Let \(\textbf{F}\) be a vector field in three dimensions such that the component functions of \(\textbf{F}\) have continuous first-order partial derivatives on the domain of \(\textbf{F}\). Suppose \(\textbf{F}\) is a conservative vector field in \(\mathbb{R}^3\):
Since \(\textbf{F}\) is conservative, there is a function \(f(x,y,z)\) such that \(∇f=\textbf{F}\). Therefore, by the definition of the gradient, \(f_x=P\), \(f_y=Q\) and \(f_z=R\). By Clairaut's theorem, \(f_{xy}=f_{yx}\), \(f_{xz}=f_{zx}\) and \(f_{yz}=f_{zy}\). This means, \(f_{xy}=P_y\) and \(f_{yx}=Q_x\), and thus \(P_y=Q_x\).
This means, \(f_{xy}=P_y\) and \(f_{yx}=Q_x\), and thus \(P_y=Q_x\).
This also means \(f_{xz}=P_z\) and \(f_{zx}=R_x\), and thus \(P_z=R_x\).
Similarly, \(f_{yz}= Q_z\) and \(f_{zy}=R_y\), and thus \(Q_z=R_y\).