Green Gradient Theorem

Green's theorem in plane is a special case of Stoke's theorem.

dxdy

■ Example 5-1: Physical Interpretation of the Divergence Theorem

Stokes Theorem

Figure 5.1: Physical Interpretation of the Divergence Theorem c)

Figure 5.1: Physical Interpretation of the Divergence Theorem

Provide a physical interpretation of the Divergence Theorem. Solution:

A fluid has a velocity field v(x, y, z) and we first seek to determine the net inflow per unit time per unit volume in a parallelepiped centered at P(x, y, z) with dimensions Ax, Ay, Az, Fig. 5.1-a.

1 dvx

The net inflow per unit time across the x planes is

Ax GHCB

Similarly

AxAyAz

AxAyAz dy

AxAyAz dz

Hence, the total increase per unit volume and unit time will be given by

Furthermore, if we consider the total of fluid crossing dS during At, Fig. 5.1-b, it will be given by (vAt)^ndS = v^ndSAt or the volume of fluid crossing dS per unit time is v-ndS.

Thus for an arbitrary volume, Fig. 5.1-c, the total amount of fluid crossing a closed surface S per unit time is / v-ndS. But this is equal to / V^vdV (Eq. 5.14), thus J s J V

which is the divergence theorem. ■

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