I was working on a problem about state-space representation, the context is of a first order differential equation that we try to linearize around a reference state vector and I came upon this notation :
$$S_{dot} = F(S,t) = F(S^*,t) + \frac {dF(t)}{dS(t)}|_{S^*} [S(t) - S^*(t)] + ... $$
Where S* is the reference state vector. The part that i don't understand is the derivative which is refered to as the first order derivative. To me it kind of looks like a Jacobian or something but I'm not sure. And I don't see how concretely you would calculate it.
Let $\dot{x} =f(x,t)$ be a state space model, the linearization around $x_R$ ($R$ for reference) is given by:
$$\Delta \dot{x} =f(x_R,t)+ \left.\dfrac{\partial f}{\partial x}\right|_{x_R}\Delta x,$$
in which $\Delta x = x - x_R$. And as you allready suspected $$\left.\dfrac{\partial f}{\partial x}\right|_{x_R}$$
is the Jacobian evaluated at $x_R$.
For a second order system, this looks like
$$\dot{x}_1 = f_1(x,t)$$ $$\dot{x}_2 = f_2(x,t)$$
$$\Delta\dot{x}_1 = f_1(x_R,t)+ \left.\dfrac{\partial f_1}{\partial x_1}\right|_{x_1=x_{R1},x_2=x_{R2}}\Delta x_1 +\left.\dfrac{\partial f_1}{\partial x_2}\right|_{x_1=x_{R1},x_2=x_{R2}}\Delta x_2$$
$$\Delta\dot{x}_2 = f_2(x_R,t)+\left.\dfrac{\partial f_2}{\partial x_1}\right|_{x_1=x_{R1},x_2=x_{R2}}\Delta x_1 +\left.\dfrac{\partial f_2}{\partial x_2}\right|_{x_1=x_{R1},x_2=x_{R2}}\Delta x_2$$