{"id":1117,"date":"2021-11-09T23:40:53","date_gmt":"2021-11-09T23:40:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1117"},"modified":"2022-11-01T05:00:15","modified_gmt":"2022-11-01T05:00:15","slug":"vector-fields","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/vector-fields\/","title":{"raw":"Vector Fields","rendered":"Vector Fields"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Recognize a vector field in a plane or in space.<\/span><\/li>\r\n \t<li><span class=\"os-abstract-content\">Sketch a vector field from a given equation.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1167793470654\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Examples of Vector Fields<\/h2>\r\n<p id=\"fs-id1167793638685\">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?\u00a0Figure 1\u00a0gives visual representations of such phenomena.<\/p>\r\n<p id=\"fs-id1167793299474\">Figure 1(a) shows a gravitational field exerted by two astronomical objects, such as a star and a planet or a planet and a moon. 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. The vectors of largest magnitude in the figure are the vectors closest to the larger object. The larger object has greater mass, so it exerts a gravitational force of greater magnitude than the smaller object.<\/p>\r\n<p id=\"fs-id1167794209630\">Figure 1(b) shows the velocity of a river at points on its surface. The vector associated with a given point on the river\u2019s surface gives the velocity of the water at that point. Since the vectors to the left of the figure are small in magnitude, the water is flowing slowly on that part of the surface. As the water moves from left to right, it encounters some rapids around a rock. The speed of the water increases, and a whirlpool occurs in part of the rapids.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3299\" align=\"aligncenter\" width=\"731\"]<img class=\"wp-image-3299 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162020\/6-1-1.jpeg\" alt=\"Two images, labeled A and B. Image A shows the gravitational field exerted by two astronomical bodies on a small object. The earth is on the left, and the moon is on the right. The earth is surrounded by long arrows pointing towards its center arranged in concentric circles. There is a break in the circle on the right, across from the moon. The moon is surrounded by smaller arrows that curve out and to the right. Image B shows the vector velocity field of water on the surface of a river with a large rock in the middle. The arrows tend to point at the same angle as the riverbank. Where the river meets the rock, the arrows point around the rock. After the rock, the some arrows point forward, and others turn back to the rock. The water flows fastest towards the middle of the river and around the rock and slowest along the riverbank.\" width=\"731\" height=\"841\" \/> Figure 1. (a) The gravitational field exerted by two astronomical bodies on a small object. (b) The vector velocity field of water on the surface of a river shows the varied speeds of water. Red indicates that the magnitude of the vector is greater, so the water flows more quickly; blue indicates a lesser magnitude and a slower speed of water flow.[\/caption]\r\n<p id=\"fs-id1167794060800\">Each figure illustrates an example of a vector field. Intuitively, a vector field is a map of vectors. In this section, we study vector fields in [latex]\\mathbb{R}^2[\/latex] and\u00a0[latex]\\mathbb{R}^3[\/latex].<\/p>\r\n\r\n<div data-type=\"note\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167794293131\">A\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term229\" data-type=\"term\"><strong>vector field [latex]F[\/latex]<\/strong> in\u00a0[latex]\\mathbb{R}^2[\/latex]\u00a0<\/span>is an assignment of a two-dimensional vector [latex]{\\bf{F}}(x,y)[\/latex]\u00a0to each point [latex](x, y)[\/latex] of a subset [latex]D[\/latex] of [latex]\\mathbb{R}^2[\/latex]. The subset [latex]D[\/latex] is the\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term230\" class=\"no-emphasis\" data-type=\"term\">domain<\/span>\u00a0of the vector field.<\/p>\r\n<p id=\"fs-id1167794328312\">A vector field <strong>[latex]F[\/latex]<\/strong> in [latex]\\mathbb{R}^3[\/latex] is an assignment of a three-dimensional vector [latex]{\\bf{F}}(x,y,z)[\/latex] to each point [latex](x, y, z)[\/latex] of a subset [latex]D[\/latex] of [latex]\\mathbb{R}^3[\/latex]. The subset [latex]D[\/latex] is the domain of the vector field.<\/p>\r\n\r\n<\/div>\r\n<h4 data-type=\"title\">Vector Fields in [latex]\\mathbb{R}^2[\/latex]<\/h4>\r\n<p id=\"fs-id1167793929736\">A vector field in [latex]\\mathbb{R}^2[\/latex] can be represented in either of two equivalent ways. The first way is to use a vector with components that are two-variable functions:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y)=\\langle{P}(x,y),Q(x,y)\\rangle}[\/latex].<\/p>\r\nThe second way is to use the standard unit vectors:\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y)=P(x,y){\\bf{i}}+Q(x,y){\\bf{j}}}[\/latex].<\/p>\r\nA vector field is said to be\u00a0<em data-effect=\"italics\">continuous<\/em>\u00a0if its component functions are continuous.\r\n<div id=\"fs-id1167794027862\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding a vector associated with a given point<\/h3>\r\nLet [latex]{\\bf{F}}(x,y)=(2y^2+x-4){\\bf{i}}+\\cos(x){\\bf{j}}[\/latex] be a vector field in [latex]\\mathbb{R}^2[\/latex]. Note that this is an example of a continuous vector field since both component functions are continuous. What vector is associated with point [latex](0, -1)[\/latex]?\r\n\r\n[reveal-answer q=\"882649457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"882649457\"]\r\n<p id=\"fs-id1167794135008\">Substitute the point values for [latex]x[\/latex] and\u00a0[latex]y[\/latex]:<\/p>\r\n[latex]\\hspace{8cm}\\begin{align}\r\n\r\n{\\bf{F}}(0,1)&amp;=(2(-1)^2+0-4){\\bf{i}}+\\cos(0){\\bf{j}} \\\\\r\n\r\n&amp;=-2{\\bf{i}}+{\\bf{j}}\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nLet [latex]{\\bf{G}}(x,y)=x^2y{\\bf{i}}-(x+y){\\bf{j}}[\/latex] be a vector field in [latex]\\mathbb{R}^2[\/latex]. What vector is associated with the point [latex](-2, 3)[\/latex]?\r\n\r\n[reveal-answer q=\"872447946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"872447946\"]\r\n\r\n[latex]12{\\bf{i}}-{\\bf{j}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Drawing a Vector Field<\/h2>\r\n<p id=\"fs-id1167794094176\">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 [latex]\\mathbb{R}^2[\/latex], as is the range. Therefore the \u201cgraph\u201d of a vector field in [latex]\\mathbb{R}^2[\/latex] lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in [latex]\\mathbb{R}^2[\/latex] in a plane itself. To do this, draw the vector associated with a given point at the point in a plane. For example, suppose the vector associated with point [latex](4, -1)[\/latex] is\u00a0[latex]\\langle3,1\\rangle[\/latex]. Then, we would draw vector [latex]\\langle3,1\\rangle[\/latex] at point\u00a0[latex](4,-1)[\/latex].<\/p>\r\n<p id=\"fs-id1167793267239\">We should plot enough vectors to see the general shape, but not so many that the sketch becomes a jumbled mess. If we were to plot the image vector at each point in the region, it would fill the region completely and is useless. 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 [latex]\\mathbb{R}^2[\/latex].<\/p>\r\n<p id=\"fs-id1167794064980\">There are two types of vector fields in [latex]\\mathbb{R}^2[\/latex] on which this chapter focuses: radial fields and rotational fields. Radial fields model certain gravitational fields and energy source fields and rotational fields model the movement of a fluid in a vortex. In a\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term231\" data-type=\"term\">radial field<\/span>, all vectors either point directly toward or directly away from the origin. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, the vector located at point [latex](x, y)[\/latex] is perpendicular to the circle centered at the origin that contains point [latex](x, y)[\/latex], and all other vectors on this circle have the same magnitude.<\/p>\r\n\r\n<div id=\"fs-id1167794336014\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: drawing a radial vector field<\/h3>\r\nSketch the vector field [latex]{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}[\/latex].\r\n\r\n[reveal-answer q=\"623977834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"623977834\"]\r\n<p id=\"fs-id1167793617583\">To sketch this vector field, choose a sample of points from each quadrant and compute the corresponding vector. The following table gives a representative sample of points in a plane and the corresponding vectors.