Talk:Curl

From Wikipedia, the free encyclopedia

WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class Mid Priority  Field: Analysis
One of the 500 most frequently viewed mathematics articles.

I was hoping to find the formula for curl in polar coordinates... cylindrical coordinates to be precise. <sigh> moink 02:10, 16 Jan 2004 (UTC)

Hooray for <math>Math markup!</math>


My understanding is that cul is actually a tensor, with n * (n-1) / 2 free components. It only comes out nice in 3 dimensions because 3*2/2 = 3.

Contents

[edit] invariance

not only does curl give a pseudovector depending if the (orthogonal) coordinate system is left or right-handed but it is clearly invariant for an arbitrary coordinate system. It would be nice if this fact was linked to the field of differential geometry and tensor calculus in which the curl operation has a far more natural counterpart. m3n0

[edit] Gilbert Strang and Curl

Gilbert Strang, in Introduction to Applied Mathematics introduces the Curl as

\begin{pmatrix} 0 & {-\frac{\partial}{\partial z}} & {\frac{\partial}{\partial y}} \\  \\
{\frac{\partial}{\partial z}} & 0 & {-\frac{\partial}{\partial x}} \\
 \\  {-\frac{\partial}{\partial y}} & {\frac{\partial}{\partial x}} & 0 \end{pmatrix}

\begin{pmatrix} F_x \\  \\
F_y \\
 \\  F_z \end{pmatrix}

A few problems are worked out using this notation for the curl before Strang introduces the more convenient method of the cross product of del and F. I'm not sure where this notation comes from or how (if at all) it's better, so I didn't want to include it on the page. --- Trevie 17:07, 23 September 2005 (UTC)

It looks like this is just another longer way of representing

\begin{pmatrix}
{\frac{\partial F_z}{\partial y}} - {\frac{\partial F_y}{\partial z}} \\  \\
{\frac{\partial F_x}{\partial z}} - {\frac{\partial F_z}{\partial x}}\\  \\
{\frac{\partial F_y}{\partial x}} - {\frac{\partial F_x}{\partial y}}
\end{pmatrix}

--- Trevie 18:56, 23 September 2005 (UTC)

I've seen that notation for general cross products. A dynamics book I have uses it a lot. Basically, it turns the cross from a binary operator to a unary operator on the first vector. That is

\vec{v}^\times = \begin{pmatrix} 0 & -v_3 & v_2 \\
v_3 & 0 & -v_1 \\
-v_2 & v_1 & 0 \end{pmatrix}

so that you get

 \vec{v} \times \vec{u}=\vec{v}^\times \vec{u} which is a simple matrix-vector multiplication. The author liked it because he could use expressions like  \vec{w}^\times \vec{w}^\times in things like the expressions for transforming between frames of reference, and they became meaningful. moink 08:18, 26 September 2005 (UTC)

[edit] Relation to Divergence

I think there are a few things that could be added to this article, they may be trivial but:

 -The curl of a gradient of a scalar function is always zero (equivalently
the curl of a conservative function is zero)
 -The divergence of the curl is zero
 -Green's Theorem can be expressed as a double integral of curl dot-producted with kdA

--205.188.116.195 00:31, 14 December 2005 (UTC) Duke of Worcestershire

I think some of that is already at del. There is also Lagrange's formula. I would encourage you to indeed add some of the stuff you suggested, and also where appropriate refer to the two articles above. (It is OK if what you add to this article is already covered in the two, a bit of duplication does not hurt as long as it is connected to the matter at hand). Oleg Alexandrov (talk) 01:42, 14 December 2005 (UTC)


[edit] Rot

I think there should be reference to the fact "rotation" is also used as a synonym for curl, with operator rot(·). —DIV

I agree. --Vladimír Fuka 22:32, 29 September 2006 (UTC)

[edit] Related to curvature?

