2313-4-2 - 50 C HAPTER 4 4.2 V ECTOR F IELDS IN T WO D...

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50 C HAPTER 4 V ECTOR F IELDS IN T WO D IMENSIONS 4.2. F LUX AND D IVERGENCE From here on, we will almost always think of vector fields as representing flows of mate- rial. In this section, we seek to quantify how this flow is moving. The first quantity we study, outward flux, measures the net flow out of a region in the plane. The second quan- tity we study, divergence, measures flux per unit of area. In the next section, we study the relationship between these two quantities (Green’s Theorem). O UTWARD F LUX Suppose that we have a vector field F which describes the velocity of moving air in a room with a lit fireplace and a glass of ice water. Near the fireplace, the air is being warmed, so it is expanding, which means that F near the fireplace is pointing out. Near the ice water, the opposite is happening, so F near the ice water is pointing in. In terms of flux, this means that the outward flux of this vector field on a small region around the fireplace is positive, while in a small region around the ice water, it is negative. Flux is the rate of flow through a unit area during a unit time. Various sorts of flux appear in physics, for example: Newton’s law of viscosity describes the rate of transfer of momentum across an area, Fourier’s law of conduction descibes the rate of heat flow across an area, Fick’s law of diffusion describes the rate of movement of molecules across an area. From our point of view, given a 2 -dimensional vector field F and a region R in the plane, the outward flux of F over R is the amount of material flowing out of the region R . Sup- pose that the boundary of R is parameterized by the smooth curve r t . Then to compute the flux of F over R , we take the line integral over r t of the amount of material flowing out of the region at each point. Consider the following region, R , whose boundary is parameterized by the vector func- tion r t . r t r t 0 n 0 r t 0 R We want to calculate the amount of material leaving the region R at each point r t 0 on the boundary. At this point, the vector r t 0 is tangent to the boundary of the region.
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S ECTION 4.2 F LUX AND D IVERGENCE 51 There are two unit vectors normal to r t 0 . Let n 0 denote the unit vector which is normal to r t 0 and points away from the region, which we refer to as the outward-pointing unit normal vector . The flow at the point r t 0 is given by the vector field, F r t 0 . We are only interested in the amount of this flow which is leaving the region, i.e., the amount of this flow that is parallel to n 0 . Therefore, the amount of this flow at the point r t 0 is the scalar projection of F r t 0 onto n 0 . Since we chose n 0 to be a unit vector, this is F r t 0 n 0 . The line integral (with respect to arc length) of this quantity gives us the total flow out of the region, Outward flux C F n ds, where C is the boundary of the region R , and n is the outward-pointing unit normal vector to C .
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