The Gauss-Ostrogradsky formula, also known as the Gauss theorem, relates the flux of a vector field through a surface to the divergence of that field in the volume. Here are the main formulas associated with this theorem:
- Gauss-Ostrogradsky formula:
[
\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiiint_{V} \nabla \cdot \mathbf{F} \, dV
]
where:
- ( \mathbf{F} ) is a vector field,
- ( S ) — closed surface,
- ( V ) — the volume bounded by the surface ( S ),
- ( d\mathbf{S} ) is the outward vector of the area,
- ( \nabla \cdot \mathbf{F} ) is the divergence of a vector field.
- Vector field divergence:
[
\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\ partial z}
]
where ( F_x, F_y, F_z ) are the components of the vector field ( \mathbf{F} ).