Is this related to the curvature of a function? - SigmaEpsilonΣΕ 11:51, 7 October 2006 (UTC)

I don't think it is. --BenWhitey 06:24, 14 January 2007 (UTC)

[edit] n dimensions

Is there a way to define curl in n dimensions? unsigned

I would think not. And even if one could define it, it would be some kind of mathematical artifact without any useful physical interpretations or properties. But I am not sure. Oleg Alexandrov (talk) 03:55, 31 October 2006 (UTC)
I think this page should point out clearly that curl only applies in three dimensions, in contrast to gradient and divergence, which apply in any dimension. It is possible to define analogous operations to curl in higher dimensions. In two dimensions we have two operations: gradient and divergence, and applying divergence to gradient gives you zero. In three dimensions, gradient, curl and divergence - and applying two in sequence gives you zero. In four dimensions, there are four operations, e.g. gradient, "twist", "spin", and divergence - and applying two in sequence gives you zero, In five dimensions, five operations and so on. While gradient turns a scalar field into a vector field, curl turns a vector field into a vector field, and divergence turns a vector field into a scalar field, "twist" turns a vector field into a skew-symmetric matrix field, and spin turns a skew-symmetric matrix field into a vector field. The number of degrees of freedom in the field outputs follow Pascal's triangle: in 2 dimensions, the gradient takes a scalar field (1 degree of freedom) to a vector field (2 degrees) which is what divergence takes to a scalar field (1). In three dimensions, gradient goes from (1) to (3) which curl takes to (3) which divergence takes to (1). In four dimensions, you get 1->4->6->4->1 because a 4x4 skew-symmetric matrix has 6 degrees of freedom. Thus is Pascal's triangle spelled out. If I knew tensor calculus, I bet there would be a neater way to say this (and you need them to go above 5 dimensions here). Qseep 06:17, 7 December 2006 (UTC)

[edit] what is curl, really?

This article should explain what curl is in terms of geometry so that people can understand what it is. This is how my professor explained it. You take a pinwheel and put it into a river parallel to the bottom. The river is the flow (flux) of a vector field. The pinwheel is the device that you are using to calculate the curl. Depending on whether the river is flowing faster on one part of the pinwheel than the other then it will either spin or not spin. That is what curl is. Does that make sense? --BenWhitey 06:27, 14 January 2007 (UTC)

Curl is how much rotation there is at a designated point in a vector field. Its length gives the amount of rotation. Its direction gives the direction of the rotation using the convention of the right-hand rule.

--Loodog 22:30, 22 May 2007 (UTC)

I think an example like this would be useful to readers, but it needs to be better explained. Whether or not the wheel will spin seems to be very directly a matter of torque (and is more indirectly related to curl). The thing is, curl has no use (that I know of) unless you're talking about vector fields. Then, all it tells you is how curvy the field lines are (generally or evaluated at a specific point) and which way they curve. In your example (or rather your professor's), the river, which can be regarded as a force field, doesn't curve at all. The torque doesn't either. What does curve is (among others), the acceleration field of the wheel. If we define a cylindrical coordinate system with the z-axis through the center of the wheel, and ρ- and φ-axes in the plane of the wheel, then:
\vec{a} = \alpha \rho \hat{\phi} and \nabla \times \vec a = \frac{1}{\rho} \frac{\partial}{\partial \rho} [\rho (\alpha \rho) ] \hat{z} = 2 \alpha \hat{z}
This tells you that the acceleration field curves constantly at , which you can then use to relate to torque or something. In fact, showing that \tau = \frac{mr^2}{2} \frac{ \nabla \times \vec{a}(\rho) }{ 2 } = \frac{r^2}{4} \nabla \times \vec{f}(\rho) (for a more or less uniform wheel) is straight-forward. I think that stating the example more like this would be better than simply stating that curl determines whether the wheel will or will not spin. Any other thoughts on this?

--Misho88 (talk) 07:15, 20 December 2007 (UTC)

[edit] Examples

I added some sorely needed mathematical exmples. Feel free to tweak with the format if you have a better aesthetic sense than I.--Loodog 03:28, 23 May 2007 (UTC)

[edit] Curl in 2D, a scalar ?

