Could someone please explain this? (HW-related)

[Jamin2112]

Homework Statement

So the last thing we got to learn before studying for finals ...

Homework Equations

... was converting something like

dx/dt = F(x,y)
dy/dt = G(x,y)

into a ...

The Attempt at a Solution

... system of equations like (x' y')^T = J * (x y)^T,

where J is the Jacobian Matrix [ (∂F/∂x ∂G/∂x)^T , (∂D/∂y ∂G/∂y)^T ].

I'm just wondering why exactly that works. Could someone show me a proof or something? Being the weird dude I am, I need to know these things.

 

[xaos]

hm, perhaps you're approximating a nonlinear system by a linear. I'm thinking 'taylor expansion', but then this may be my response to everything. do you happen to know if the initial point is [F(x_o,y_o),G(x_o,y_o)]=[0,0]? that might help clarify things a bit...

 

[rhoparkour]

xaos is right, this "conversion" of the problem is a local approximation of your F,G functions to an linear (or affine) function. In general the set of equations you give is not linear, the only way to "linearize" it is to do this Taylor expansion (assuming the F and G were not already linear).

 

[Jamin2112]

Alright. I'll just roll with it.

 

[HallsofIvy]

In other words, at some point $(x_0,y_0)$, F(x,y) is replaced by its "tangent plane approximation",
$$\frac{\partial F}{\partial x}(x_0,y_0)(x- x_0)+ \frac{\partial F(x_0, y_0)}{\partial y}(y- y_0)+ F(x_0, y_0)$$
and G(x, y) is replaced by its "tangent plane approximation",
$$\frac{\partial G}{\partial x}(x_0,y_0)(x- x_0)+ \frac{\partial G}{\partial y}(x_0, y_0)(y- y_0)+ G(x_0, y_0)$$

Now, you have the two equations
$$\frac{dx}{dt}= \frac{\partial F}{\partial x}(x_0,y_0)(x- x_0)+ \frac{\partial F(x_0, y_0)}{\partial y}(y- y_0)+ F(x_0, y_0)$$
$$\frac{dy}{dt}= \frac{\partial G}{\partial x}(x_0,y_0)(x- x_0)+ \frac{\partial G}{\partial y}(x_0, y_0)(y- y_0)+ G(x_0, y_0)$$

which we can write as the matrix equation
$$\frac{d}{dt}\begin{pmatrix}x \\ y\end{pmatrix}= \begin{pmatrix}\frac{\partial F}{\partial x}(x_0, y_0) & \frac{\partial F}{\partial y}(x_0, y_0) \\ \frac{\partial G}{\partial x}(x_0, y_0) & \frac{\partial G}{\partial y}(x_0, y_0)\end{pmatrix}\begin{pmatrix}x- x_0 \\ y- y_0\end{pmatrix}+ \begin{pmatrix}F(x_0, y_0) \\ G(x_0, y_0)\end{pmatrix}$$