2.9 Divergence of a Vector Field

Before defining the divergence of a vector field let us specify a scalar field f at point P in terms of a vector field as

where the point P is enclosed by volume v bounded by surface s . Although ? v can be of any shape, we construct a parallelepiped with sides ? x, y, and ? z, as shown in Figure 2.29 in order to evaluate (2.80). Note that defines the outward flow of the vector field through the surface as the unit normal to ds points away from the volume enclosed. Thus, gives the net outward flow of flux of a vector field from the volume ? v. However, the outward flow of a vector field through the face in the positive x direction, using the Taylor series expansion and neglecting the higher-order terms, is

Figure 2.29: A differential volume in the rectangular coordinate system

The outward flow of the vector field through the surface in the negative x direction is

Therefore, the net outward flow of the vector field through both the surfaces in the x direction is

We can similarly obtain expressions for the net outward flow of the vector field through the surfaces in the y and z directions. The net outward flow of the vector field through all the surfaces enclosing the volume v