The article begins stating "In vector calculus, curl is a vector operator ...". In the article however, without saying that explicitely, everything is in 3D. I have the feeling that there should be at least a 2D equivalent: Imagine a vector valued function R3 − > R3 where z'=0 everywhere. If we now look into the z=0 plane and have a rotation in that plane, we will get a (3D) curl vector in z direction.

No, you're crazy.--Loodog 23:44, 5 November 2007 (UTC)

Example: F(x,y,z) = (-y, x, 0)

Now, obviously, the interesting things happen in the xy planes (e.g. z=0). If we forget about the z-dimension because the result does not depend on z (i.e. consider our function R2 − > R2), we might ask, how much in-plane rotation we have in 2D, which should be a scalar like dfy/dx-dfx/dy. This embeds into 3D and we always have a curl vector (0,0,x) where x is the 2D curl. By the way, this is the same as the simple example in the text. Such 2D vector fields appear on surfaces or in image processing. So what about 2D curl definition ?

Anyway, the article would benefit from a clear statement about the assumed 3D dimensionality.89.53.7.199 23:39, 5 November 2007 (UTC)

Hi. Points:
  1. Curl for two-dimensional vector fields is already defined, and it's given an orientation that requires three dimensions, and with good reason.
  2. If you want a scalar to represent the curl of a two-dimensional vector field, why commit original research and invent bizarre new math when you can just use:
(\vec{\nabla} \times \vec{F}) \cdot  \boldsymbol{\hat{z}}

--Loodog 19:21, 6 November 2007 (UTC)

This is exactly what I was saying, although I am not used to thinking in \vec{\nabla}-notation. Sorry if I was unclear. I am not proposing anything or inventing bizarre math, but I am looking for a 2D curl definition... I dont have a function F:R3 − > R3 but a function F:R2 − > R2. The definition on the wikipedia curl page uses the cross product which doesnt exist in two dimensions, so it implies at least 3 dimensions, which, however, isnt stated anywhere. It's perhaps obvious to all those people who know what curl is. On the other hand people always tend to think "SOMETHING is in 3D", because they always work with SOMETHING in 3D and do not understand the math behind it. So basically you're saying that curl is an intrinsically 3D phenomenon and you need "artificial" 3D embedding (which results in a 3D curl vector = (0,0,dy/dx-dx/dy) ) to define it for 2D  ?
What do you mean with "Curl for two-dimensional vector fields is already defined". Where ? This is what I am looking for... --134.245.253.140 17:44, 7 November 2007 (UTC)
Take a look at the examples on the page. All of those given are 2D vector fields. To talk about the direction of their curl, you need another dimension perpendicular to those. E.g. Do time lapse photography on a second hand as it runs around the clock. Take direction of the velocity at various points in time to be the vector field in question. Obviously, the direction of the curl will be clockwise. How are you going to draw the vector representing "clockwise" on the 2D face of a clock? You don't. You define, by convention, clockwise curl to be out of the clock coming towards you, requiring a third dimension.--Loodog 19:24, 7 November 2007 (UTC)
I have already seen the examples, all of these are handled in 3D. The basic difference between rotation in 2D and 3D is, that in 3D rotation is around an axis (with a direction), while in 2D rotation is around a point. Therefore the direction is a concept of 3D and makes no sense in 2D. Here the only relevant thing is the angle. In 4D curl might be an entity with even much higher dimension. Your example makes sense and so do the examples on the curl page, but they are biased towards 3D and do not give a 2d definition. The meaning of divergence and curl in the 2D plane explains what I was looking for:curl is a scalar (dfy/dx-dfx/dy) in 2D. Nevertheless, thanks for your feedback ! --89.53.33.139 22:20, 7 November 2007 (UTC)
Look. Curl is a mapping: R3->R3. In the event that your domain is R2, you still get a mapping into R3 since R2 is a subset of R3. Curl of a 2D field is a 3D vector confined to one dimension perpendicular to the plane, but can have dependence on x and y. Curl of a 3D field is a vector in 3D that can have dependence on x, y, and z. Curl of a 4D field has no meaning. I don't know what else you want.--Loodog 23:57, 7 November 2007 (UTC)

[edit] smoothness conditions?

