Talk:Gradient

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1 The recent move
2 Jargon heavy?
3 Question
4 Gradient
5 Question about wikibook on calculus
6 Should gradient be a disambigutation page?
7 Gradient as normal
8 The Gradient in other coordinate systems

[edit] The recent move

I disagree with gradient having been moved to gradient (calculus) (and making gradient into disambiguation) for three reasons:

It was done without prior consultation on the talk page.
The links pointing to gradient were not fixed to point to the new page.
There is little need for a disambiguation page.
It looks to me that the meaning of gradient as vector field is by far more dominant than the meaning of the gradient as slope. And either way, the second meaning is also mathematical, in spite of what the created disambiguation page at gradient seemed to imply.

gradient (calculus) is is now moved back to gradient. Discussion welcome. Oleg Alexandrov 23:41, 14 Jun 2005 (UTC)

I made a gradient (disambiguation) page, listing the two meanings. Oleg Alexandrov 23:52, 14 Jun 2005 (UTC)

[edit] Jargon heavy?

Right now this article is far too jargon heavy and i can't make any sense of it. In lower levels of maths at least gradiant reffers to slope that is the m in y=mx+c or more generally the value of dy/dx. Plugwash 01:47, 20 July 2005 (UTC)

I added a sentence in the introduction saying that the gradient becomes the derivative in 1D. Now anybody could follow the derivative article to read more about that. Is that better, or you want more? :) Oleg Alexandrov 01:57, 20 July 2005 (UTC)

thats certainly better though i think we should get rid of the disabig and bring it all together in one article since the concepts are so closely related. Plugwash 18:13, 20 July 2005 (UTC)

I would not agree with that. In math the gradient is primarily exactly as described in the article (a vector field of partial derivatives), while what you mean by gradient is called slope. Also, most articles linkin to gradient do seem to mean the vector field definition. But this is an opinion. If you can do a good job at combining the things without being too biased towards the slope definition, then please go ahead. Oleg Alexandrov 18:38, 20 July 2005 (UTC)

mmm ok slope is a big article. lemme have a think about this. Plugwash 19:25, 20 July 2005 (UTC)

I don't agree with the recent changes. I believe the article now does a poor job at explaining what the gradient is. All this because there are two gradients, one is the vector field, the other is a number. They are of course related, via the dot product, but they are not the same. That's why I think it is better to have a disambiguation page explaining the similarities and differences, and keep this page only for the main meaning, which is the vector field. Oleg Alexandrov 00:54, 22 July 2005 (UTC)

You keep claiming that the vector calculus meaning is the main meaning yet almost everyone has heared and uses the word gradiant (at least here in britan, americans seem to prefer grade) yet only a comparitively small number will have even heared of vector calculus. Plugwash 02:15, 22 July 2005 (UTC)

Fair enough. At least now we agree that two meanings are involved. I understand that most people in Britan think of gradient as a slope, however, very few of the articles linking here mean that by gradient (I checked around 20 randomly chosen ones, maybe you are willing to do some research yourself and let me know what you think).

I believe one can write a short note mentioning the British usage, then link to grade (geography), which is the article describing the meaning as you want it.

Either way, I find the first paragraph in the text very clumsily written. Oleg Alexandrov 03:10, 22 July 2005 (UTC)

to me at least the vector calculus definition seems a strictening up of the laymans definition. That is it defines that the gradiant is measured in the direction where it is greatest and defines that the direction of greatest slope is stored as the angle part of the gradiant. Plugwash 13:02, 22 July 2005 (UTC)

Plugwash, good point. I still think however that the introduction is too clumsy now. I would like the first paragraph be split in two, as too many things are there. I will look at it later today. Oleg Alexandrov 16:06, 22 July 2005 (UTC)

Yeah, it's a little problematic that in British English, what we use "gradient" for "slope", and "gradient vector" for what is described in this article... Enochlau 11:00, 8 November 2005 (UTC)

One way to settle this discussion is to find some references (published textbooks) on the subject where gradient is defined one way or the other. Given these references, it should be straightforward to present two or three definitions (a vector field, slope, the general case for higher level mathematics) and discuss their relations. Right now, I only see some argumentation without any backup in the established literature.--KYN 17:56, 17 December 2005 (UTC)

