Sometimes called "Jacobian Matrix", or "differential".
Common definition
Given a function f:Rm→Rn (I use the convention of m first because it takes it from m to n) the:
- differential of f at x, denoted dfx or Dfx which I prefer, as you often find df in a fraction involving dx
- Jacobian matrix of f at x often denoted Jf(x)
Are given by:
Dfx:Rm→Rn, Dfx=(∂f1∂x1⋯∂f1∂xm⋮⋱⋮∂fn∂x1⋯∂fn∂xm)
This is a n-by-m matrix using my convention.
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How to remember which way round this matrix goes
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