How to read the Jacobian (determinant) shorthand notation, and why is it so cryptic?
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?
1 Answer
$\begingroup$It is a little awkward, but using the partial derivative notation with vectors basically means:
$$\begin{align}\dfrac{\partial (a, b, c)}{\partial (x, y, z)} ~=~& \det\left((\dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z})^\top(a,b,c) \right)^\top \\[1ex] =~& \begin{vmatrix}\dfrac{\partial a}{\partial x}&\dfrac{\partial a}{\partial y}&\dfrac{\partial a}{\partial z}\\ \dfrac{\partial b}{\partial x}&\dfrac{\partial b}{\partial y}&\dfrac{\partial b}{\partial z}\\\dfrac{\partial c}{\partial x}&\dfrac{\partial c}{\partial y}&\dfrac{\partial c}{\partial z}\end{vmatrix} \end{align}$$
The notation summarises the essentials of the Jacobian determinant. You are taking the partial derivatives of $a, b, c$ each with respect to $x,y,z$, constructing a matrix of the result and evaluating its determinant.
$$\dfrac{\partial (a, b, c)}{\partial (x, y, z)}$$
$\endgroup$ 1
