4612: 4608: 4601: 4598: 4593: 4589: 4585: 4582: 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3719: 3716: 3713: 3710: 3688: 3684: 3680: 3675: 3671: 3667: 3662: 3658: 3649: 3648: 3647: 3645: 3641: 3637: 3636:129.11.76.230 3633: 3613: 3609: 3588: 3583: 3579: 3575: 3570: 3566: 3540: 3536: 3532: 3527: 3523: 3519: 3514: 3510: 3497: 3493: 3489: 3485: 3480: 3479: 3478: 3477: 3473: 3469: 3468:ThinkerFeeler 3460: 3458: 3451: 3443: 3440: 3436: 3432: 3425: 3421: 3414: 3410: 3406: 3402: 3398: 3394: 3390: 3386: 3382: 3378: 3374: 3370: 3366: 3362: 3358: 3354: 3349: 3345: 3341: 3336: 3332: 3328: 3324: 3319: 3315: 3311: 3305: 3301: 3297: 3292: 3288: 3284: 3280: 3279: 3277: 3273: 3269: 3265: 3259: 3255: 3251: 3247: 3243: 3239: 3235: 3231: 3227: 3223: 3219: 3215: 3210: 3209: 3208: 3205: 3201: 3197: 3196: 3195: 3193: 3189: 3185: 3181: 3176: 3170: 3167: 3163: 3159: 3155: 3151: 3142: 3141: 3137: 3133: 3129: 3123: 3121: 3117: 3113: 3107: 3106: 3102: 3098: 3092: 3090: 3086: 3082: 3081:71.249.100.27 3078: 3070: 3069: 3065: 3061: 3053: 3049: 3045: 3041: 3036: 3032: 3031: 3028: 3024: 3020: 3016: 3013:Let H be the 3012: 3011: 3010: 3009: 3005: 3001: 3000:JeepAssembler 2997: 2993: 2992:JeepAssembler 2984: 2982: 2981: 2977: 2973: 2951: 2948: 2942: 2939: 2936: 2933: 2927: 2924: 2921: 2915: 2908: 2907: 2906: 2889: 2886: 2875: 2871: 2864: 2858: 2855: 2848: 2847: 2846: 2828: 2824: 2818: 2813: 2808: 2804: 2800: 2797: 2789: 2788: 2787: 2781: 2779: 2777: 2773: 2769: 2765: 2756: 2752: 2748: 2744: 2740: 2736: 2732: 2716: 2713: 2707: 2701: 2678: 2672: 2669: 2665: 2662: 2653: 2649: 2648: 2647: 2646: 2643: 2635: 2633: 2632: 2628: 2624: 2623:84.75.181.226 2619: 2618: 2615: 2611: 2605: 2601: 2597: 2593: 2588: 2570: 2558: 2554: 2543: 2539: 2526: 2515: 2511: 2500: 2496: 2481: 2476: 2471: 2458: 2454: 2443: 2439: 2426: 2415: 2411: 2400: 2396: 2382: 2377: 2372: 2360: 2356: 2333: 2322: 2318: 2294: 2285: 2284: 2282: 2281: 2280: 2271: 2267: 2263: 2262:92.34.134.152 2259: 2257: 2254: 2250: 2249: 2248: 2247: 2246: 2245: 2240: 2236: 2232: 2228: 2212: 2205: 2195: 2183: 2179: 2175: 2174: 2173: 2172: 2169: 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1143: 1139: 1129: 1123: 1104: 1074: 1047: 1044: 1040: 1032: 1031: 1030: 1028: 1027:grad operator 1024: 1016: 1014: 1013: 987: 984: 981: 965: 962: 959: 946: 941: 936: 931: 928: 923: 908: 906: 902: 898: 880: 876: 872: 869: 856: 848: 844: 832: 824: 820: 