1. ## Taylor Expansion

I have a couple of questions relating to Taylor expansions.

1) I have a function f(s) and I'm trying to use a Taylor expansion to look at the change in value of f over a small change in s, $\displaystyle \delta s$, how would I represent
$\displaystyle f(s + \delta s)$ as a Taylor expansion?

2) If I have a function of 2 variables, how would I find a Taylor expansion for f(s,t)?

2. Originally Posted by bigdoggy
I have a couple of questions relating to Taylor expansions.

1) I have a function f(s) and I'm trying to use a Taylor expansion to look at the change in value of f over a small change in s, $\displaystyle \delta s$, how would I represent
$\displaystyle f(s + \delta s)$ as a Taylor expansion?

2) If I have a function of 2 variables, how would I find a Taylor expansion for f(s,t)?
1) $\displaystyle f(s + \delta s) = f(s) + f'(s) \delta s + \frac{f''(s)}{2!} (\delta s)^2 + \frac{f'''(s)}{3!} (\delta s)^3 + \cdots$

2) $\displaystyle f(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ $\displaystyle + \frac{1}{2!} f_{xx}(a,b)(x-a)^2 + f_{xy}(a,b)(x-a)(y-b) + \frac{1}{2!} f_{yy}(a,b)(y-b)^2 + \cdots$

3. Originally Posted by Danny
1) $\displaystyle f(s + \delta s) = f(s) + f'(s) \delta s + \frac{f''(s)}{2!} (\delta s)^2 + \frac{f'''(s)}{3!} (\delta s)^3 + \cdots$

2) $\displaystyle f(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ $\displaystyle + \frac{1}{2!} f_{xx}(a,b)(x-a)^2 + f_{xy}(a,b)(x-a)(y-b) + \frac{1}{2!} f_{yy}(a,b)(y-b)^2 + \cdots$
thanks, but why is this?

4. Originally Posted by bigdoggy
thanks, but why is this?
Which one are you refering to?