Lets say we have a function $f : \mathbb{R}^3\rightarrow \mathbb{R}^3$, as defined below, with its value being denoted as $(a, b, c)$ for convenient reference.
$$f(x,y,z) = (x^2, y^2, z^2) = (a, b, c)$$
The Jacobian matrix of $f$ and subsequently the Jacobian determinant would then be:
$$ \begin{bmatrix} a'\\ b'\\ c' \end{bmatrix} = \begin{bmatrix} \frac {\partial a}{\partial x} & \frac {\partial a}{\partial y} & \frac {\partial a}{\partial z}\\ \frac {\partial b}{\partial x} &\frac {\partial b}{\partial y} & \frac {\partial b}{\partial z}\\ \frac {\partial c}{\partial x} &\frac {\partial c}{\partial y} & \frac {\partial c}{\partial z} \end{bmatrix} = \begin{bmatrix} 2x & 0 &0 \\ 0 & 2y &0 \\ 0 & 0 &2z \end{bmatrix} $$
$$ \begin{vmatrix} a'\\ b'\\ c' \end{vmatrix} = \begin{vmatrix} \frac {\partial a}{\partial x} & \frac {\partial a}{\partial y} & \frac {\partial a}{\partial z}\\ \frac {\partial b}{\partial x} &\frac {\partial b}{\partial y} & \frac {\partial b}{\partial z}\\ \frac {\partial c}{\partial x} &\frac {\partial c}{\partial y} & \frac {\partial c}{\partial z} \end{vmatrix} = \begin{vmatrix} 2x & 0 &0 \\ 0 & 2y &0 \\ 0 & 0 &2z \end{vmatrix} = 2x2y2z $$ Ok sure, this makes sense. It's kind of just like normal calculus but expanding everything out into a matrix.
Now I look at the shorthand notation for the Jacobian determinant: $$ \frac {\partial(a,b,c)}{\partial(x,y,z)} = 2x2y2z $$ Where did this even come from? Why are there partials there. How does it convey the same amount of information? How do I even read this "shorthand notation". It just seems so left field - out of nowhere.
- How do I read it
- How did this arguably rather cryptic notation come about?