Consider a non linear Transformation $$T:R^2\rightarrow R^2$$ $$T(x,y)=(P(x,y),Q(x,y))$$ Where $P$ and $Q$ are differential function.
Consider $(x_0,y_0)$, since $P$ and $Q$ are differentiable functions,around $(x_0,y_0)$ the linear approximation of this functions are $$P(x,y)=P(x_0,y_0)+P_x(x-x_0)+P_y(y-y_0)$$ where $P_x$ and $P_y$ are partial derivatives with respect to x and y evaluated at $(x_0,y_0)$ and $(x,y)$ is a point arbitrarily close to $(x_0,y_0)$
Similarly, $$Q(x,y)=Q(x_0,y_0)+Q_x(x-x_0)+Q_y(y-y_0)$$
Therefore for a point $(x,y)$ close to $(x_0,y_0)$
$$T(x,y)=(P(x,y),Q(x,y))=(P(x_0,y_0)+P_x(x-x_0)+P_y(y-y_0),Q(x_0,y_0)+Q_x(x-x_0)+Q_y(y-y_0))$$
For $T$ to be linear around $(x_0,y_0)$ ,$(x_0,y_0)$ should be equal to $(0,0)$ therefore $T(x_0,y_0)=(0,0)$ Therefore let $T'$ be a the linear approximation of T around $(x_0,y_0)$ such that $$T'(x, y)=(P_xx+P_yy,Q_xx+Q_yy)$$ Where $(x_0,y_0)=(0,0),T':R^2\rightarrow R^2$ Therefore matrix of $T'$ with respect to standard basis is : $$\mathcal M(T')=\begin{pmatrix} P_x & P_y\\ Q_x & Q_y \end{pmatrix}$$ Which is the Jacobian Matrix at $(x_0,y_0)$
Is this interpretation correct?