Let $F(x,y,z): \mathbb{R}^{3} \rightarrow \mathbb{R}$, and $dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy + \frac{\partial F}{\partial z}dz$.
If $x,y,z$ are linear independent, how to use the defination of the linear independent to proof $\nexists\left(a,b,c\right) \neq \left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right): dF = adx+bdy+cdz$.
Linear independence is defined on vectors. $dx,dy,dz$ , which are infintesimals, are scalars. Is there any method for appling the defination of the linear independence on them?
Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.