Taylor expansion for f(x,y) about (x0,y0) ?

[izzy93]

Can someone please explain the Taylor expansion for f(x,y) about (x0,y0) ?

Would really appreciate some sort of step by step answer :)

thankyou

 

[hunt_mat]

Simple, just take a Taylor expansion in $x$ around $x_{0}$
$$f(x_{0}+h_{x},y_{0}+h_{y})=f(x_{0},y_{0}+h_{y})+h_{x} \partial_{x}f(x_{0},y_{0}+h_{y})+h_{x}^{2}\partial_{x}^{2}f(x_{0},y_{0}+h_{y})+\cdots$$
Then take the Tavlor expansion in $y$ of each term. Simple but tedious.

 

[izzy93]

Thanks for the reply, it looks logical but I'm stuck on how it all comes together/
and where does the h^2(x) term in the 3rd term on the right come from

 

[hunt_mat]

I am applying the 1D Taylor expansion to the x-variable. I am assuming, you know about the 1D Taylor expansion right?

 

[izzy93]

yes I do using this equation, f(x) = f(x0) + f '(x0)/1! (x-x0) ...

 

[hunt_mat]

The notation I am using:
$$h_{x}=x-x_{0}$$

 

[izzy93]

ok, I think i get it, bit slow atm! thankyou!