![]() Since liquids are incompressible, the amount of liquid inside a closed volume is constant if there are no sources or sinks inside the volume then the flux of liquid out of S is zero. The flux of liquid out of the volume at any time is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface. ![]() Consider an imaginary closed surface S inside a body of liquid, enclosing a volume of liquid. A moving liquid has a velocity-a speed and a direction-at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid. In two dimensions, it is equivalent to Green's theorem. In one dimension, it is equivalent to integration by parts. However, it generalizes to any number of dimensions. In these fields, it is usually applied in three dimensions. The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. ![]() Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region inside the surface. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
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