Jacobian

Sometimes called "Jacobian Matrix", or "differential".

Common definition

Given a function

$$f:\mathbb{R}^m\rightarrow\mathbb{R}^n$$ (I use the convention of m first because it takes it from m to n) the:

• differential of $f$ at $x$, denoted $$df_x$$ or $$Df_x$$ which I prefer, as you often find $$df$$ in a fraction involving $$dx$$
• Jacobian matrix of $f$ at $x$ often denoted $$J_{f(x)}$$

Are given by:

$$Df_x:\mathbb{R}^m\rightarrow\mathbb{R}^n$$ , $$Df_x=\begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_m} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \cdots & \frac{\partial f_n}{\partial x_m} \end{pmatrix}$$

This is a n-by-m matrix using my convention.

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How to remember which way round this matrix goes

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I have trouble remembering this but it is trivial to deduce. Notice $$f:\mathbb{R}^m\rightarrow\mathbb{R}^n$$ this means:

(m length vector) $$\mapsto$$ (n length vector)

So for $$x\in \mathbb{R}^m$$ we must have $$Df_x x\in\mathbb{R}^n$$ (in the matrix multiplication sense)

As we go across $$Df_x$$ and down $$x$$ when multiplying, we must have columns (the width of the matrix) = length of the vector, so it is $m$ across, and thus $n$ down.

Furthermore

$$\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \cdots \\ \vdots & \ddots & \vdots \\ \cdots & \cdots & \cdots \end{pmatrix}\times\begin{pmatrix}x\\y\\ \vdots \end{pmatrix}$$ $$=\begin{pmatrix} \frac{\partial f_1}{\partial x}x+\frac{\partial f_1}{\partial y}y+\cdots \\ \frac{\partial f_2}{\partial x}x+\frac{\partial f_2}{\partial y}y+\cdots \\ \vdots \end{pmatrix}$$ $$=\begin{pmatrix} \text{change in }f_1\text{per }x\text{ times }x +\text{change in }f_1\text{per }y\text{ times }y + \cdots \\ \text{change in }f_2\text{per }x\text{ times }x +\text{change in }f_2\text{per }y\text{ times }y + \cdots \\ \vdots \end{pmatrix}$$

Which makes perfect dimensional sense