<\/p>\r\n\r\n<div id=\"fs-id1167793952941\" class=\"os-table\">\r\n<table class=\"unnumbered\" style=\"height: 121px;\" data-id=\"fs-id1167793952941\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,\\frac12\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,-\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\" valign=\"top\">\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-\\frac12\\rangle[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,-\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p id=\"fs-id1167794329899\">Figure 2(a) shows the vector field. To see that each vector is perpendicular to the corresponding circle,\u00a0Figure 2(b) shows circles overlain on the vector field.<\/p>\r\n\r\n[caption id=\"attachment_3301\" align=\"aligncenter\" width=\"663\"]<img class=\"wp-image-3301 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162053\/6-1-2-663x1024.jpeg\" alt=\"Visual representations of a radial vector field on a coordinate field. The arrows are stretching away from the origin in a radial pattern. The magnitudes increase the further the arrows are from the origin, so the lines are longer. The second version shows concentric circles around the origin to highlight the radial pattern.\" width=\"663\" height=\"1024\" \/> Figure 2. (a) A visual representation of the radial vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}}[\/latex]. (b) The radial vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}}[\/latex] with overlaid circles. Notice that each vector is perpendicular to the circle on which it is located.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nDraw the radial field [latex]\\small{{\\bf{F}}(x,y)=-\\frac{x}3{\\bf{i}}-\\frac{y}3{\\bf{j}}}[\/latex].\r\n\r\n[reveal-answer q=\"320974724\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"320974724\"]\r\n\r\n[caption id=\"attachment_3302\" align=\"aligncenter\" width=\"717\"]<img class=\"wp-image-3302 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162127\/6-1-tryitans1.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/> Figure 3. A visual representation of the radial vector field\u00a0[latex]\\small{{\\bf{F}}(x,y)=-\\frac{x}3{\\bf{i}}-\\frac{y}3{\\bf{j}}}[\/latex].[\/caption][\/hidden-answer]<\/div>\r\nIn contrast to radial fields, in a\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term232\" data-type=\"term\">rotational field<\/span>, the vector at point [latex](x, y)[\/latex] is tangent (not perpendicular) to a circle with radius [latex]r=\\sqrt{x^2+y^2}[\/latex]. In a standard rotational field, all vectors point either in a clockwise direction or in a counterclockwise direction, and the magnitude of a vector depends only on its distance from the origin. Both of the following examples are clockwise rotational fields, and we see from their visual representations that the vectors appear to rotate around the origin.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: chapter opener: drawing a Rotational vector field<\/h3>\r\n[caption id=\"attachment_5227\" align=\"aligncenter\" width=\"325\"]<img class=\"size-full wp-image-5227\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31212332\/6.4.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/dc1c17f35ff4f1559d82e4922f942bee41e04cce&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A photograph of a hurricane, showing the rotation around its eye.&quot; id=&quot;11&quot;&gt;\" width=\"325\" height=\"215\" \/> Figure 4. (credit: modification of work by NASA)[\/caption]\r\n\r\nSketch the vector field\u00a0[latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex].\r\n\r\n[reveal-answer q=\"774923650\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"774923650\"]\r\n<h2 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h2>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1167794060214\">Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors.\u00a0Figure 5\u00a0shows the resulting vector field.<\/p>\r\n\r\n<div id=\"fs-id1167794034119\" class=\"os-table\">\r\n<table class=\"unnumbered\" data-id=\"fs-id1167793952941\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-2\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,-1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle2,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,2\\rangle[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\" valign=\"top\">\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-2,0\\rangle[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\r\n<td style=\"height: 25px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,-1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_5230\" align=\"aligncenter\" width=\"740\"]<img class=\"size-full wp-image-5230\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31212724\/6.5.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/0262ed9020ac9975013b56fb2d2ffcaf3369ea2f&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A visual representation of the given vector field in a coordinate plane with two additional diagrams with notation. The first representation shows the vector field. The arrows are circling the origin in a clockwise motion. The second representation shows concentric circles, highlighting the radial pattern. The The third representation shows the concentric circles. It also shows arrows for the radial vector &lt;a,b&gt; for all points (a,b). Each is perpendicular to the arrows in the given vector field.&quot; id=&quot;12&quot;&gt;\" width=\"740\" height=\"728\" \/> Figure 5. (a) A visual representation of vector field [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex]. (b) Vector field [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex] with circles centered at the origin. (c) [latex]{\\bf{F}}(a,b)[\/latex] is perpendicular to radial vector [latex]\\langle{a},b\\rangle[\/latex] at point [latex](a, b)[\/latex].[\/caption]\r\n<h2 id=\"13\" data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h2>\r\n<p id=\"fs-id1167793932094\">Note that vector [latex]{\\bf{F}}(a,b)=\\langle{b},-a\\rangle[\/latex] points clockwise and is perpendicular to radial vector [latex]\\langle{a},b\\rangle[\/latex]. (We can verify this assertion by computing the dot product of the two vectors:[latex]\\langle{a},b\\rangle\\cdot\\langle{-b},a\\rangle=-ab+ab=0[\/latex].) Furthermore, vector [latex]\\langle{b},-a\\rangle[\/latex] has length [latex]r=\\sqrt{a^2+b^2}[\/latex]. Thus, we have a complete description of this rotational vector field: the vector associated with point [latex](a, b)[\/latex] is the vector with length [latex]r[\/latex] tangent to the circle with radius [latex]r[\/latex], and it points in the clockwise direction.<\/p>\r\n<p id=\"fs-id1167793931853\">Sketches such as that in\u00a0Figure 6\u00a0are often used to analyze major storm systems, including\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term233\" class=\"no-emphasis\" data-type=\"term\">hurricanes<\/span>\u00a0and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an effect caused by Earth\u2019s rotation about its axis and is called the Coriolis Effect.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: sketching a vector field<\/h3>\r\nSketch vector field\u00a0[latex]{\\bf{F}}(x,y)=\\frac{y}{x^2+y^2}{\\bf{i}}-\\frac{x}{x^2+y^2}{\\bf{j}}[\/latex].\r\n\r\n[reveal-answer q=\"190865665\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190865665\"]\r\n<p id=\"fs-id1167793452931\">To visualize this vector field, first note that the dot product [latex]{\\bf{F}}(a,b)\\cdot(a{\\bf{i}}+b{\\bf{j}})[\/latex] is zero for any point [latex](a,b)[\/latex]. Therefore, each vector is tangent to the circle on which it is located. Also, as [latex](a,b)\\to(0,0)[\/latex], the magnitude of [latex]{\\bf{F}}(a,b)[\/latex] goes to infinity. To see this, note that<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{||{\\bf{F}}(a,b)||=\\sqrt{\\dfrac{a^2+b^2}{(a^2+b^2)^2}}=\\sqrt{\\dfrac{1}{a^2+b^2}}}[\/latex].<\/p>\r\nSince [latex]\\frac1{a^2+b^2}\\to\\infty[\/latex]\u00a0as [latex](a,b)\\to(0,0)[\/latex] then [latex]||{\\bf{F}}(a,b)||\\to\\infty[\/latex] as [latex](a,b)\\to(0,0)[\/latex]. This vector field looks similar to the vector field in\u00a0Example \"Chapter Opener: Drawing a Rotational Vector Field\", but in this case the magnitudes of the vectors close to the origin are large. The table below shows a sample of points and the corresponding vectors, and\u00a0Figure 6\u00a0shows the vector field. Note that this vector field models the whirlpool motion of the river in\u00a0Figure 1(b). The domain of this vector field is all of [latex]\\mathbb{R}^2[\/latex] except for point [latex](0, 0)[\/latex].