Smooth functions
Posted: Thu Jul 14, 2016 1:08 pm
Let \( f:\mathbb{R}^n \rightarrow \mathbb{R} \) be a smooth function (meaning that it has continuous partial derivatives of any order).
Prove that there exist smooth functions \( g_1, \dots, g_n \) such that:
$$f\left ( x_1, \dots, x_n \right )-f\left ( 0, \dots, 0 \right )=\sum_{i=1}^{n}x_i g_i \left ( x_1, \dots, x_n \right )$$
Prove that there exist smooth functions \( g_1, \dots, g_n \) such that:
$$f\left ( x_1, \dots, x_n \right )-f\left ( 0, \dots, 0 \right )=\sum_{i=1}^{n}x_i g_i \left ( x_1, \dots, x_n \right )$$