![]() ![]() 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.If your browser is so outdated or unusual that the linked advice doesn't work, consider these ideas.You will need to install a UserScript loader first.You will see formatted as in a textbook if the MathJax UserScript is installed and working. Type this as an example (replace the [- with [ when typing): ![]() being a jerk (Jerks get banned.) Using LaTeX.image or video link-posts (Links to articles, Desmos, Wolfram|Alpha, and the like are fine.).one-sentence posts (Have some respect for people who take time to answer your question and follow the posting rules.).posting of quizzes or lists of questions.offers or solicitations of payment in any form. ![]()
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