\r\n<div id=\"fs-id1167793610554\" class=\"os-table\">\r\n<table class=\"unnumbered\" style=\"height: 115px; width: 734px;\" data-id=\"fs-id1167793610554\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\r\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\r\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\r\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-\\frac12\\rangle[\/latex]<\/td>\r\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\r\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,-\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\r\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\r\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,0\\rangle[\/latex]<\/td>\r\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\r\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\r\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\r\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\r\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,\\frac12\\rangle[\/latex]<\/td>\r\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\r\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\" valign=\"top\">\r\n<td style=\"width: 103.734px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\r\n<td style=\"width: 105.188px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\r\n<td style=\"width: 103.734px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\r\n<td style=\"width: 105.047px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,0\\rangle[\/latex]<\/td>\r\n<td style=\"width: 104.141px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\r\n<td style=\"width: 143.094px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,-\\frac12\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_3303\" align=\"aligncenter\" width=\"493\"]<img class=\"wp-image-3303 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162212\/6-1-3.jpeg\" alt=\"A visual representation of the given vector field in a coordinate plane. The magnitude is larger closer to the origin. The arrows are rotating the origin clockwise. It could be use to model whirlpool motion of a fluid.\" width=\"493\" height=\"497\" \/> Figure 6. A visual representation of vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{y}{x^2+y^2}{\\bf{i}}-\\frac{x}{x^2+y^2}{\\bf{j}}}[\/latex]. This vector field could be used to model whirlpool motion of a fluid.[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSketch vector field [latex]{\\bf{F}}(x,y)=\\langle-2y,2x\\rangle[\/latex]. Is the vector field radial, rotational, or neither?\r\n\r\n[reveal-answer q=\"458017451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458017451\"]\r\n\r\nRotational.\r\n\r\n[caption id=\"attachment_3304\" align=\"aligncenter\" width=\"717\"]<img class=\"wp-image-3304 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162230\/6-1-tryitans2.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/> Figure 7. A virtual representation of vector field\u00a0[latex]{\\bf{F}}(x,y)=\\langle-2y,2x\\rangle[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: velocity field of a fluid<\/h3>\r\nSuppose that [latex]{\\bf{v}}(x,y)=-\\frac{2y}{x^2+y^2}{\\bf{i}}+\\frac{2x}{x^2+y^2}{\\bf{j}}[\/latex] is the velocity field of a fluid. How fast is the fluid moving at point [latex](1, -1)[\/latex]?<span style=\"white-space: nowrap;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">(Assume the units of speed are meters per second.)<\/span>\r\n\r\n[reveal-answer q=\"850298742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"850298742\"]\r\n<p id=\"fs-id1167793420548\">To find the velocity of the fluid at point [latex](1, -1)[\/latex], substitute the point into [latex]\\bf{v}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{v}}(1,-1)=-\\frac{2(-1)}{1+1}{\\bf{i}}+\\frac{2(1)}{1+1}{\\bf{j}}={\\bf{i}}+{\\bf{j}}}[\/latex].<\/p>\r\n<p id=\"fs-id1167793559136\">The speed of the fluid at [latex](1, -1)[\/latex] is the magnitude of this vector. Therefore, the speed is [latex]||{\\bf{i}}+{\\bf{j}}||=\\sqrt2[\/latex] m\/sec.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nVector field [latex]v(x,y)=\\langle4|x|,1\\rangle[\/latex] models the velocity of water on the surface of a river. What is the speed of the water at point [latex](2, 3)[\/latex]? Use meters per second as the units.\r\n\r\n[reveal-answer q=\"239547128\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"239547128\"]\r\n\r\n[latex]\\sqrt{65}[\/latex] m\/sec.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250310&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DwYz5K5V70A&amp;video_target=tpm-plugin-bbimm0bj-DwYz5K5V70A\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.4_transcript.html\">transcript for \u201cCP 6.4\u201d here (opens in new window).<\/a><\/center>We have examined vector fields that contain vectors of various magnitudes, but just as we have unit vectors, we can also have a unit vector field. A vector field [latex]\\bf{F}[\/latex]\u00a0is a\u00a0<strong><span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term234\" data-type=\"term\">unit vector field<\/span><\/strong>\u00a0if the magnitude of each vector in the field is [latex]1[\/latex]. In a unit vector field, the only relevant information is the direction of each vector.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: a unit vector field<\/h3>\r\nShow that vector field [latex]{\\bf{F}}(x,y)=\\left\\langle\\frac{y}{\\sqrt{x^2+y^2}},-\\frac{x}{\\sqrt{x^2+y^2}}\\right\\rangle[\/latex] is a unit vector field.\r\n\r\n[reveal-answer q=\"144934750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"144934750\"]\r\n<p id=\"fs-id1167793936279\">To show that\u00a0[latex]\\bf{F}[\/latex]\u00a0is a unit field, we must show that the magnitude of each vector is [latex]1[\/latex]. Note that<\/p>\r\n[latex]\\hspace{4cm}\\large{\\begin{align}\r\n\r\n\\sqrt{\\left(\\frac{y}{\\sqrt{x^2+y^2}}\\right)^2+\\left(-\\frac{x}{\\sqrt{x^2+y^2}}\\right)^2}&amp;=\\sqrt{\\frac{y^2}{\\sqrt{x^2+y^2}}+\\frac{x^2}{\\sqrt{x^2+y^2}}} \\\\\r\n\r\n&amp;=\\sqrt{\\frac{x^2+y^2}{x^2+y^2}} \\\\\r\n\r\n&amp;=1.\r\n\r\n\\end{align}}[\/latex]\r\n<p id=\"fs-id1167793371583\">Therefore,\u00a0[latex]\\bf{F}[\/latex]\u00a0is a unit vector field.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nIs vector field [latex]{\\bf{F}}(x,y)=\\langle-y,x\\rangle[\/latex] a unit vector field?\r\n\r\n[reveal-answer q=\"594329457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594329457\"]\r\n\r\nNo.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1167793290986\">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. 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. Therefore, the unit vector field associated with velocity is the field we would study.<\/p>\r\n<p id=\"fs-id1167793926293\">If [latex]{\\bf{F}}(x,y)=\\langle{P,Q,R}\\rangle[\/latex] is a vector field, then the corresponding unit vector field is [latex]\\left\\langle\\frac{P}{||{\\bf{F}}||},\\frac{Q}{||{\\bf{F}}||},\\frac{R}{||{\\bf{F}}||}\\right\\rangle[\/latex]. Notice that if [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex] is the vector field from\u00a0Example \"Chapter Opener: Drawing a Rotational Vector Field\", then the magnitude of\u00a0[latex]\\bf{F}[\/latex]\u00a0is [latex]\\sqrt{x^2+y^2}[\/latex], and therefore the corresponding unit vector field is the field\u00a0[latex]\\bf{G}[\/latex]\u00a0from the previous example.<\/p>\r\n<p id=\"fs-id1167793355186\">If\u00a0[latex]\\bf{F}[\/latex]\u00a0is a vector field, then the process of dividing\u00a0[latex]\\bf{F}[\/latex]\u00a0by its magnitude to form unit vector field [latex]{\\bf{F}}\/||{\\bf{F}}||[\/latex] is called\u00a0<em data-effect=\"italics\">normalizing<\/em>\u00a0the field\u00a0[latex]\\bf{F}[\/latex].<\/p>\r\n\r\n<section id=\"fs-id1167794210544\" data-depth=\"2\">\r\n<h4 data-type=\"title\">Vector Fields in [latex]\\mathbb{R}^3[\/latex]<\/h4>\r\n<p id=\"fs-id1167793640104\">We have seen several examples of vector fields in [latex]\\mathbb{R}^2[\/latex]; let\u2019s now turn our attention to vector fields in [latex]\\mathbb{R}^3[\/latex]. These vector fields can be used to model gravitational or electromagnetic fields, and they can also be used to model fluid flow or heat flow in three dimensions. A two-dimensional vector field can really only model the movement of water on a two-dimensional slice of a river (such as the river\u2019s surface). Since a river flows through three spatial dimensions, to model the flow of the entire depth of the river, we need a vector field in three dimensions.<\/p>\r\n<p id=\"fs-id1167793518838\">The extra dimension of a three-dimensional field can make vector fields in [latex]\\mathbb{R}^3[\/latex] more difficult to visualize, but the idea is the same. To visualize a vector field in [latex]\\mathbb{R}^3[\/latex], plot enough vectors to show the overall shape. We can use a similar method to visualizing a vector field in [latex]\\mathbb{R}^2[\/latex] by choosing points in each octant.