Note that as taught in British high schools, "gradient" means the tangent of the angle of inclination, not the sine. I.e. grade (geography) describes an ambiguity that doesn't exist in practice in the British usage. Also, the concept is taught in terms of the derivative of a function, not as being specific to geography. -- David Hopwood 82.42.16.20 17:19, 19 December 2005 (UTC)

[edit] Question

Shouldn't

$\nabla \phi = \begin{pmatrix} {\frac{\partial \phi}{\partial x}}, {\frac{\partial \phi}{\partial y}}, {\frac{\partial \phi}{\partial z}} \end{pmatrix}$

$\nabla \phi = \begin{pmatrix} {\frac{\partial \phi}{\partial x}}, {\frac{\partial \phi}{\partial y}}, {\frac{\partial \phi}{\partial z}} \end{pmatrix}^T$ ,

the gradient being (at least in engineering) usually thought as a column vector?

Andrea Censi 15:18, 22 March 2006 (UTC)

Does it matter that much if it is a row or column vector? I think this is a more general, vector calculus issue, and I would argue that it is not worth the trouble mentioning that a vector may be written both horizontally and vertically. Oleg Alexandrov (talk) 05:53, 23 March 2006 (UTC)

Well, the convention that it is used is that $\frac{\partial f}{\partial \mathbf{x}}$ and

$\nabla_\mathbf{x} f$ are each other's transpose, such that

$\frac{\partial f}{\partial x} = \begin{bmatrix} \frac{\partial f}{\partial {x_1}} & \dots & \frac{\partial f}{\partial {x_n}} \end{bmatrix}$

is a row vector. In fact this is formally important because so you can write

$d f = \frac{\partial f}{\partial x} dx$

and the multiplication is well defined ( first is a row, second is a column). And then it's better that the gradient being a column vector because it is of the same "kind" as $x$ . So I propose to write the following definition for gradient:

$\nabla_\mathbf{x} f := \left(\frac{\partial f}{\partial x}\right)^T = \begin{bmatrix} \frac{\partial f}{\partial {x_1}} \\ \vdots \\ \frac{\partial f}{\partial {x_n}} \end{bmatrix}$

So it's not a question that an object is "written" horizontally or vertically, but in calculus a row vector and a column vector are really of different type. This is not an "aestetic" point, but having consistency in such matters really makes calculation easier. In fact, I see that you used the same convention in Gradient_descent

Andrea Censi 17:43, 23 March 2006 (UTC)

The distinction is between contravariant vectors and covariant vectors or forms. The fact that a contra- and a covector can be contracted to a scalar which can be represented by matrix multiplication of the components arranged in a column and a row matrix, does not imply a preference for representing a (contra)vector as either row or cloumn matrix. That is just convention, it doesn't mean anything. For me the gradient of a function f is $\mathrm{d}f = \sum_i \frac{\partial f}{\partial c^i}\mathrm{d}c^i$ where c is any set of curvilinear coordinates. Only if I suppress the choice of coordinates do I need to choose between listing the components $\{\frac{\partial f}{\partial c^i}\}_i$ as a row or column matrix. --MarSch 11:47, 11 April 2006 (UTC)

[edit] Gradient

If you ask any mathematician to define gradient, the defintion that he gives will involve partial derivatives. The use of the word gradient to mean 'slope' results from the fact the slope is a special case of a gradient that is commonly encountered in lower level mathematics. Since mathematicians are forever generalizing, I believe that the gradient article should simply mention the word confusion and point out that the slope is a special case. This should take no more than two sentences. The article should describe the gradient in its full meaning. In that respect, the article is perhaps jargon-light as oppsed to jargon-heavy. --anon

That the slope is a special case is the second paragraph in the introduction. Should there be more than that? Oleg Alexandrov (talk) 23:40, 17 November 2005 (UTC)

And ask any photoshop user to define gradient you'll get a gradual blend. Needs a mention? ...dave souza, talk 16:06, 22 March 2006 (UTC)

I guess that could go in gradient (disambiguation), as this article seems to be about the vector calculus concept. Oleg Alexandrov (talk) 05:53, 23 March 2006 (UTC)

Which redirects to gradient. Is it worth a separate page? As an attempt, I'll just add a note here. The illustration shows nice circular and linear gradients..dave souza, talk 06:40, 23 March 2006 (UTC)

The redirect can be transformed in a disambig, if necessary.

