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:

  1. 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.
  1. 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} ).

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