<\/p>\r\n<p id=\"fs-id1167794137223\">Just as with vector fields in [latex]\\mathbb{R}^2[\/latex], we can represent vector fields in [latex]\\mathbb{R}^3[\/latex] with component functions. We simply need an extra component function for the extra dimension. We write either<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y,z)=\\langle{P}(x,y,z),Q(x,y,z),R(x,y,z)\\rangle}[\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}}+Q(x,y,z){\\bf{j}}+R(x,y,z){\\bf{k}}}[\/latex].<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: sketching a vector field in three dimensions<\/h3>\r\nDescribe vector field [latex]{\\bf{F}}(x,y,z)=\\langle1,1,z\\rangle[\/latex].\r\n\r\n[reveal-answer q=\"698341485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698341485\"]\r\n\r\nFor this vector field, the [latex]x[\/latex] and [latex]y[\/latex] components are constant, so every point in [latex]\\mathbb{R}^3[\/latex] has an associated vector with [latex]x[\/latex] and [latex]y[\/latex] components equal to one. To visualize\u00a0[latex]\\bf{F}[\/latex], we first consider what the field looks like in the [latex]xy[\/latex]-plane. In the [latex]xy[\/latex]-plane, [latex]z=0[\/latex]. Hence, each point of the form [latex](a, b, 0)[\/latex] has vector [latex]\\langle 1,1,0\\rangle[\/latex] associated with it. For points not in the [latex]xy[\/latex]-plane but slightly above it, the associated vector has a small but positive [latex]z[\/latex] component, and therefore the associated vector points slightly upward. For points that are far above the [latex]xy[\/latex]-plane, the [latex]z[\/latex] component is large, so the vector is almost vertical.\u00a0Figure 8\u00a0shows this vector field.\r\n\r\n[caption id=\"attachment_3305\" align=\"aligncenter\" width=\"388\"]<img class=\"wp-image-3305 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162304\/6-1-4.jpeg\" alt=\"A visual representation of the given vector field in three dimensions. The arrows always have x and y components of 1. The z component changes according to the height. The closer z comes to 0, the smaller the z component becomes, and the further away z is from 0, the larger the z component becomes.\" width=\"388\" height=\"383\" \/> Figure 8. A visual representation of vector field [latex]{\\bf{F}}(x,y,z)=\\langle1,1,z\\rangle[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSketch vector field [latex]{\\bf{G}}(x,y,z)=\\langle2,\\frac{z}2,1\\rangle[\/latex].\r\n\r\n[reveal-answer q=\"234184589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"234184589\"]\r\n\r\n[caption id=\"attachment_5236\" align=\"aligncenter\" width=\"388\"]<img class=\"size-full wp-image-5236\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31215426\/6-1-tryitans31.jpeg\" alt=\"\" width=\"388\" height=\"383\" \/> Figure 9. A visual representation of vector field [latex]{\\bf{G}}(x,y,z)=\\langle2,z\/2,1\\rangle[\/latex].[\/caption][\/hidden-answer]<\/div>\r\nIn the next example, we explore one of the classic cases of a three-dimensional vector field: a gravitational field.\r\n<div id=\"fs-id1167793977405\" data-type=\"equation\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Describing a gravitational vector field<\/h3>\r\nNewton\u2019s law of gravitation states that [latex]{\\bf{F}}=-G\\frac{m_1m_2}{r^2}{\\bf{\\hat{r}}}[\/latex], where [latex]G[\/latex] is the universal gravitational constant. It describes the gravitational field exerted by an object (object 1) of mass [latex]m_1[\/latex] located at the origin on another object (object 2) of mass [latex]m_2[\/latex] located at point [latex](x, y, z)[\/latex]. Field [latex]{\\bf{F}}[\/latex]\u00a0denotes the gravitational force that object 1 exerts on object 2, [latex]r[\/latex] is the distance between the two objects, and [latex]{\\bf{\\hat{r}}}[\/latex] indicates the unit vector from the first object to the second. The minus sign shows that the gravitational force attracts toward the origin; that is, the force of object 1 is attractive. Sketch the vector field associated with this equation.\r\n\r\n[reveal-answer q=\"587238475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587238475\"]\r\n<p id=\"fs-id1167794222795\">Since object 1 is located at the origin, the distance between the objects is given by [latex]r=\\sqrt{x^2+y^2+z^2}[\/latex]. The unit vector from object 1 to object 2 is [latex]{\\bf{\\hat{r}}}=\\frac{\\langle{x},y,z\\rangle}{||\\langle{x},y,z\\rangle||}[\/latex], and hence [latex]{\\bf{\\hat{r}}}=\\langle\\frac{x}r,\\frac{y}r,\\frac{z}r\\rangle[\/latex]. Therefore, gravitational vector field\u00a0[latex]{\\bf{F}}[\/latex]\u00a0exerted by object 1 on object 2 is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}=-Gm_1m_2\\langle\\frac{x}{r^3},\\frac{y}{r^3},\\frac{z}{r^3}\\rangle}[\/latex].<\/p>\r\n<p id=\"fs-id1167793268303\">This is an example of a radial vector field in [latex]\\mathbb{R}^3[\/latex].<\/p>\r\n<p id=\"fs-id1167793409039\">Figure 10\u00a0shows what this gravitational field looks like for a large mass at the origin. Note that the magnitudes of the vectors increase as the vectors get closer to the origin.<\/p>\r\n\r\n[caption id=\"attachment_5234\" align=\"aligncenter\" width=\"368\"]<img class=\"size-full wp-image-5234\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31215057\/6.8.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/402d536f7b8d9535d73cedf0749b381641280ce3&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A visual representation of the given gravitational vector field in three dimensions. The magnitudes of the vectors increase as the vectors get closer to the origin. The arrows point in, towards the mass at the origin.&quot; id=&quot;27&quot;&gt;\" width=\"368\" height=\"399\" \/> Figure 10. A visual representation of gravitational vector field [latex]\\large{{\\bf{F}}=-Gm_1m_2\\langle\\frac{x}{r^3},\\frac{y}{r^3},\\frac{z}{r^3}\\rangle}[\/latex] for a large mass at the origin.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nThe mass of asteroid 1 is [latex]750,000[\/latex] kg and the mass of asteroid 2 is [latex]130,000[\/latex] kg. Assume asteroid 1 is located at the origin, and asteroid 2 is located at [latex](15, -5, 10)[\/latex], measured in units of 10 to the eighth power kilometers. Given that the universal gravitational constant is [latex]G=6.67384\\times10^{-11}\\text{m}^3\\text{kg}^{-1}\\text{s}^{-2}[\/latex], find the gravitational force vector that asteroid 1 exerts on asteroid 2.\r\n\r\n[reveal-answer q=\"834187034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834187034\"]\r\n\r\n[latex]1.49063\\times10^{-18},4.96876\\times10^{-19},9.93752\\times10^{-19}\\text{N}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250311&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=oPljuQWQZ6s&amp;video_target=tpm-plugin-iieqfnyr-oPljuQWQZ6s\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.7_transcript.html\">transcript for \u201cCP 6.7\u201d here (opens in new window).<\/a><\/center><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Recognize a vector field in a plane or in space.<\/span><\/li>\n<li><span class=\"os-abstract-content\">Sketch a vector field from a given equation.<\/span><\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1167793470654\" data-depth=\"1\">\n<h2 data-type=\"title\">Examples of Vector Fields<\/h2>\n<p id=\"fs-id1167793638685\">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?\u00a0Figure 1\u00a0gives visual representations of such phenomena.<\/p>\n<p id=\"fs-id1167793299474\">Figure 1(a) shows a gravitational field exerted by two astronomical objects, such as a star and a planet or a planet and a moon. 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. The vectors of largest magnitude in the figure are the vectors closest to the larger object. The larger object has greater mass, so it exerts a gravitational force of greater magnitude than the smaller object.<\/p>\n<p id=\"fs-id1167794209630\">Figure 1(b) shows the velocity of a river at points on its surface. The vector associated with a given point on the river\u2019s surface gives the velocity of the water at that point. Since the vectors to the left of the figure are small in magnitude, the water is flowing slowly on that part of the surface. As the water moves from left to right, it encounters some rapids around a rock. The speed of the water increases, and a whirlpool occurs in part of the rapids.