Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a
gradient vector field and divergence-free (solenoidal) vector field under
suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD),
which has been applied to analyze the topological features of vector fields. In
this study, we apply the HHD to study certain types of vector fields. In
particular, we investigate the existence of strictly orthogonal HHDs, which
assure an effective analysis. The first object of the study is linear vector
fields. We demonstrate that a strictly orthogonal HHD for a vector field of the
form $\textbf{F}(\textbf{x}) = A \textbf{x}$ can be obtained by solving an
algebraic Riccati equation. Subsequently, a method to explicitly construct a
Lyapunov function is established. In particular, if A is normal, there exists
an easy solution to this equation. Next, we study planar vector fields. In this
case, the HHD yields a complex potential, which is a generalization of the
notion in hydrodynamics with the same name. We demonstrate the convenience of
the complex potential formalism by analyzing vector fields given by homogeneous
quadratic polynomials.