Divergence Vector

8 The divergence of a vector field of a body b with boundary Q, Fig. 3.5 is defined by considering that each point of the surface has a normal n, and that the body is surrounded by a vector field v(x). The volume of the body is v(b).

Figure 3.5: Vector Field Crossing a Solid Region

The divergence of the vector field is thus defined as div v(x) = lim

where v.n is often referred as the flux and represents the total volume of "fluid" that passes through dA in unit time, Fig. 3.6 This volume is then equal to the base of the cylinder dA times the height of

3.3 Divergenc n

Div Vectors Field

Figure 3.6: Flux Through Area dA

Figure 3.6: Flux Through Area dA

the cylinder v-n. We note that the streamlines which are tangent to the boundary do not let any fluid out, while those normal to it let it out most efficiently.

10 The divergence thus measure the rate of change of a vector field.

11 The definition is clearly independent of the shape of the solid region, however we can gain an insight into the divergence by considering a rectangular parallelepiped with sides Axi, Ax2, and Ax3, and with normal vectors pointing in the directions of the coordinate axies, Fig. 3.7. If we also consider the corner

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