<\/p>\n<div id=\"attachment_3299\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3299\" class=\"wp-image-3299 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162020\/6-1-1.jpeg\" alt=\"Two images, labeled A and B. Image A shows the gravitational field exerted by two astronomical bodies on a small object. The earth is on the left, and the moon is on the right. The earth is surrounded by long arrows pointing towards its center arranged in concentric circles. There is a break in the circle on the right, across from the moon. The moon is surrounded by smaller arrows that curve out and to the right. Image B shows the vector velocity field of water on the surface of a river with a large rock in the middle. The arrows tend to point at the same angle as the riverbank. Where the river meets the rock, the arrows point around the rock. After the rock, the some arrows point forward, and others turn back to the rock. The water flows fastest towards the middle of the river and around the rock and slowest along the riverbank.\" width=\"731\" height=\"841\" \/><\/p>\n<p id=\"caption-attachment-3299\" class=\"wp-caption-text\">Figure 1. (a) The gravitational field exerted by two astronomical bodies on a small object. (b) The vector velocity field of water on the surface of a river shows the varied speeds of water. Red indicates that the magnitude of the vector is greater, so the water flows more quickly; blue indicates a lesser magnitude and a slower speed of water flow.<\/p>\n<\/div>\n<p id=\"fs-id1167794060800\">Each figure illustrates an example of a vector field. Intuitively, a vector field is a map of vectors. In this section, we study vector fields in [latex]\\mathbb{R}^2[\/latex] and\u00a0[latex]\\mathbb{R}^3[\/latex].<\/p>\n<div data-type=\"note\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167794293131\">A\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term229\" data-type=\"term\"><strong>vector field [latex]F[\/latex]<\/strong> in\u00a0[latex]\\mathbb{R}^2[\/latex]\u00a0<\/span>is an assignment of a two-dimensional vector [latex]{\\bf{F}}(x,y)[\/latex]\u00a0to each point [latex](x, y)[\/latex] of a subset [latex]D[\/latex] of [latex]\\mathbb{R}^2[\/latex]. The subset [latex]D[\/latex] is the\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term230\" class=\"no-emphasis\" data-type=\"term\">domain<\/span>\u00a0of the vector field.<\/p>\n<p id=\"fs-id1167794328312\">A vector field <strong>[latex]F[\/latex]<\/strong> in [latex]\\mathbb{R}^3[\/latex] is an assignment of a three-dimensional vector [latex]{\\bf{F}}(x,y,z)[\/latex] to each point [latex](x, y, z)[\/latex] of a subset [latex]D[\/latex] of [latex]\\mathbb{R}^3[\/latex]. The subset [latex]D[\/latex] is the domain of the vector field.<\/p>\n<\/div>\n<h4 data-type=\"title\">Vector Fields in [latex]\\mathbb{R}^2[\/latex]<\/h4>\n<p id=\"fs-id1167793929736\">A vector field in [latex]\\mathbb{R}^2[\/latex] can be represented in either of two equivalent ways. The first way is to use a vector with components that are two-variable functions:<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y)=\\langle{P}(x,y),Q(x,y)\\rangle}[\/latex].<\/p>\n<p>The second way is to use the standard unit vectors:<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y)=P(x,y){\\bf{i}}+Q(x,y){\\bf{j}}}[\/latex].<\/p>\n<p>A vector field is said to be\u00a0<em data-effect=\"italics\">continuous<\/em>\u00a0if its component functions are continuous.<\/p>\n<div id=\"fs-id1167794027862\" class=\"ui-has-child-title\" data-type=\"example\">\n<div class=\"textbox exercises\">\n<h3>Example: finding a vector associated with a given point<\/h3>\n<p>Let [latex]{\\bf{F}}(x,y)=(2y^2+x-4){\\bf{i}}+\\cos(x){\\bf{j}}[\/latex] be a vector field in [latex]\\mathbb{R}^2[\/latex]. Note that this is an example of a continuous vector field since both component functions are continuous. What vector is associated with point [latex](0, -1)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q882649457\">Show Solution<\/span><\/p>\n<div id=\"q882649457\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794135008\">Substitute the point values for [latex]x[\/latex] and\u00a0[latex]y[\/latex]:<\/p>\n<p>[latex]\\hspace{8cm}\\begin{align}    {\\bf{F}}(0,1)&=(2(-1)^2+0-4){\\bf{i}}+\\cos(0){\\bf{j}} \\\\    &=-2{\\bf{i}}+{\\bf{j}}    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Let [latex]{\\bf{G}}(x,y)=x^2y{\\bf{i}}-(x+y){\\bf{j}}[\/latex] be a vector field in [latex]\\mathbb{R}^2[\/latex]. What vector is associated with the point [latex](-2, 3)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q872447946\">Show Solution<\/span><\/p>\n<div id=\"q872447946\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]12{\\bf{i}}-{\\bf{j}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 data-type=\"title\">Drawing a Vector Field<\/h2>\n<p id=\"fs-id1167794094176\">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 [latex]\\mathbb{R}^2[\/latex], as is the range. Therefore the \u201cgraph\u201d of a vector field in [latex]\\mathbb{R}^2[\/latex] lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in [latex]\\mathbb{R}^2[\/latex] in a plane itself. To do this, draw the vector associated with a given point at the point in a plane. For example, suppose the vector associated with point [latex](4, -1)[\/latex] is\u00a0[latex]\\langle3,1\\rangle[\/latex]. Then, we would draw vector [latex]\\langle3,1\\rangle[\/latex] at point\u00a0[latex](4,-1)[\/latex].<\/p>\n<p id=\"fs-id1167793267239\">We should plot enough vectors to see the general shape, but not so many that the sketch becomes a jumbled mess. If we were to plot the image vector at each point in the region, it would fill the region completely and is useless. 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 [latex]\\mathbb{R}^2[\/latex].<\/p>\n<p id=\"fs-id1167794064980\">There are two types of vector fields in [latex]\\mathbb{R}^2[\/latex] on which this chapter focuses: radial fields and rotational fields. Radial fields model certain gravitational fields and energy source fields and rotational fields model the movement of a fluid in a vortex. In a\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term231\" data-type=\"term\">radial field<\/span>, all vectors either point directly toward or directly away from the origin. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, the vector located at point [latex](x, y)[\/latex] is perpendicular to the circle centered at the origin that contains point [latex](x, y)[\/latex], and all other vectors on this circle have the same magnitude.<\/p>\n<div id=\"fs-id1167794336014\" class=\"ui-has-child-title\" data-type=\"example\">\n<div class=\"textbox exercises\">\n<h3>Example: drawing a radial vector field<\/h3>\n<p>Sketch the vector field [latex]{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q623977834\">Show Solution<\/span><\/p>\n<div id=\"q623977834\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793617583\">To sketch this vector field, choose a sample of points from each quadrant and compute the corresponding vector. The following table gives a representative sample of points in a plane and the corresponding vectors.<\/p>\n<div id=\"fs-id1167793952941\" class=\"os-table\">\n<table class=\"unnumbered\" style=\"height: 121px;\" data-id=\"fs-id1167793952941\" data-label=\"\">\n<tbody>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,\\frac12\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,-\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\" valign=\"top\">\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\n<td style=\"height: 25px; width: 178.672px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-\\frac12\\rangle[\/latex]<\/td>\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\n<td style=\"height: 25px; width: 153.703px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\n<td style=\"height: 25px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,-\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"fs-id1167794329899\">Figure 2(a) shows the vector field. To see that each vector is perpendicular to the corresponding circle,\u00a0Figure 2(b) shows circles overlain on the vector field.<\/p>\n<div id=\"attachment_3301\" style=\"width: 673px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3301\" class=\"wp-image-3301 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162053\/6-1-2-663x1024.jpeg\" alt=\"Visual representations of a radial vector field on a coordinate field. The arrows are stretching away from the origin in a radial pattern. The magnitudes increase the further the arrows are from the origin, so the lines are longer. The second version shows concentric circles around the origin to highlight the radial pattern.\" width=\"663\" height=\"1024\" \/><\/p>\n<p id=\"caption-attachment-3301\" class=\"wp-caption-text\">Figure 2. (a) A visual representation of the radial vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}}[\/latex]. (b) The radial vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{x}2{\\bf{i}}+\\frac{y}2{\\bf{j}}}[\/latex] with overlaid circles. Notice that each vector is perpendicular to the circle on which it is located.