The problem with the photoshop definition is that I don't see how it relates to the definition of gradient as slope, inclination. It seems to me a separate independent definition, and if this is so, it appears confusing inserted in the article where it is. That's when disambiguation pages are handy I think. Wonder what you think. Oleg Alexandrov (talk) 14:34, 23 March 2006 (UTC)

Good question, I've tried rephrasing the wording to clarify this, and also pointed out that gradient meaning road or surface slope is the usual usage in the UK. As this is the generic "gradient" page the various meanings should be outlined on it: of course it could be logical to move this page to Gradient (calculus) if it's to be confined to a particular mathematical meaning. ..dave souza, talk 18:39, 23 March 2006 (UTC)

I don't quite see how the "Photoshop gradient" implies the vector calculus gradient, so now there is a contradiction in paragraph 3. Oleg Alexandrov (talk) 04:34, 24 March 2006 (UTC)

Didn't mean to imply that, so have had another go at improving my phrasing. The difficulty is that vector calculus is over my head, but gradient is a term and concept I've often used in construction and imaging. Try thinking of photoshop and other graphics applications (vector as well as bitmap) as mathematical transformations on the values of pixels. This is what a transform layer in photoshop does: a gradient transform gradually changes values from one colour to another. Hope that helps. ..dave souza, talk 06:45, 24 March 2006 (UTC)

The "gradient" in Photoshop is a particularization. In fact, it means "produce an image whose gradient is fixed". For example, consider an image which goes from black to white from left to right. Let

f (x, y)

be the gray value at each pixel x,y. The gradient of this image is then

(1,0)

and is constant. If you set the gradient as

(0,1)

then it's black at the top and white at the bottom. if you set the gradient as

(1,1)

you get black at top-left and white at the bottom-right. So I think that this is the reason why this pattern was called "gradient" in photoshop. As for me, I'd mention that the other meanings are derived from the original meaning (calculus). Andrea Censi 10:26, 24 March 2006 (UTC)

Dear experts, how about a image gradient article, with pictures, references, and all that? And then we may link to it from gradient. Oleg Alexandrov (talk) 04:56, 25 March 2006 (UTC)

I agree Andrea Censi 15:49, 25 March 2006 (UTC)

Gradient is the logical page for disambiguating, and if a brief paragraph about the related graphics usage can't be accommodated then logically the calculus aspect should become Gradient (calculus), the graphics version could be Gradient (image) and the related term Grade also included on the disambig page. If you're desperate to keep this exotic page as is, then at the least it should have a For other uses disambiguation paragraph at the top. Mathematicians, please be logical. ...dave souza, talk 08:40, 26 March 2006 (UTC)

Yes, making multiple pages also seems a good solution. But please, Dave, please be clear to whom you are referring to as "mathematicians", because it could be an insult to some :-) Andrea Censi 09:31, 26 March 2006 (UTC)

[edit] Question about wikibook on calculus

I discovered that this concept is well covered on a Wikibook http://en.wikibooks.org/wiki/Calculus:Multivariable_calculus#Gradient_vectors - what is the wikipedia policy in this case, just copy the material, summarize it or link to it? Andrea Censi 16:08, 26 March 2006 (UTC)

[edit] Should gradient be a disambigutation page?

The current version of gradient is a bird of many feathers, part disambiguation page, and part the vector calculus gradient, which is a sorry mess.

There are two options now. Either

Make gradient into disambig, and move the vector calculus meaning to gradient (vector calculus)
Or use gradient (disambiguation) for the other meanings, and leave gradient for the vector calculus meaning.