811: 807: 801: 800: 796: 790: 789: 784: 761: 757: 753: 750: 747: 742: 738: 725: 707: 696: 693: 690: 687: 684: 681: 678: 675: 672: 669: 666: 663: 660: 652: 649: 646: 640: 637: 634: 631: 628: 625: 622: 619: 616: 613: 610: 602: 596: 593: 590: 587: 584: 581: 578: 575: 572: 569: 566: 551: 550: 549: 546: 543: 541: 540:Michael Hardy 522: 511: 508: 505: 502: 499: 496: 493: 490: 487: 484: 481: 478: 475: 467: 464: 461: 455: 452: 449: 446: 443: 440: 437: 434: 431: 428: 425: 417: 411: 408: 405: 402: 399: 396: 393: 390: 387: 384: 381: 366: 365: 364: 362: 358: 334: 330: 326: 323: 320: 315: 311: 300: 299: 298: 295: 293: 283: 279: 275: 274:84.75.181.226 270: 269: 267: 265: 262: 258: 254: 253: 252: 250: 249:Michael Hardy 245: 243: 237: 235: 226: 224: 223: 219: 215: 210: 206: 204: 198: 194: 187: 185: 183: 178: 171: 169: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 6801: 6794: 6775: 6759:Barçaforlife 6560: 6500: 6465: 6459: 6455:differential 6424: 6418: 6400: 6396: 6393: 6374: 6369: 6366: 6325:(non-random) 6324: 6320: 6316: 6308: 6301: 6271: 5983: 5898: 5732: 5729: 5725: 5696: 5591: 5547:— Preceding 5543: 5492:— Preceding 5488: 5465:— Preceding 5462: 5445:92.19.54.161 5423:— Preceding 5420: 5384: 5221: 5201: 5178:— Preceding 5175: 5150:— Preceding 5146: 5143: 5118:Blacklemon67 5083: 5079: 5075: 5071: 5046:Blacklemon67 5005:Blacklemon67 4996: 4992: 4988: 4959: 4955: 4953: 4934: 4914: 4874: 4371: 3866: 3862: 3858: 3856: 3853: 3791: 3772: 3768: 3765:Applications 3749: 3737:Jitse Niesen 3557: 3488:Jitse Niesen 3483: 3464: 3455: 3434: 3427: 3423: 3416: 3412: 3408: 3404: 3400: 3396: 3392: 3388: 3384: 3380: 3376: 3372: 3368: 3364: 3360: 3356: 3352: 3347: 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5989:talk 5867:and 5712:talk 5686:talk 5657:talk 5622:talk 5600:talk 5592:some 5577:talk 5557:talk 5535:talk 5502:talk 5475:talk 5449:talk 5433:talk 5407:talk 5392:talk 5371:talk 5340:talk 5318:talk 5259:talk 5232:talk 5209:talk 5205:Doug 5188:talk 5160:talk 5122:talk 5092:talk 5050:talk 5027:talk 5009:talk 4968:talk 4941:talk 4922:talk 4901:talk 3838:talk 3820:talk 3804:talk 3778:talk 3756:talk 3741:talk 3640:talk 3601:and 3492:talk 3472:talk 3363:) = 3272:talk 3252:sin 3188:talk 3158:talk 3136:talk 3101:talk 3085:talk 3064:talk 3044:talk 3023:talk 3004:talk 2996:talk 2976:talk 2772:talk 2751:talk 2627:talk 2596:talk 2266:talk 2235:talk 2137:talk 1368:The 1219:? -- 812:iff 545:--- 278:talk 218:talk 6799:). 