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$f(x,y) = 3x-x^2-xy$

$g(x,y) = y+y^2-3xy$

Find the linearization of the system at the critical point (1,2).

I know the Jacobin can be used for this problem, but I was wondering if it's permissible to drop the quadratic terms and then do the Jacobin on the system

$f(x,y) = 3x-xy$

$g(x,y) = y-3xy$

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Try it. Do you get the same answer as when you use the Jacobian? – André Nicolas Dec 9 '12 at 1:23
up vote 1 down vote accepted

If you know what the linearization is (as it sounds like you do), then you can answer your own question by calculating a couple of partial derivatives and plugging in some points. That is, calculate the partial derivatives of your original $f$ and $g$, plug in $(1,2)$, and do the same for your second $f$ and $g$ and see if you get the same thing.

In general, if your critical point is $(0,0)$ then you can ignore the quadratic and higher order terms (note that I would consider $xy$ a quadratic term). This is because the partial derivative of such a higher order term will still have some variable in it, and it will die when you plug in $0$ for that variable. But for other critical points, this won't always work.

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So, essentially, quadratic terms shouldn't be dropped unless the critical point being analyzed is (0,0)? – Bailor Tow Dec 9 '12 at 1:21

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