I presume that some smoothness (e.g. must be a differentiable, or perhaps continuously differentiable, field??) conditions are required so that the two limits used for the curl definition (one using the surface, and the other using the path, integral) can (a) exist; or (b) be equal to each other. I guess such conditions must be required e.g. on the vector field itself, whose curl is being defined, and (perhaps less important to know) on the set of surfaces of the volumes used to produce the limit as V -> 0. Can anyone who knows add brief inserts on these requirement? If necessary and sufficient conditions are too long, complex and distracting then maybe some well-known sufficient conditions would be nice instead. E.g. is it enough to guarantee existence and equivalence of curl defined in these ways that the vector field should be differentiable (in the sense of a function between Banach spaces - is this called Frechet - forgotten)? Thanks in advance.

[edit] Notation Issue

In the section where the curl operator is represented by Einstein notation, shouldn't one of the l's and one of the m's be superscripts not subscripts? Isn't that how the summation part works?--SurrealWarrior 00:04, 1 December 2007 (UTC)

No. All Einstein Notation says is to not explicitly write summation signs when you have a repeated index in term since it's redundant.--Loodog 06:45, 1 December 2007 (UTC)

[edit] Deleted a Section

I stopped by this site to check out what info it had on the curl, and found that someone mistakenly offered the definition of the divergence as one definition of the curl, so I deleted it, leaving the "other" definition, which was correct. —Preceding unsigned comment added by 128.95.43.202 (talk) 05:32, 8 February 2008 (UTC)

[edit] CURL IS NOT DIVERGENCE

Perhaps, this section was here again. The article started from the "definition of curl", which is actually the Divergence Theorem, and the picture was wrong as well. Check it:


// I deleted it

A cross-sectional demonstration of curl as a surface integral for a simple vector field.  F in black, n in blue, and n  F in red
A cross-sectional demonstration of curl as a surface integral for a simple vector field. F in black, n in blue, and n \times F in red

The curl of a vector field \mathbf{F} is defined as the limit of the ratio of the surface integral of the cross product of \mathbf{F} with the normal \mathbf{n} of closed surface S(2), over a closed surface S^{(2)}=\partial V^{(3)}, the boundary to the volume V(3) enclosed by the surface S(2), as the volume goes to zero:

 \operatorname{curl}(\mathbf{F}) = \lim_{V^{(3)} \to 0} \frac{1}{|V^{(3)}|} \,\iint_{\partial V^{(3)}}\mathbf{n}\times\mathbf{F}\,dS^{(2)}

More precisely, at each point p in three-dimensional space, \operatorname{curl}(\mathbf{F})(p) is given by the above limit, where the closed surfaces \partial V^{(3)} all enclose p and the diameter, not just the volume, of the enclosed region tends to zero.

This definition is rather complex and not very useful, and the following alternative equivalent definition gives better measures to calculate components of \operatorname{curl}(\mathbf{F}).


Commentor (talk) 03:15, 9 March 2008 (UTC)


[edit] Definition

I think that curl should be defined in the standard classical way in Cartesian coordinates (which is now in the second section). The current definition has a few issues. Foremost amongst these is that each component of curl(F) is only well defined modulo a factor of +/- 1. Aside from this however, it is also unclear what is meant by S^2 -> 0. I agree that the equivalence of this type of definition to the classical definition should be discussed because the current one does make the rotational nature of the curl apparent, but this should be reserved for a later section.

On another issue, I think that the relationship between curl and the exterior derivative should be given (which naturally shows that curl can be defined in a coordinate independent way).

Holmansf (talk) 05:09, 20 May 2008 (UTC)

User, please post new talk topics at the bottom.--Loodog (talk) 01:45, 21 May 2008 (UTC)