6457:of 5797:sin 5759:sin 5647:in 5645::-) 5612:by 5363:EoM 5224:EoM 5078:to 4995:to 4653:cos 4599:cos 4432:sin 4179:cos 4125:cos 3969:sin 3415:) | 3355:by 2940:sin 2856:sin 2798:sin 2694:if 2062:d) 2022:c) 1803:b) 1592:a) 1215:or 895:is 808:is 203:not 142:Mid 6862:: 6850:) 6830:π 6810:θ 6784:) 6765:) 6740:→ 6726:→ 6717:→ 6701:→ 6658:∇ 6618:∂ 6601:∂ 6549:) 6541:. 6510:) 6489:) 6481:. 6479:}} 6473:{{ 6439:) 6408:) 6383:) 6350:) 6335:) 6285:) 6260:) 6252:. 6236:) 6219:) 6211:. 6203:b} 6104:∫ 6072:∫ 5991:) 5930:∈ 5911:∫ 5907:∫ 5884:π 5800:⁡ 5762:⁡ 5754:π 5742:∫ 5714:) 5688:) 5659:) 5624:) 5602:) 5579:) 5559:) 5537:) 5504:) 5477:) 5451:) 5435:) 5409:) 5394:) 5373:) 5342:) 5320:) 5261:) 5253:. 5234:) 5226:. 5211:) 5190:) 5166:) 5162:• 5124:) 5094:) 5074:: 5052:) 5029:) 5011:) 5003:-- 4991:: 4970:) 4943:) 4924:) 4903:) 4835:40 4774:⋅ 4656:⁡ 4637:− 4602:⁡ 4583:− 4551:⟹ 4435:⁡ 4426:− 4332:40 4329:− 4271:⋅ 4255:− 4182:⁡ 4163:− 4128:⁡ 4109:− 4048:⟹ 3972:⁡ 3963:− 3865:→ 3861:: 3840:) 3822:) 3806:) 3780:) 3758:) 3743:) 3723:θ 3717:ϕ 3642:) 3494:) 3474:) 3424:dx 3422:| 3407:= 3405:dy 3387:= 3346:→ 3342:: 3316:→ 3312:: 3302:→ 3298:: 3274:) 3244:, 3240:, 3232:, 3228:, 3220:, 3216:, 3190:) 3164:) 3160:• 3138:) 3114:: 3103:) 3087:) 3066:) 3046:) 3025:) 3006:) 2978:) 2970:? 2952:θ 2943:⁡ 2928:θ 2876:∘ 2872:45 2859:⁡ 2809:∘ 2805:45 2801:⁡ 2774:) 2753:) 2629:) 2598:) 2551:∂ 2536:∂ 2527:⋯ 2508:∂ 2493:∂ 2482:⋮ 2477:⋱ 2472:⋮ 2451:∂ 2436:∂ 2427:⋯ 2408:∂ 2393:∂ 2353:∂ 2343:∂ 2334:⋯ 2315:∂ 2305:∂ 2268:) 2237:) 2210:∂ 2200:⊤ 2192:∂ 2139:) 2096:⊤ 2088:∂ 2078:⊤ 2070:∂ 2041:∂ 2030:∂ 1986:∂ 1968:∂ 1963:⋯ 1948:∂ 1930:∂ 1919:⋮ 1902:∂ 1884:∂ 1879:⋯ 1864:∂ 1846:∂ 1830:⊤ 1822:∂ 1811:∂ 1771:∂ 1753:∂ 1748:⋯ 1733:∂ 1715:∂ 1708:⋮ 1691:∂ 1673:∂ 1668:⋯ 1653:∂ 1635:∂ 1618:∂ 1608:⊤ 1600:∂ 1550:∂ 1532:∂ 1527:⋯ 1512:∂ 1494:∂ 1483:⋮ 1466:∂ 1448:∂ 1443:⋯ 1428:∂ 1410:∂ 1393:∂ 1382:∂ 1365:. 1351:⊤ 1329:∂ 1318:∂ 1313:⋯ 1298:∂ 1287:∂ 1269:∂ 1258:∂ 1162:∂ 1136:∂ 1127:∇ 1124:⊗ 975:∂ 953:∂ 870:− 751:… 685:… 650:… 635:… 591:… 500:… 465:… 450:… 406:… 324:… 280:) 220:) 6846:( 6780:( 6761:( 6743:f 6736:f 6732:, 6723:u 6713:u 6709:, 6705:R 6697:R 6674:i 6670:f 6663:T 6636:) 6627:j 6623:x 6610:i 6606:f 6595:( 6574:) 6570:f 6545:( 6506:( 6485:( 6466:f 6460:f 6435:( 6404:( 6379:( 6346:( 6331:( 6281:( 6256:( 6232:( 6215:( 6197:b 6191:a 6170:b 6150:a 6130:S 6108:S 6081:b 6076:a 6051:1 