# What exactly does $\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$ refer to?

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.

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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. Jul 25 '11 at 5:35
Wolfram MathWorld seems confused as to what exactly the notation means too. See mathworld.wolfram.com/Jacobian.html, equations (4), (7) and (10). –  Josh Chen Jul 25 '11 at 6:13
Well at the very least, I see that equation three there uses brackets and not vertical bars... –  Ｊ. Ｍ. Jul 25 '11 at 6:15
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 8:54
I'm pretty sure Do Carmo uses it as Jacobian determinant in his book on Riemannian geometry. –  Simon Jul 25 '11 at 10:08

## 1 Answer

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.

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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 Chen Jul 25 '11 at 23:07