13

I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be...

Question: Does this notation $$\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$$ refer to the Jacobian matrix $$ J = \begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix},$$ or the Jacobian determinant $\det J$?

This answer seems to support the latter interpretation, while this (and Wikipedia) both support the former.

I am aware of the ambiguity of "Jacobian" being used to refer to either the determinant or the matrix itself, is this a similar case? It's really a bit annoying because when I see things like $$ \left| \frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)} \right| $$ I don't know if it means the absolute value of the Jacobian determinant, or the determinant of the Jacobian matrix.

Community
  • 1
Josh
  • 2,539
  • 2
    It's the matrix; that being said, $\left| \frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)} \right|$ is very unfortunate notation; I'd use $\det$ myself if I have to talk about the determinant in this context... – J. M. ain't a mathematician Jul 25 '11 at 05:35
  • Wolfram MathWorld seems confused as to what exactly the notation means too. See http://mathworld.wolfram.com/Jacobian.html, equations (4), (7) and (10). – Josh Jul 25 '11 at 06:13
  • Well at the very least, I see that equation three there uses brackets and not vertical bars... – J. M. ain't a mathematician Jul 25 '11 at 06:15
  • I'd say it's the matrix, whereas $\frac{d(y_1,\dots,y_m)}{d(x_1,\dots,x_n)}$ is the determinant. – Hans Lundmark Jul 25 '11 at 08:54
  • I'm pretty sure Do Carmo uses it as Jacobian determinant in his book on Riemannian geometry. – Simon Jul 25 '11 at 10:08
  • I would say determinant. But we see here it is ambiguous, so you should say what you mean when you use it. – GEdgar Jul 25 '11 at 13:35
  • @Hans: So the one with $\partial$ is the matrix while the one with $d$ is the determinant? *Goes woozy* – Josh Jul 25 '11 at 23:08
  • @GEdgar: Yeah I can see it's really ambiguous.. Just wanted to know which was the more common interpretation, if any. Thanks :) – Josh Jul 25 '11 at 23:10
  • To see which is more common, find it in a few textbooks. (Not Wikipedia.) I know of no textbook that has the notation mentioned by Hans. – GEdgar Jul 26 '11 at 02:14
  • @GEdgar: Funny. That notation is what I learned from a standard Swedish calculus book long ago, and I've never given it much thought since then. Maybe it's a local thing. I see now that Rudin's Principles, for example, uses $\partial$ for the determinant. – Hans Lundmark Jul 26 '11 at 08:04

1 Answers1

6

Every time that I use it and have seen it, I have used it to refer to the matrix itself. I will typically use det J to refer to the determinant, but I admit that I have used the term Jacobian to refer to both the matrix and the determinant.

This is another case where math is precise, but the language of math is often not. I would recommend that you take care to be specific in your usage - use Jacobian matrix and Jacobian determinant so that there is no confusion.

davidlowryduda
  • 91,687
  • 1
    Yes, I agree with using the specific phrases "Jacobian matrix" and "Jacobian determinant", but with regard to the "partial" notation? I suppose I should just state what I mean it to be at the outset? – Josh Jul 25 '11 at 23:07