6048:= 6045:n 6017:y 6014:d 6009:x 6006:d 5987:( 5969:y 5966:d 5961:x 5958:d 5953:) 5950:y 5947:, 5944:x 5941:( 5938:f 5933:S 5927:) 5924:y 5921:, 5918:x 5915:( 5881:2 5878:= 5875:x 5855:0 5852:= 5849:x 5829:x 5809:) 5806:x 5803:( 5777:x 5774:d 5771:) 5768:x 5765:( 5751:2 5746:0 5710:( 5684:( 5655:( 5620:( 5598:( 5575:( 5555:( 5533:( 5500:( 5473:( 5447:( 5431:( 5405:( 5390:( 5369:( 5338:( 5316:( 5300:) 5297:y 5294:, 5291:x 5288:( 5285:f 5282:= 5279:z 5257:( 5230:( 5207:( 5186:( 5158:( 5120:( 5090:( 5084:F 5080:R 5076:R 5072:F 5048:( 5025:( 5007:( 4997:R 4993:R 4989:f 4966:( 4960:R 4956:R 4939:( 4920:( 4899:( 4889:. 4860:. 4855:2 4851:x 4845:1 4841:x 4832:+ 4827:= 4822:| 4814:2 4810:x 4802:3 4798:x 4790:0 4785:5 4779:| 4769:1 4765:x 4761:8 4758:+ 4755:= 4750:| 4742:2 4738:x 4730:3 4726:x 4720:0 4711:0 4702:5 4693:0 4682:) 4677:3 4673:x 4667:2 4663:x 4659:( 4648:2 4644:x 4640:2 4628:) 4623:3 4619:x 4613:2 4609:x 4605:( 4594:3 4590:x 4586:2 4572:1 4568:x 4564:8 4556:| 4542:3 4538:x 4532:2 4528:x 4524:= 4515:3 4511:y 4499:2 4495:x 4491:5 4488:= 4477:1 4473:y 4461:) 4456:3 4452:x 4446:2 4442:x 4438:( 4429:2 4421:2 4416:1 4412:x 4408:4 4405:= 4394:2 4390:y 4357:. 4352:2 4348:x 4342:1 4338:x 4324:= 4319:| 4311:2 4307:x 4299:3 4295:x 4287:0 4282:5 4276:| 4266:1 4262:x 4258:8 4252:= 4247:| 4239:2 4235:x 4227:3 4223:x 4217:0 4208:) 4203:3 4199:x 4193:2 4189:x 4185:( 4174:2 4170:x 4166:2 4154:) 4149:3 4145:x 4139:2 4135:x 4131:( 4120:3 4116:x 4112:2 4098:1 4094:x 4090:8 4079:0 4070:5 4061:0 4053:| 4039:3 4035:x 4029:2 4025:x 4021:= 4012:3 4008:y 3998:) 3993:3 3989:x 3983:2 3979:x 3975:( 3966:2 3958:2 3953:1 3949:x 3945:4 3942:= 3931:2 3927:y 3913:2 3909:x 3905:5 3902:= 3891:1 3887:y 3867:R 3863:R 3859:F 3836:( 3818:( 3802:( 3776:( 3754:( 3739:( 3720:, 3714:, 3711:r 3689:3 3685:x 3681:, 3676:2 3672:x 3668:, 3663:1 3659:x 3638:( 3614:3 3610:x 3589:, 3584:2 3580:x 3576:, 3571:1 3567:x 3541:3 3537:x 3533:, 3528:2 3524:x 3520:, 3515:1 3511:x 3490:( 3484:p 3470:( 3430:T 3428:J 3419:T 3417:J 3413:x 3411:( 3409:g 3403:) 3401:y 3399:( 3397:f 3393:x 3391:( 3389:T 3385:y 3381:y 3377:x 3373:x 3371:( 3369:T 3367:( 3365:f 3361:x 3359:( 3357:g 3353:g 3348:D 3344:E 3340:T 3331:T 3327:D 3323:E 3318:R 3314:E 3310:g 3304:R 3300:D 3296:f 3270:( 3254:θ 3250:r 3246:φ 3242:θ 3238:r 3234:z 3230:y 3226:x 3222:z 3218:y 3214:x 3200:F 3186:( 3156:( 3134:( 3128:z 3120:R 3116:R 3112:F 3099:( 3083:( 3062:( 3042:( 3021:( 3002:( 2994:( 2974:( 2955:) 2949:2 2946:( 2937:r 2934:= 2931:) 2925:, 2922:r 2919:( 2916:f 2890:1 2887:= 2884:) 2881:) 2868:( 2865:2 2862:( 2841:. 