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Unformatted text preview: 16. 1 Chapter Sixteen Integrating Vector Functions 16.1 Introduction Suppose water (or some other incompressible fluid ) flows at a constant velocity v in space (through a pipe, for instance), and we wish to know the rate at which the water flows across a rectangular surface S that is normal to the stream lines: What is the rate at which the fluid flows through S ? Let M t ( ) denote the total volume of fluid that has passed through the surface at time t. The amount of fluid that flows through during the time between t t t and + is simply M t t M t a t ( ) ( ) | | +- = v , where a is the area of S. Thus, the rate of flow through S is dM dt a = | | v . The result is slightly more complicated when various exciting changes are made. Clearly there is nothing special about the surface's being a rectangle. But suppose that S is placed at an angle to the stream lines instead of being placed normal to the them. Then we have dM dt a = v n , where n is a unit normal to the surface S . 16. 2 Observe that matters which unit normal to the plane surface we choose. If we choose the other normal (- n ), then our rate will be the negative of this one. We must thus specify an orientation of the surface. We are computing the rate of flow from one side of the surface to the other, and so we have to specify the "sides", so to speak. 16.2 Flux Now, let's look at the general situation. The surface is not restricted to being a plane surface, and the velocity of the flow is not restricted to being constant in space; it may vary with position as well as time. Specifically, suppose S is a surface, together with an orientationthat is, some means of specifying two "sides"and suppose F r ( ) is a function F R R 3 3 : , which is the velocity of the incompressible fluid. , which is the velocity of the incompressible fluid....
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This note was uploaded on 01/29/2012 for the course ENG 2zz3 taught by Professor Proff during the Spring '11 term at McMaster University.
- Spring '11