TITLE: What is a gradient matrix?
QUESTION [5 upvotes]: I know what the gradient of a function is, but I have this problem which seems to have something else in mind.  
The problem reads

Write the oscillator equation $y''-\cos y=0$ as a system of first-order equations with $x_1=y$ and $x_2=y'$ and find the gradient matrix $\nabla f$, and compute its eigenvalue as a function of $x$.  Draw some typical trajectories in phase space and indicate where you expect divergence, based on eigenvalue analysis.

Does this just mean to turn the system into a matrix like 
$$\begin{pmatrix}x_1\\x_2\end{pmatrix}'=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}+\begin{pmatrix}0\\-\cos x_1\end{pmatrix}$$
Would the coefficient matrix be the gradient matrix?  Or would it be the fundamental matrix solution?  I've tried googling the term and nothing seems to be what this problem is referring to.

REPLY [2 votes]: The first order system has a sign error, it should read as 
$$
\binom{\dot x_1}{\dot x_2}=F(x_1,x_2)=\binom{x_2}{\cos x_1}
$$
The gradient matrix you are to compute is now the Jacobian matrix of $F$. For that task, you can forget the context of a differential equation and just compute the partial derivatives
$$
F'(x)=\begin{pmatrix}0&1\\-\sin x_1&0\end{pmatrix}.
$$