taylor expansion of function with a vector as variable 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!
 A: 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 $e_i$ be an element of a basis $\lbrace e_1,\dots, e_n \rbrace$ of $\mathbb R^n$ and $t$ a scalar parameter.  Let $x_0$, the point you want to expand around, be given by $x_0 = x - t e_i$ or $x = x_0 + t e_i$.  There is only one direction connecting $x$ and $x_0$, and the magnitude can always be calculated (which fixes $t$).  Then you can say
$$
f(x) = f(x_0) + Df(x_0)(te_i) + \frac{1}{2}D^2f(x_0)(te_i,te_i) + o(t^2)\,,
$$
namely
$$f(x) = f(x_0 + t e_i) = f(x_0) + \left. \frac{\partial f}{\partial x_i} \right|_{x_0} t + \frac{1}{2} \left. \frac{\partial^2 f}{\partial x_i^2} \right|_{x_0} t^2 + o(t^2)\,.$$
Now, identify $\partial f/\partial x_i$ as $e_i \cdot \nabla f$.  In addition, see that $t e_i = x - x_0$.  Some clever recombining of terms gives
$$f(x) = f(x_0) + (x-x_0) \cdot \nabla f|_{x_0} + \frac{1}{2} ((x - x_0) \cdot \nabla)^2 f|_{x_0} + o(t^2)\,.$$
This is suitably general to cover any point $x$.