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Draw the radial field [latex]\\small{{\\bf{F}}(x,y)=-\\frac{x}3{\\bf{i}}-\\frac{y}3{\\bf{j}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q320974724\">Show Solution<\/span><\/p>\n<div id=\"q320974724\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"attachment_3302\" style=\"width: 727px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3302\" class=\"wp-image-3302 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162127\/6-1-tryitans1.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/><\/p>\n<p id=\"caption-attachment-3302\" class=\"wp-caption-text\">Figure 3. A visual representation of the radial vector field\u00a0[latex]\\small{{\\bf{F}}(x,y)=-\\frac{x}3{\\bf{i}}-\\frac{y}3{\\bf{j}}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In contrast to radial fields, in a\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term232\" data-type=\"term\">rotational field<\/span>, the vector at point [latex](x, y)[\/latex] is tangent (not perpendicular) to a circle with radius [latex]r=\\sqrt{x^2+y^2}[\/latex]. In a standard rotational field, all vectors point either in a clockwise direction or in a counterclockwise direction, and the magnitude of a vector depends only on its distance from the origin. Both of the following examples are clockwise rotational fields, and we see from their visual representations that the vectors appear to rotate around the origin.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: chapter opener: drawing a Rotational vector field<\/h3>\n<div id=\"attachment_5227\" style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5227\" class=\"size-full wp-image-5227\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31212332\/6.4.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/dc1c17f35ff4f1559d82e4922f942bee41e04cce&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A photograph of a hurricane, showing the rotation around its eye.&quot; id=&quot;11&quot;&gt;\" width=\"325\" height=\"215\" \/><\/p>\n<p id=\"caption-attachment-5227\" class=\"wp-caption-text\">Figure 4. (credit: modification of work by NASA)<\/p>\n<\/div>\n<p>Sketch the vector field\u00a0[latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q774923650\">Show Solution<\/span><\/p>\n<div id=\"q774923650\" class=\"hidden-answer\" style=\"display: none\">\n<h2 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h2>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1167794060214\">Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors.\u00a0Figure 5\u00a0shows the resulting vector field.<\/p>\n<div id=\"fs-id1167794034119\" class=\"os-table\">\n<table class=\"unnumbered\" data-id=\"fs-id1167793952941\" data-label=\"\">\n<tbody>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-2\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,-1\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle2,0\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,1\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\n<td style=\"height: 24px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,2\\rangle[\/latex]<\/td>\n<td style=\"height: 24px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\n<td style=\"height: 24px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,1\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\" valign=\"top\">\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\n<td style=\"height: 25px; width: 103.734px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\n<td style=\"height: 25px; width: 122px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-2,0\\rangle[\/latex]<\/td>\n<td style=\"height: 25px; width: 103.719px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\n<td style=\"height: 25px; width: 161px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,-1\\rangle[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"attachment_5230\" style=\"width: 750px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5230\" class=\"size-full wp-image-5230\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31212724\/6.5.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/0262ed9020ac9975013b56fb2d2ffcaf3369ea2f&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A visual representation of the given vector field in a coordinate plane with two additional diagrams with notation. The first representation shows the vector field. The arrows are circling the origin in a clockwise motion. The second representation shows concentric circles, highlighting the radial pattern. The The third representation shows the concentric circles. It also shows arrows for the radial vector &lt;a,b&gt; for all points (a,b). Each is perpendicular to the arrows in the given vector field.&quot; id=&quot;12&quot;&gt;\" width=\"740\" height=\"728\" \/><\/p>\n<p id=\"caption-attachment-5230\" class=\"wp-caption-text\">Figure 5. (a) A visual representation of vector field [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex]. (b) Vector field [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex] with circles centered at the origin. (c) [latex]{\\bf{F}}(a,b)[\/latex] is perpendicular to radial vector [latex]\\langle{a},b\\rangle[\/latex] at point [latex](a, b)[\/latex].<\/p>\n<\/div>\n<h2 id=\"13\" data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis<\/span><\/h2>\n<p id=\"fs-id1167793932094\">Note that vector [latex]{\\bf{F}}(a,b)=\\langle{b},-a\\rangle[\/latex] points clockwise and is perpendicular to radial vector [latex]\\langle{a},b\\rangle[\/latex]. (We can verify this assertion by computing the dot product of the two vectors:[latex]\\langle{a},b\\rangle\\cdot\\langle{-b},a\\rangle=-ab+ab=0[\/latex].) Furthermore, vector [latex]\\langle{b},-a\\rangle[\/latex] has length [latex]r=\\sqrt{a^2+b^2}[\/latex]. Thus, we have a complete description of this rotational vector field: the vector associated with point [latex](a, b)[\/latex] is the vector with length [latex]r[\/latex] tangent to the circle with radius [latex]r[\/latex], and it points in the clockwise direction.<\/p>\n<p id=\"fs-id1167793931853\">Sketches such as that in\u00a0Figure 6\u00a0are often used to analyze major storm systems, including\u00a0<span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term233\" class=\"no-emphasis\" data-type=\"term\">hurricanes<\/span>\u00a0and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an effect caused by Earth\u2019s rotation about its axis and is called the Coriolis Effect.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: sketching a vector field<\/h3>\n<p>Sketch vector field\u00a0[latex]{\\bf{F}}(x,y)=\\frac{y}{x^2+y^2}{\\bf{i}}-\\frac{x}{x^2+y^2}{\\bf{j}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190865665\">Show Solution<\/span><\/p>\n<div id=\"q190865665\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793452931\">To visualize this vector field, first note that the dot product [latex]{\\bf{F}}(a,b)\\cdot(a{\\bf{i}}+b{\\bf{j}})[\/latex] is zero for any point [latex](a,b)[\/latex]. Therefore, each vector is tangent to the circle on which it is located. Also, as [latex](a,b)\\to(0,0)[\/latex], the magnitude of [latex]{\\bf{F}}(a,b)[\/latex] goes to infinity. To see this, note that<\/p>\n<p style=\"text-align: center;\">[latex]\\large{||{\\bf{F}}(a,b)||=\\sqrt{\\dfrac{a^2+b^2}{(a^2+b^2)^2}}=\\sqrt{\\dfrac{1}{a^2+b^2}}}[\/latex].<\/p>\n<p>Since [latex]\\frac1{a^2+b^2}\\to\\infty[\/latex]\u00a0as [latex](a,b)\\to(0,0)[\/latex] then [latex]||{\\bf{F}}(a,b)||\\to\\infty[\/latex] as [latex](a,b)\\to(0,0)[\/latex]. This vector field looks similar to the vector field in\u00a0Example &#8220;Chapter Opener: Drawing a Rotational Vector Field&#8221;, but in this case the magnitudes of the vectors close to the origin are large. The table below shows a sample of points and the corresponding vectors, and\u00a0Figure 6\u00a0shows the vector field. Note that this vector field models the whirlpool motion of the river in\u00a0Figure 1(b). The domain of this vector field is all of [latex]\\mathbb{R}^2[\/latex] except for point [latex](0, 0)[\/latex].