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Unformatted text preview: LESSON 19 Vector Fields and Line Integrals READ: Sections 17.1, 17.2. NOTES: In this chapter, a new type of function is introduced that is especially important in physics, and particularly in the study of electricity and magnetism. You have no doubt heard of the earths magnetic field, or the electric field surrounding electric transmission line. The functions of this chapter provide a mathematical way to describe such fields. The functions are called vector fields , and denoted by symbols such as- F . The input for a vector field is either a point in 2-space, ( x,y ), or a point in 3-space, ( x,y,z ). The output is, in the first case, a two dimensional vector, and in the second case, a three dimensional vector. So we will write- F ( x,y ) = F 1 ( x,y )- + F 2 ( x,y )- or- F ( x,y,z ) = F 1 ( x,y,z )- + F 2 ( x,y,z )- + F 3 ( x,y,z )- k . The functions F 1 , F 2 , and F 3 are real-valued functions of two or three variables, called the component functions . When dealing with a vector field, it is customary to think of the points in the domain as vectors, so ( x,y ) and ( x,y,z ) are denoted by- x . Then the vector fields are written compactly as- F (- x ) = F 1- + F 2- or- F (- x ) = F 1- + F 2- + F 3- k . A vector field cant be graphed in the usual sort of way. After all, to graph a vector field defined for points in 2-space, we would need a four dimensional diagram, and for a vector field defined for points in three space, a six dimensional picture would be needed. Not much chance of that. Nevertheless, there is a way of building a picture that gives an impression of the nature of the vector field. Think physically: imagine that for the point- x , the vector- F (- x ) represents a force acting at the point- x . As a vector,- F (- x ) can be represented as an arrow giving the magnitude and direction of the force. If many of these arrows are drawn in, we can get a feeling for the way the vector field (or force field ) acts. Such arrow diagrams appear on page 924 of the text. A good way to think about these pictures is to imagine the arrows represent wind direction and speed. So if were standing at a spot in the picture, we would feel the wind pushing in the direction of the arrow with a force proportional to the length of the arrow. Moreover, if we were wearing roller skates, we would be pushed along in the path indicated by the various arrows. For example, for the vector field- G of figure 3, page 924, if we were standing down at the top left corner of the diagram, we would roll along...
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