taylor expansion of function with a vector as variable

Question

I know how to do a taylor expansion of a function from R to R. I dont know how to do taylor expansion of functions which have 3D vectors as variable. How can I do this? I would appreciate it if someone also worked out an example. Thank you!

Answer 1

Then what you're looking for is the Taylor expansion of a scalar field--a function $f$ that maps $\mathbb R^n$ to $\mathbb R$.

An easy way to build up intuition about this is to do the expansion only in one direction. Let $\hat n$ be a unit vector and $t$ a scalar parameter. Let $x_0$, the point you want to expand around, be given by $x_0 = x - t \hat n$ or $x = x_0 + t \hat n$. There is only one direction connecting $x$ and $x_0$, and the magnitude can always be calculated (which fixes $t$). Then you can say

$$f(x) = f(x_0 + t \hat n) = f(x_0) + \left. \frac{\partial f}{\partial t} \right|_{x_0} t + \frac{1}{2} \left. \frac{\partial^2 f}{\partial t^2} \right|_{x_0} t^2 + \ldots$$

Now, identify $\partial f/\partial t$ as $\hat n \cdot \nabla f$. In addition, see that $t\hat n = x - x_0$. Some clever recombining of terms gives

$$f(x) = f(x_0) + (x-x_0) \cdot \nabla f|_{x_0} + \frac{1}{2} ([x - x_0] \cdot \nabla)^2 f|_{x_0} + \ldots$$

This is suitably general to cover any point $x$.