Get ready Part 4

Define functions $h_i(x)=\frac{1}{2}\sum_{j=1}^{n}c_{ij}x_j-f_i(x)$ for $1\leq i\leq n$ . Observe that $\frac{\partial h_i}{\partial x_j}-\frac{\partial h_j}{\partial x_i}=0$ for all $i,j$ (we are using the fact that $c_{ji}=-c_{ij}$ , by definition). Since the 1-form $\omega=h_1(x)dx_1+...+h_n(x)dx_n$ is closed and the domain $\mathbb{R}^n$ is simply connected, the form will be exact and have a potential $g$ (we can let $g(x)=\int_{0}^{x}\omega$ ), meaning that $h_i(x)=\frac{\partial g}{\partial x_i}$ . Rearranging the terms in the equation defining $h_i(x)$ , we find that $f_i(x)+\frac{\partial g}{\partial x_i}=\frac{1}{2}\sum_{j=1}^{n}c_{ij}x_j$ , a linear function, as claimed.

A great problem that I will be happy to assign as a homework or exam question in my calculus classes (if you allow). It really tests the understanding of the basic concepts. Thank you for posting, comrade!