The Gauss Theorem. Let's learn math together!

The Gauss Theorem

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Denote by G the three-dimensional solid bounded piece-wise continuous, smooth, closed surface S with exterior normal. Suppose that specify a vector field:

whose components have continuous partial derivatives.

According to the formula of Gauss,

where through

denotes the divergence of a vector field F (it is indicated also by the symbol div F), and the surface integral is evaluated over the closed surface.

The formula of Gauss binds the surface integrals of the second kind with the corresponding triple integrals.

This formula can be written also in the coordinate form:

In the particular case, considering , we obtain a formula for the calculation of body volume G:

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