# Multi-variate Taylor Series Expansion

I understand how to use Taylor series to expand basic functions. However, I am trying to work out how to expand Taylor series with more than one variable.

So far I have the equation with the two variables $x_1$ and $x_2$: $f(x_1,x_2)=x_1x_2+x_1^2$.

I expanded: $f({\bf x}-{\bf y})=(x_1-y_1)(x_2-y_2)+(x_1-y_1)^2$.

However, for another question I have ${\bf x}=(x_1,...,x_n)$. I have $\bf x$ and $\bf y$ are fixed in $R^n$.

I am defining a function $g(k)=f({\bf x}+k(2{\bf y}-{\bf x}))$, for $k$ belonging to $R$. I am unsure how to expand $g(k)$ using Taylor series. Also in this instance I am confused whether or not $f$ would be a function of ($x$ and $y$) as these are both fixed, or a function of $k$ which can vary?

• What is your $**x**$? – Babai Nov 26 '14 at 11:20
• @Susobhan, it was apparently supposed to be bold face x in math mode. I edited the question so that it renders more reasonably. – Joonas Ilmavirta Nov 26 '14 at 11:47