<\/p>\n<div id=\"fs-id1167793610554\" class=\"os-table\">\n<table class=\"unnumbered\" style=\"height: 115px; width: 734px;\" data-id=\"fs-id1167793610554\" data-label=\"\">\n<tbody>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](x, y)[\/latex]<\/td>\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]F(x, y)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 0)[\/latex]<\/td>\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-1\\rangle[\/latex]<\/td>\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](2, 0)[\/latex]<\/td>\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,-\\frac12\\rangle[\/latex]<\/td>\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](1, 1)[\/latex]<\/td>\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,-\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 1)[\/latex]<\/td>\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle1,0\\rangle[\/latex]<\/td>\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](0, 2)[\/latex]<\/td>\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,0\\rangle[\/latex]<\/td>\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 1)[\/latex]<\/td>\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle\\frac12,\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, 0)[\/latex]<\/td>\n<td style=\"width: 105.188px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,1\\rangle[\/latex]<\/td>\n<td style=\"width: 103.734px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-2, 0)[\/latex]<\/td>\n<td style=\"width: 105.047px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle0,\\frac12\\rangle[\/latex]<\/td>\n<td style=\"width: 104.141px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex](-1, -1)[\/latex]<\/td>\n<td style=\"width: 143.094px; height: 24px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\" valign=\"top\">\n<td style=\"width: 103.734px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -1)[\/latex]<\/td>\n<td style=\"width: 105.188px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-1,0\\rangle[\/latex]<\/td>\n<td style=\"width: 103.734px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](0, -2)[\/latex]<\/td>\n<td style=\"width: 105.047px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,0\\rangle[\/latex]<\/td>\n<td style=\"width: 104.141px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex](1, -1)[\/latex]<\/td>\n<td style=\"width: 143.094px; height: 19px;\" data-valign=\"top\" data-align=\"left\">[latex]\\langle-\\frac12,-\\frac12\\rangle[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"attachment_3303\" style=\"width: 503px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3303\" class=\"wp-image-3303 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162212\/6-1-3.jpeg\" alt=\"A visual representation of the given vector field in a coordinate plane. The magnitude is larger closer to the origin. The arrows are rotating the origin clockwise. It could be use to model whirlpool motion of a fluid.\" width=\"493\" height=\"497\" \/><\/p>\n<p id=\"caption-attachment-3303\" class=\"wp-caption-text\">Figure 6. A visual representation of vector field [latex]\\small{{\\bf{F}}(x,y)=\\frac{y}{x^2+y^2}{\\bf{i}}-\\frac{x}{x^2+y^2}{\\bf{j}}}[\/latex]. This vector field could be used to model whirlpool motion of a fluid.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Sketch vector field [latex]{\\bf{F}}(x,y)=\\langle-2y,2x\\rangle[\/latex]. Is the vector field radial, rotational, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q458017451\">Show Solution<\/span><\/p>\n<div id=\"q458017451\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rotational.<\/p>\n<div id=\"attachment_3304\" style=\"width: 727px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3304\" class=\"wp-image-3304 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162230\/6-1-tryitans2.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/><\/p>\n<p id=\"caption-attachment-3304\" class=\"wp-caption-text\">Figure 7. A virtual representation of vector field\u00a0[latex]{\\bf{F}}(x,y)=\\langle-2y,2x\\rangle[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: velocity field of a fluid<\/h3>\n<p>Suppose that [latex]{\\bf{v}}(x,y)=-\\frac{2y}{x^2+y^2}{\\bf{i}}+\\frac{2x}{x^2+y^2}{\\bf{j}}[\/latex] is the velocity field of a fluid. How fast is the fluid moving at point [latex](1, -1)[\/latex]?<span style=\"white-space: nowrap;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">(Assume the units of speed are meters per second.)<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q850298742\">Show Solution<\/span><\/p>\n<div id=\"q850298742\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793420548\">To find the velocity of the fluid at point [latex](1, -1)[\/latex], substitute the point into [latex]\\bf{v}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{v}}(1,-1)=-\\frac{2(-1)}{1+1}{\\bf{i}}+\\frac{2(1)}{1+1}{\\bf{j}}={\\bf{i}}+{\\bf{j}}}[\/latex].<\/p>\n<p id=\"fs-id1167793559136\">The speed of the fluid at [latex](1, -1)[\/latex] is the magnitude of this vector. Therefore, the speed is [latex]||{\\bf{i}}+{\\bf{j}}||=\\sqrt2[\/latex] m\/sec.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Vector field [latex]v(x,y)=\\langle4|x|,1\\rangle[\/latex] models the velocity of water on the surface of a river. What is the speed of the water at point [latex](2, 3)[\/latex]? Use meters per second as the units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q239547128\">Show Solution<\/span><\/p>\n<div id=\"q239547128\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\sqrt{65}[\/latex] m\/sec.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250310&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DwYz5K5V70A&amp;video_target=tpm-plugin-bbimm0bj-DwYz5K5V70A\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.4_transcript.html\">transcript for \u201cCP 6.4\u201d here (opens in new window).<\/a><\/div>\n<p>We have examined vector fields that contain vectors of various magnitudes, but just as we have unit vectors, we can also have a unit vector field. A vector field [latex]\\bf{F}[\/latex]\u00a0is a\u00a0<strong><span id=\"4ea1e6ac-b330-4b3a-a84a-b899b4fc0de5_term234\" data-type=\"term\">unit vector field<\/span><\/strong>\u00a0if the magnitude of each vector in the field is [latex]1[\/latex]. In a unit vector field, the only relevant information is the direction of each vector.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: a unit vector field<\/h3>\n<p>Show that vector field [latex]{\\bf{F}}(x,y)=\\left\\langle\\frac{y}{\\sqrt{x^2+y^2}},-\\frac{x}{\\sqrt{x^2+y^2}}\\right\\rangle[\/latex] is a unit vector field.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q144934750\">Show Solution<\/span><\/p>\n<div id=\"q144934750\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793936279\">To show that\u00a0[latex]\\bf{F}[\/latex]\u00a0is a unit field, we must show that the magnitude of each vector is [latex]1[\/latex]. Note that<\/p>\n<p>[latex]\\hspace{4cm}\\large{\\begin{align}    \\sqrt{\\left(\\frac{y}{\\sqrt{x^2+y^2}}\\right)^2+\\left(-\\frac{x}{\\sqrt{x^2+y^2}}\\right)^2}&=\\sqrt{\\frac{y^2}{\\sqrt{x^2+y^2}}+\\frac{x^2}{\\sqrt{x^2+y^2}}} \\\\    &=\\sqrt{\\frac{x^2+y^2}{x^2+y^2}} \\\\    &=1.    \\end{align}}[\/latex]<\/p>\n<p id=\"fs-id1167793371583\">Therefore,\u00a0[latex]\\bf{F}[\/latex]\u00a0is a unit vector field.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Is vector field [latex]{\\bf{F}}(x,y)=\\langle-y,x\\rangle[\/latex] a unit vector field?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594329457\">Show Solution<\/span><\/p>\n<div id=\"q594329457\" class=\"hidden-answer\" style=\"display: none\">\n<p>No.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793290986\">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. 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. Therefore, the unit vector field associated with velocity is the field we would study.<\/p>\n<p id=\"fs-id1167793926293\">If [latex]{\\bf{F}}(x,y)=\\langle{P,Q,R}\\rangle[\/latex] is a vector field, then the corresponding unit vector field is [latex]\\left\\langle\\frac{P}{||{\\bf{F}}||},\\frac{Q}{||{\\bf{F}}||},\\frac{R}{||{\\bf{F}}||}\\right\\rangle[\/latex]. Notice that if [latex]{\\bf{F}}(x,y)=\\langle{y},-x\\rangle[\/latex] is the vector field from\u00a0Example &#8220;Chapter Opener: Drawing a Rotational Vector Field&#8221;, then the magnitude of\u00a0[latex]\\bf{F}[\/latex]\u00a0is [latex]\\sqrt{x^2+y^2}[\/latex], and therefore the corresponding unit vector field is the field\u00a0[latex]\\bf{G}[\/latex]\u00a0from the previous example.<\/p>\n<p id=\"fs-id1167793355186\">If\u00a0[latex]\\bf{F}[\/latex]\u00a0is a vector field, then the process of dividing\u00a0[latex]\\bf{F}[\/latex]\u00a0by its magnitude to form unit vector field [latex]{\\bf{F}}\/||{\\bf{F}}||[\/latex] is called\u00a0<em data-effect=\"italics\">normalizing<\/em>\u00a0the field\u00a0[latex]\\bf{F}[\/latex].