Talk:Jacobian
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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, Wile E. Heresiarch 21:28, 23 Mar 2004 (UTC)
Isnt' there an error in the simplification of the determinant given as example? Where is the term x3 cos(x1) gone? -- looxix 00:32 Mar 24, 2003 (UTC)
I've always heard the initial consonant pronounced as a affricate, as in "John". Michael Hardy 01:23 Mar 24, 2003 (UTC)
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 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. Kevin Baas 2003.03.26
Why in the world do you call
- <math>(x_1,\dots,x_n)<math>
a "basis" of an n-space? A basis would be something like this:
- <math>\{\,(1,0,0,0,\dots,0),\,(0,1,0,0,\dots,0),\dots,\,(0,0,0,0,\dots,0,1)\,\}.<math>
(By the way, the Latin phrase "per se" doesn't have an "a" or a "y" in it.) Michael Hardy 20:36 Mar 26, 2003 (UTC)
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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:
- <math>\{\,(1,0,0,0,\dots,0),\,(0,1,0,0,\dots,0),\dots,\,(0,0,0,0,\dots,0,1)\,\}.<math>
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 the same as (1,0,0) in another system.
Thus, when it is said that <math>(x_1,\dots,x_n)<math> is a basis, i interpret this as saying that x1, 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.
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. -Kevin Baas
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, and are based off of the eigenvalues of that system. -Kevin Baas
Would it be correct to say f is conformal iff <math>J_f(x)|J_f(x)|^{-1/n}<math> is orthogonal (where n is the dimension)? 142.177.126.230 23:49, 4 Aug 2004 (UTC)