I would be for the second, as it appears to me that most pages linking here are about the vector gradient or related, and the "Photoshop gradient" needs forking to its own article. Oleg Alexandrov (talk) 17:17, 26 March 2006 (UTC)

I favor number 2. I think the main use of the word is the mathematical use, and other uses are just variations on a theme. -lethe ^{talk +} 17:25, 26 March 2006 (UTC)

Yes there is a need for Gradient (image processing), it not just photoshop a very common concept, worthy of more extensive treatment. Slope does a better job of providing a discription of the concept for non mathematicians and this page is better for mathematicians. In the light of this I favor the first so readers can find the right article for them quickly. --Salix alba (talk) 21:04, 26 March 2006 (UTC)

Naturally I favour the first, as a non mathematician searching for "gradient" to explain a road sign or description of a river gets hit by university level equations that are worse than useless. Nice to know that the graphics usage, which I thought a minor aside, is worthy of an article: that emphasises the need for the first port of call to provide disambiguation. The number of articles linking here was reduced by one when I noticed Khyber Pass which, as suspected, was referring to what some (but not all) in the US call a grade. Yosemite National Park and Inclined plane also seem to expect that usage, to name but two. Obviously there's effort involved in changing the mathematical links to gradient (calculus) or whatever, but it's worth it to make Wikipedia more accessible, and I'm willing to work on it. Note that gradient (disambiguation) currently redirects to gradient. ...dave souza, talk 22:54, 26 March 2006 (UTC)

Are you willing to fix the links to point in the appropriate places? If so, I would also agree with making this a disambig. Oleg Alexandrov (talk) 00:13, 27 March 2006 (UTC)

That's what I'm offering to do, though some help with it would be welcome. ..dave souza, talk 12:35, 27 March 2006 (UTC)

As far as I'm concerned, the revision to this page of 02:56, 4 April 2006 by Oleg Alexandrov, and splitting off image gradient for the graphics term, meets the need for disambiguation on this page. ..dave souza, talk 09:53, 4 April 2006 (UTC)

It is wrong for the term gradient to go to Gradient (calculus). The best example of the term gradient, that is nearly universal, are the winds: caused by the gradient, the potential, the "slope of the curve(line) between the H (High pressure) and the L (Low pressure). All the weatherPeople out there, will never stop referring to the Pressure "gradient" between the High and Low pressure centers.--- MMcAnnis,YumaAZMmcannis 21:43, 8 April 2006 (UTC)

I just got to this page by Climax community: ...."species distributed themselves along nutrient and other envirnmental gradients." I was going to wikify the term Gradient, instead I have to say all this stuff. (and Not wikify the term;Alas.) Gradient is not just a term for Calculus.--- MMcAnnis,YumaAZMmcannis 21:43, 8 April 2006 (UTC)

As I stated, i wanted to wikify Gradient, but was unable to for obvious reasons:
..."species distributed themselves along nutrient and other envirnmental gradients." ....i cannot wiki this over to a: gradient (mathematics); it makes no sense,...Alas. ---I am trying to state my case, without, anyone guessing what I am trying to say.--(See Climax community)--MMcAnnis////Mmcannis 02:25, 9 April 2006 (UTC)

But the mathematical usage is exactly the intended (only?) usage, but "along" should probably be changed to "according to".--MarSch 11:23, 11 April 2006 (UTC)

[edit] Gradient as normal

Nowhere in the talk page did I find a discussion as to the fact that the gradient of a scalar field at a point is equal normal to the field at that point. I believe this is a great use of (vector) gradient and ought to be incorporated. Shall I go ahead and do so? -unsigned

What do you mean by "normal to a scalar field"? All I know is normal to a surface. Oleg Alexandrov (talk) 03:23, 16 August 2006 (UTC)

I have no idea what he was talking about. But I thought of something similar that might be mentioned in the article. The gradient is perpendicular to the level set. --Spoon! 23:30, 31 August 2006 (UTC)

[edit] The Gradient in other coordinate systems

I'm wondering if the expressions for the gradient in the other common 3D coordinate systems - such as spherical polar, and cylindrical coordinates - should be included in the article on the gradient? I think situations where these forms are convenient arise commonly enough that they should be at least mentioned and stated.

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