<\/p>\n<section id=\"fs-id1167794210544\" data-depth=\"2\">\n<h4 data-type=\"title\">Vector Fields in [latex]\\mathbb{R}^3[\/latex]<\/h4>\n<p id=\"fs-id1167793640104\">We have seen several examples of vector fields in [latex]\\mathbb{R}^2[\/latex]; let\u2019s now turn our attention to vector fields in [latex]\\mathbb{R}^3[\/latex]. These vector fields can be used to model gravitational or electromagnetic fields, and they can also be used to model fluid flow or heat flow in three dimensions. A two-dimensional vector field can really only model the movement of water on a two-dimensional slice of a river (such as the river\u2019s surface). Since a river flows through three spatial dimensions, to model the flow of the entire depth of the river, we need a vector field in three dimensions.<\/p>\n<p id=\"fs-id1167793518838\">The extra dimension of a three-dimensional field can make vector fields in [latex]\\mathbb{R}^3[\/latex] more difficult to visualize, but the idea is the same. To visualize a vector field in [latex]\\mathbb{R}^3[\/latex], plot enough vectors to show the overall shape. We can use a similar method to visualizing a vector field in [latex]\\mathbb{R}^2[\/latex] by choosing points in each octant.<\/p>\n<p id=\"fs-id1167794137223\">Just as with vector fields in [latex]\\mathbb{R}^2[\/latex], we can represent vector fields in [latex]\\mathbb{R}^3[\/latex] with component functions. We simply need an extra component function for the extra dimension. We write either<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y,z)=\\langle{P}(x,y,z),Q(x,y,z),R(x,y,z)\\rangle}[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}(x,y,z)=P(x,y,z){\\bf{i}}+Q(x,y,z){\\bf{j}}+R(x,y,z){\\bf{k}}}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: sketching a vector field in three dimensions<\/h3>\n<p>Describe vector field [latex]{\\bf{F}}(x,y,z)=\\langle1,1,z\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698341485\">Show Solution<\/span><\/p>\n<div id=\"q698341485\" class=\"hidden-answer\" style=\"display: none\">\n<p>For this vector field, the [latex]x[\/latex] and [latex]y[\/latex] components are constant, so every point in [latex]\\mathbb{R}^3[\/latex] has an associated vector with [latex]x[\/latex] and [latex]y[\/latex] components equal to one. To visualize\u00a0[latex]\\bf{F}[\/latex], we first consider what the field looks like in the [latex]xy[\/latex]-plane. In the [latex]xy[\/latex]-plane, [latex]z=0[\/latex]. Hence, each point of the form [latex](a, b, 0)[\/latex] has vector [latex]\\langle 1,1,0\\rangle[\/latex] associated with it. For points not in the [latex]xy[\/latex]-plane but slightly above it, the associated vector has a small but positive [latex]z[\/latex] component, and therefore the associated vector points slightly upward. For points that are far above the [latex]xy[\/latex]-plane, the [latex]z[\/latex] component is large, so the vector is almost vertical.\u00a0Figure 8\u00a0shows this vector field.<\/p>\n<div id=\"attachment_3305\" style=\"width: 398px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3305\" class=\"wp-image-3305 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162304\/6-1-4.jpeg\" alt=\"A visual representation of the given vector field in three dimensions. The arrows always have x and y components of 1. The z component changes according to the height. The closer z comes to 0, the smaller the z component becomes, and the further away z is from 0, the larger the z component becomes.\" width=\"388\" height=\"383\" \/><\/p>\n<p id=\"caption-attachment-3305\" class=\"wp-caption-text\">Figure 8. A visual representation of vector field [latex]{\\bf{F}}(x,y,z)=\\langle1,1,z\\rangle[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Sketch vector field [latex]{\\bf{G}}(x,y,z)=\\langle2,\\frac{z}2,1\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q234184589\">Show Solution<\/span><\/p>\n<div id=\"q234184589\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"attachment_5236\" style=\"width: 398px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5236\" class=\"size-full wp-image-5236\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31215426\/6-1-tryitans31.jpeg\" alt=\"\" width=\"388\" height=\"383\" \/><\/p>\n<p id=\"caption-attachment-5236\" class=\"wp-caption-text\">Figure 9. A visual representation of vector field [latex]{\\bf{G}}(x,y,z)=\\langle2,z\/2,1\\rangle[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we explore one of the classic cases of a three-dimensional vector field: a gravitational field.<\/p>\n<div id=\"fs-id1167793977405\" data-type=\"equation\">\n<div class=\"textbox exercises\">\n<h3>Example: Describing a gravitational vector field<\/h3>\n<p>Newton\u2019s law of gravitation states that [latex]{\\bf{F}}=-G\\frac{m_1m_2}{r^2}{\\bf{\\hat{r}}}[\/latex], where [latex]G[\/latex] is the universal gravitational constant. It describes the gravitational field exerted by an object (object 1) of mass [latex]m_1[\/latex] located at the origin on another object (object 2) of mass [latex]m_2[\/latex] located at point [latex](x, y, z)[\/latex]. Field [latex]{\\bf{F}}[\/latex]\u00a0denotes the gravitational force that object 1 exerts on object 2, [latex]r[\/latex] is the distance between the two objects, and [latex]{\\bf{\\hat{r}}}[\/latex] indicates the unit vector from the first object to the second. The minus sign shows that the gravitational force attracts toward the origin; that is, the force of object 1 is attractive. Sketch the vector field associated with this equation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587238475\">Show Solution<\/span><\/p>\n<div id=\"q587238475\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794222795\">Since object 1 is located at the origin, the distance between the objects is given by [latex]r=\\sqrt{x^2+y^2+z^2}[\/latex]. The unit vector from object 1 to object 2 is [latex]{\\bf{\\hat{r}}}=\\frac{\\langle{x},y,z\\rangle}{||\\langle{x},y,z\\rangle||}[\/latex], and hence [latex]{\\bf{\\hat{r}}}=\\langle\\frac{x}r,\\frac{y}r,\\frac{z}r\\rangle[\/latex]. Therefore, gravitational vector field\u00a0[latex]{\\bf{F}}[\/latex]\u00a0exerted by object 1 on object 2 is<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\bf{F}}=-Gm_1m_2\\langle\\frac{x}{r^3},\\frac{y}{r^3},\\frac{z}{r^3}\\rangle}[\/latex].<\/p>\n<p id=\"fs-id1167793268303\">This is an example of a radial vector field in [latex]\\mathbb{R}^3[\/latex].<\/p>\n<p id=\"fs-id1167793409039\">Figure 10\u00a0shows what this gravitational field looks like for a large mass at the origin. Note that the magnitudes of the vectors increase as the vectors get closer to the origin.<\/p>\n<div id=\"attachment_5234\" style=\"width: 378px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5234\" class=\"size-full wp-image-5234\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/31215057\/6.8.jpg\" alt=\"&lt;img src=&quot;\/apps\/archive\/20220422.171947\/resources\/402d536f7b8d9535d73cedf0749b381641280ce3&quot; data-media-type=&quot;image\/jpeg&quot; alt=&quot;A visual representation of the given gravitational vector field in three dimensions. The magnitudes of the vectors increase as the vectors get closer to the origin. The arrows point in, towards the mass at the origin.&quot; id=&quot;27&quot;&gt;\" width=\"368\" height=\"399\" \/><\/p>\n<p id=\"caption-attachment-5234\" class=\"wp-caption-text\">Figure 10. A visual representation of gravitational vector field [latex]\\large{{\\bf{F}}=-Gm_1m_2\\langle\\frac{x}{r^3},\\frac{y}{r^3},\\frac{z}{r^3}\\rangle}[\/latex] for a large mass at the origin.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>The mass of asteroid 1 is [latex]750,000[\/latex] kg and the mass of asteroid 2 is [latex]130,000[\/latex] kg. Assume asteroid 1 is located at the origin, and asteroid 2 is located at [latex](15, -5, 10)[\/latex], measured in units of 10 to the eighth power kilometers. Given that the universal gravitational constant is [latex]G=6.67384\\times10^{-11}\\text{m}^3\\text{kg}^{-1}\\text{s}^{-2}[\/latex], find the gravitational force vector that asteroid 1 exerts on asteroid 2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834187034\">Show Solution<\/span><\/p>\n<div id=\"q834187034\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1.49063\\times10^{-18},4.96876\\times10^{-19},9.93752\\times10^{-19}\\text{N}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250311&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=oPljuQWQZ6s&amp;video_target=tpm-plugin-iieqfnyr-oPljuQWQZ6s\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP6.7_transcript.html\">transcript for \u201cCP 6.7\u201d here (opens in new window).<\/a><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1117\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 6.4. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>CP 6.7. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":428269,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 6.4\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"CP 6.7\",\"author\":\"Ryan 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