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Considering and as representing currents, the first equation expresses that the amount which flows out of a small region is equal to the amount which flows in. The second equation expresses that the total flow from a portion of a conducting surface into space and along the surface is zero. The third equation expresses that the flow from one sub-region across a curve on a conducting surface is equal to the flow into the adjacent sub-region. These three equations thus express that and , considered as space and surface currents, represent a flow of something which is conserved. For and to have this property the above discussion shows it is necessary and sufficient that A be everywhere solenoidal. In hydrodynamics, the field corresponds to vorticity and it clearly is solenoidal because it is the curl of velocity.
The previous section showed that a vector field can be determined from the divergence and curl of the vector field and the values on surfaces of discontinuities and bounding surfaces. The integral equations are useful for developing analytical solutions for simple systems. However, in hydrodynamics the vorticity is generally an unknown quantity. Thus it is useful to express the potentials as differential equations that are solved simultaneously for the potentials and vorticity. The differential equations are derived by substituting the potentials into the expressions for the divergence and curl.
In two-dimensional vector fields the vector potential and the vector has a nonzero component only in the third direction. In hydrodynamics the nonzero component of the vector potential is the stream function corresponds to the vorticity, which has only one nonzero component.
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