2829:2 2825:/ 2819:2 2814:= 2770:( 2749:( 2739:f 2735:f 2731:x 2717:0 2714:= 2711:) 2708:x 2705:( 2702:f 2682:) 2679:x 2676:( 2673:f 2670:= 2666:′ 2663:x 2652:x 2625:( 2594:( 2571:] 2559:n 2555:x 2544:m 2540:f 2516:1 2512:x 2501:m 2497:f 2459:n 2455:x 2444:1 2440:f 2416:1 2412:x 2401:1 2397:f 2383:[ 2378:= 2373:] 2361:n 2357:x 2347:f 2323:1 2319:x 2309:f 2295:[ 2264:( 2233:( 2213:x 2206:/ 2196:f 2135:( 2101:= 2092:x 2084:/ 2074:f 2047:= 2044:x 2037:/ 2033:f 2002:] 1994:n 1990:x 1982:/ 1976:m 1972:f 1956:1 1952:x 1944:/ 1938:m 1934:f 1910:n 1906:x 1898:/ 1892:1 1888:f 1872:1 1868:x 1860:/ 1854:1 1850:f 1840:[ 1835:= 1826:x 1818:/ 1814:f 1787:] 1779:n 1775:x 1767:/ 1761:m 1757:f 1741:n 1737:x 1729:/ 1723:1 1719:f 1699:1 1695:x 1687:/ 1681:m 1677:f 1661:1 1657:x 1649:/ 1643:1 1639:f 1629:[ 1624:= 1621:x 1614:/ 1604:f 1566:] 1558:n 1554:x 1546:/ 1540:m 1536:f 1520:1 1516:x 1508:/ 1502:m 1498:f 1474:n 1470:x 1462:/ 1456:1 1452:f 1436:1 1432:x 1424:/ 1418:1 1414:f 1404:[ 1399:= 1396:x 1389:/ 1385:f 1345:] 1337:n 1333:x 1325:/ 1321:f 1306:1 1302:x 1294:/ 1290:f 1281:[ 1275:= 1272:x 1265:/ 1261:f 1170:j 1166:x 1157:) 1153:x 1149:( 1144:i 1140:f 1130:= 1121:) 1117:x 1113:( 1109:f 1105:= 1102:) 1099:) 1095:x 1091:( 1087:f 1083:( 1079:J 1075:= 1072:) 1069:) 1065:x 1061:( 1057:f 1053:( 1048:j 1045:i 1041:J 991:) 988:g 985:, 982:y 979:( 969:) 966:g 963:, 960:f 957:( 947:= 942:g 937:) 932:y 929:f 924:( 901:n 881:n 877:/ 873:1 865:| 860:) 857:x 854:( 849:f 845:J 840:| 836:) 833:x 830:( 825:f 821:J 806:f 781:1 767:) 762:n 758:x 754:, 748:, 743:1 739:x 735:( 708:. 705:} 700:) 697:1 694:, 691:0 688:, 682:, 679:0 676:, 673:0 670:, 667:0 664:, 661:0 658:( 653:, 647:, 644:) 641:0 638:, 632:, 629:0 626:, 623:0 620:, 617:1 614:, 611:0 608:( 603:, 600:) 597:0 594:, 588:, 585:0 582:, 579:0 576:, 573:0 570:, 567:1 564:( 559:{ 523:. 520:} 515:) 512:1 509:, 506:0 503:, 497:, 494:0 491:, 488:0 485:, 482:0 479:, 476:0 473:( 468:, 462:, 459:) 456:0 453:, 447:, 444:0 441:, 438:0 435:, 432:1 429:, 426:0 423:( 418:, 415:) 412:0 409:, 403:, 400:0 397:, 394:0 391:, 388:0 385:, 382:1 379:( 374:{ 357:n 340:) 335:n 331:x 327:, 321:, 316:1 312:x 308:( 276:( 216:( 154:. 53::

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