**15. Gauss's law for force flux**

The surface integral of the normal component of intensely of the gravitational field gives the flux through the closed surface.

$$
\overrightarrow{\Phi} = \left[ \overrightarrow{\text{g}} . \overrightarrow{ds} \right] \tag{51}
$$

*ds* ! = vector normal to the surface (ds) and magnitude equaled to the area (ds).

The integral over the whole surface gives Gauss's law for total mass enclosed by the surface.

(The surface integral )volume integral) divergences theorem

$$\overrightarrow{\Phi} = \left\{ \overrightarrow{\text{g}} \, . \overrightarrow{ds} \Rightarrow \int\_{\overrightarrow{v}} \overrightarrow{\nabla} \, . \overrightarrow{\text{g}} \, dv \right. \tag{52}$$

$$\overrightarrow{\nabla} \cdot \overrightarrow{\mathbf{g}} = \frac{\partial \mathbf{g} \mathbf{x}}{\partial \mathbf{x}} + \frac{\partial \mathbf{g}\_y}{\partial y} + \frac{\partial \mathbf{g}\_x}{\partial \mathbf{z}} \tag{53}$$

$$\mathcal{L}\vec{\cdot}\phi = \int\_{\nu} \vec{\nabla} \cdot \vec{\mathbf{g}}\,\,\,d\nu = -\int\_{\nu} \vec{\nabla} \nabla \mathbf{U} d\nu = -4\vec{\kappa}\mathbf{G}\mathbf{M} \tag{54}$$

*Gravity Field Theory DOI: http://dx.doi.org/10.5772/intechopen.99959*

$$=\int\_{v} \overrightarrow{\nabla^2} U \, dv = +4\aleph \text{G}M\tag{55}$$

$$=\int\_{v} \overrightarrow{\nabla^2} U.dv = +4\lambda G \int\_{v} \sigma dv\tag{56}$$

Where σ = The density of mass distribution in the volume V Poisson's equation

$$
\overrightarrow{\nabla}.\overrightarrow{\nabla}U = 4\aleph G\sigma\tag{57}
$$

Laplace equation of the potential (U) of the gravitational field is equal to a constant times the density of the distribution matter in the field.
