Notice that if F ( x, y ) = 〈 y, − x 〉 F ( x, y ) = 〈 y, − x 〉 is the vector field from Example 6.3, then the magnitude of F is x 2 + y 2, x 2 + y 2, and therefore the corresponding unit vector field is the field G from the previous example. If F = 〈 P, Q, R 〉 F = 〈 P, Q, R 〉 is a vector field, then the corresponding unit vector field is 〈 P | | F | |, Q | | F | |, R | | F | | 〉. Therefore, the unit vector field associated with velocity is the field we would study. In this case, the speed of the fluid (which is the magnitude of the corresponding velocity vector) is irrelevant, because all we care about is the direction of each vector. Why are unit vector fields important? Suppose we are studying the flow of a fluid, and we care only about the direction in which the fluid is flowing at a given point. Is vector field F ( x, y ) = 〈 − y, x 〉 F ( x, y ) = 〈 − y, x 〉 a unit vector field? In a radial field, the vector located at point ( x, y ) ( x, y ) is perpendicular to the circle centered at the origin that contains point ( x, y ), ( x, y ), and all other vectors on this circle have the same magnitude. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, all vectors either point directly toward or directly away from the origin. Radial fields model certain gravitational fields and energy source fields, and rotational fields model the movement of a fluid in a vortex.
There are two types of vector fields in ℝ 2 ℝ 2 on which this chapter focuses: radial fields and rotational fields. Instead, we can choose points at the intersections of grid lines and plot a sample of several vectors from each quadrant of a rectangular coordinate system in ℝ 2.
If we were to plot the image vector at each point in the region, it would fill the region completely and is useless. We should plot enough vectors to see the general shape, but not so many that the sketch becomes a jumbled mess. Then, we would draw vector 〈 3, 1 〉 〈 3, 1 〉 at point ( 4, −1 ). For example, suppose the vector associated with point ( 4, −1 ) ( 4, −1 ) is 〈 3, 1 〉. To do this, draw the vector associated with a given point at the point in a plane. Since we cannot represent four-dimensional space visually, we instead draw vector fields in ℝ 2 ℝ 2 in a plane itself. Therefore the “graph” of a vector field in ℝ 2 ℝ 2 lives in four-dimensional space. We can now represent a vector field in terms of its components of functions or unit vectors, but representing it visually by sketching it is more complex because the domain of a vector field is in ℝ 2, ℝ 2, as is the range. What vector is associated with the point ( −2, 3 ) ? ( −2, 3 ) ? Drawing a Vector Field Let G ( x, y ) = x 2 y i − ( x + y ) j G ( x, y ) = x 2 y i − ( x + y ) j be a vector field in ℝ 2. The speed of the water increases, and a whirlpool occurs in part of the rapids. As the water moves from left to right, it encounters some rapids around a rock. Since the vectors to the left of the figure are small in magnitude, the water is flowing slowly on that part of the surface. The vector associated with a given point on the river’s surface gives the velocity of the water at that point. The larger object has greater mass, so it exerts a gravitational force of greater magnitude than the smaller object.įigure 6.2(b) shows the velocity of a river at points on its surface. The vectors of largest magnitude in the figure are the vectors closest to the larger object.
At any point in the figure, the vector associated with a point gives the net gravitational force exerted by the two objects on an object of unit mass. How can we model the gravitational force exerted by multiple astronomical objects? How can we model the velocity of water particles on the surface of a river? Figure 6.2 gives visual representations of such phenomena.įigure 6.2(a) shows a gravitational field exerted by two astronomical objects, such as a star and a planet or a planet and a moon. In this section, we examine the basic definitions and graphs of vector fields so we can study them in more detail in the rest of this chapter. They are also useful for dealing with large-scale behavior such as atmospheric storms or deep-sea ocean currents. Vector fields are an important tool for describing many physical concepts, such as gravitation and electromagnetism, which affect the behavior of objects over a large region of a plane or of space. 6.1.3 Identify a conservative field and its associated potential function.6.1.2 Sketch a vector field from a given equation.6.1.1 Recognize a vector field in a plane or in space.
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