Ch 1.6e (Ch1.7) Introduction of Divergence F 講者： 許永昌 老師

Contents Formula Example A Physical Interpretation Flow rate Continuity Equation

Formula we can get In the physical interpretation, it is related to the net flow out per unit volume and per unit time. Comparison between Gradient and divergence f V Name Gradient Divergence Result Scalar  Vector Vector  Scalar In Cartesian 3D

Example P44e (P39) Divergence of a central force field: F=rf(r). Therefore, If f=rn-1, we get (rf(r))=(n+2)rn-1. If n=-2, the divergence vanishes except at r=0. Electric field built by a charge q located at r=0 point, its electric field is and  E=0 except at r=0. Sometimes we use

A Physical Interpretation Volumetric flow rate: Reference: http://en.wikipedia.org/wiki/Volumetric_flow_rate Definition: The volume flow through an area A per unit time. Volume = AvDtcosq. q q Area A vDt

Volumetric flow rate Volumetric flow rate=Avcosq. Volumetric flow rate A: area  A= v: fluid velocity q: the angle between the surface normal and the fluid velocity. Volumetric flow rate =Av for a flat plane =v ds for a curved surface. ds : differential surface:

Mass flow rate, charge flow rate Since we get the volumetric flow rate: Vflow/Dt=v ds, we also can get Mass flow rate: mflow/Dt=  rm dV/Dt= rmv ds Charge flow rate: Qflow/Dt=  r dV/Dt= rv ds Charge density: r :dQ= r dV. Current density: jrv. j j q j j j j Area A

Net flow out Each surface normal is required to point out of the region which is covered by these surfaces. Consider a small rectangular parallelepiped whose length AB, depth BF and height AC are dy, dx and dz. The net flow out per unit time from this volume is x y z A B C D E F G H

Continuity Equation Net flow out per unit time: dQ/dt=-jdt. (Why does this equation need a “-”?) dQ=rdt. The divergence appears in a wide variety of physical problems, ranging from a probability current density in quantum mechanics to neutron leakage in a nuclear reactor.

Summary 雖然V在各點上都有值，但是，他代表的是該點鄰近的vector field V 的net flow out的訊息. V dt 對應的是該小區域的 net flow out。 事實上，這個小體積的形狀並沒有要求一定要四四方方的。後面會講。

Homework 1.6.1e (1.7.1) 1.6.2e (1.7.3)

Nouns Volumetric flow rate Divergence Charge density: r Current density: rv Continuity Equation Maclaurin Expansion and Taylor’s Expansion. P46e