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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
50 CHAPTER 4V ECTOR FIELDS IN TWO DIMENSIONS 4.2. F LUX AND D IVERGENCE From here on, we will almost always think of vector ±elds as representing ²ows of mate- rial. In this section, we seek to quantify how this ²ow is moving. The ±rst quantity we study, outward ²ux, measures the net ²ow out of a region in the plane. The second quan- tity we study, divergence, measures ²ux 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 ±eld F which describes the velocity of moving air in a room with a lit ±replace and a glass of ice water. Near the ±replace, the air is being warmed, so it is expanding, which means that F near the ±replace is pointing out. Near the ice water, the opposite is happening, so F near the ice water is pointing in. In terms of ²ux, this means that the outward ²ux of this vector ±eld on a small region around the ±replace is positive, while in a small region around the ice water, it is negative. Flux is the rate of ²ow through a unit area during a unit time. Various sorts of ²ux 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 ²ow 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 ±eld F and a region R in the plane, the outward ²ux of F over R is the amount of material ²owing out of the region R .Sup - pose that the boundary of R is parameterized by the smooth curve r t .Thentocompu te the ²ux of F over R ,wetakethelineintegralover r t of the amount of material ²owing out of the region at each point. Consider the following region, R ,whoseboundaryisparameterizedbythevectorfunc- 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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SECTION 4.2 FLUX AND DIVERGENCE 51 There are two unit vectors normal to r t 0 .Le t 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 ±ow at the point r t 0 is given by the vector ²eld, F r t 0 .Weareon lyin teres ted in the amount of this ±ow which is leaving the region, i.e., theamountofthis±owthatis parallel to n 0 .Therefore ,theamountofth is±owatthepo int r t 0 is the scalar projection of F r t 0 onto n 0 .S incewechose n 0 to be a unit vector, this is F r t 0 n 0 .Thel ineintegra l (with respect to arc length) of this quantity gives us the total ±ow out of the region, Outward ±ux C F n ds, where C is the boundary of the region R ,and n is the outward-pointing unit normal vector to C .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/10/2011 for the course MAC 2313 taught by Professor Keeran during the Fall '08 term at University of Florida.

Page1 / 9

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online