I would like to obtain the longitudinal component of a vector $\mathbf {F}$ at a specific point. What Helmholtz decomposition gives me is
$${\displaystyle \mathbf {F} =-\nabla \Phi +\nabla \times \mathbf {A} ,}$$
$$\mathbf {F} _{l}=-\nabla \Phi =-{\frac {1}{4\pi }}\nabla \int _{V}{\frac {\nabla '\cdot \mathbf {F} }{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'$$
Well, $\mathbf {F}_{l}$ is obtained. That's what I want. But I assume this component of $\mathbf {F}$ is a local property of this vector field. But it requires me to integrate over the entire space. This integration is not practical in many engineering problems. Is there any method for calculation of $\mathbf {F}_{l}$ which does not require such an integral and only depends on the local area where we calculate $\mathbf {F}_{l}$ at?