I've got the following definition (the definition of the function of lyapunov), but I don't understand it very well:
Let we have $D \subset \mathbb{R}^n$ open where non $0 \in D$, $f_k \in C^1(D)$ where $f_k(0)=0$ for every $1\leq k \leq n$, $V\in C^1(D)$ and $V':D \to \mathbb{R}$ being $$V'(x)=\nabla V(x)\cdot f(x)=\sum_{k=1}^{n}\frac{\partial V(x)}{\partial x_k} f_k (x_1,...,x_n)$$ if $V$ is a positive definite function and $V'$ is a negative semidefinite function in $D$-n, $V$ is the Lyapunov function of the autonomous system $x'=f(x)$.
What I don't understand is, which is the relation between $f$ and $f_k$. Could anyone explain it to me, please??
Is it $f=(f_1,f_2,...,f_n)$ ??