Notation on vector calculus

Question

Just trying to figure out the proper representations of some vector/matrix systems. So Let's say I have a nonlinear system of ODEs

\begin{align} \frac{dx_i}{dt} = f_i(x_1,x_2, \dots x_n,t) \quad i = 1,\dots,n \tag{A} \end{align}

May I write (and is this conventional) to write system as

$$F(x) \tag{B}$$

where $x$ is the $n$-dimensional solution vector with $i$-th entry $x_i$. And where $F$ represents the vector of functions $f_i$ applied to the system above?

Furthermore if I wanted to take the Jacobian of $F(x)$ would I write

$$\frac{\partial F}{\partial x} \tag{C}$$

because I'm not sure if I'm correct about how it's applied in terms of vector operations. I believe it could be written

$$\left[\partial /\partial x_1, \dots, \partial /\partial x_1\right]^T \left[f_1(x), \dots, f_2(x)\right]$$

Answer 1

The notation is reasonable,

$$ \frac{{\rm d}{\bf x}}{{\rm d}t} = {\bf F}({\bf x}, t) $$

represents the dynamical system, and

$$ {\bf J} = \frac{\partial {\bf F}}{\partial {\bf x}} $$

its Jacobian, which can be also written as

$$ {\bf J} = \begin{pmatrix} \partial f_1/\partial x_1 & \cdots & \partial f_1/\partial x_n \\ \vdots & \ddots & \vdots \\ \partial f_n/\partial x_1 & \cdots & \partial f_m/\partial x_n \\ \end{pmatrix} $$

The last expression clearly depends on they way you arrange the components of your vectors (column vs. row)

Answer 2

Usually, one writes $\dot{x}=\mathbb f(x)$ with $x(t)\in\mathbb{R}^n$ for the explicit system, and $0=F( x,\dot{x})$ for the implicit system. The Jacobian of a function $f$ can be written as $\frac{\partial f}{\partial x}$. The most common definition for the Jacobian is $$ \frac{\partial f}{\partial x}= \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \dots & \frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \dots & \frac{\partial f_2}{\partial x_n}\\ \vdots& &\ddots & \vdots\\ \frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \dots & \frac{\partial f_n}{\partial x_n}\\ \end{pmatrix} \neq \begin{pmatrix} \frac{\partial}{\partial x_1}\\ \frac{\partial}{\partial x_2}\\ \vdots\\ \frac{\partial}{\partial x_n} \end{pmatrix} \begin{pmatrix} f_1(x)& f_2(x)& \dots& f_n(x) \end{pmatrix}. $$ But I believe some people define it as the transposed of what I've written (which may work, too, and would match your equation).