{"id":744,"date":"2025-10-09T15:35:55","date_gmt":"2025-10-09T15:35:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=744"},"modified":"2025-11-13T18:49:09","modified_gmt":"2025-11-13T18:49:09","slug":"introduction-to-vector-fields-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-vector-fields-fresh-take\/","title":{"raw":"Introduction to Vector Fields: Fresh Take","rendered":"Introduction to Vector Fields: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize a vector field in a plane or in space<\/li>\r\n \t<li>Sketch a vector field from a given equation<\/li>\r\n \t<li>Identify a conservative field and its associated potential function<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Examples of Vector Fields<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A vector field assigns a vector to each point in space, describing phenomena where both magnitude and direction matter at every location. Think of a weather map showing wind velocity\u2014at each point, an arrow indicates wind direction and its length shows speed. Vector fields model gravitational forces, fluid flow, electromagnetic fields, and many other physical systems.<\/p>\r\n<p class=\"whitespace-normal break-words\">In two dimensions, a vector field [latex]\\mathbf{F}[\/latex] assigns a two-dimensional vector [latex]\\mathbf{F}(x,y)[\/latex] to each point <a class=\"underline\" href=\"x,y\">latex<\/a>[\/latex] in its domain. In three dimensions, it assigns [latex]\\mathbf{F}(x,y,z)[\/latex] to each point <a class=\"underline\" href=\"x,y,z\">latex<\/a>[\/latex]. You can think of a vector field as a function whose inputs are points and outputs are vectors.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Representing Vector Fields<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A vector field in [latex]\\mathbb{R}^2[\/latex] can be written in two equivalent forms:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Component form:<\/strong> [latex]\\mathbf{F}(x,y) = \\langle P(x,y), Q(x,y) \\rangle[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Standard unit vector form:<\/strong> [latex]\\mathbf{F}(x,y) = P(x,y)\\mathbf{i} + Q(x,y)\\mathbf{j}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The functions [latex]P(x,y)[\/latex] and [latex]Q(x,y)[\/latex] are the component functions\u2014[latex]P[\/latex] gives the horizontal component and [latex]Q[\/latex] gives the vertical component. A vector field is continuous if both component functions are continuous.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Physical Interpretation<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Vector fields naturally describe physical situations. In a gravitational field, each vector shows the force acting on a unit mass at that point\u2014larger vectors near massive objects indicate stronger gravitational pull. In a velocity field representing water flow, each vector shows the water's velocity at that location\u2014longer vectors indicate faster flow, and the direction shows which way the water moves.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Visualizing Vector Fields<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">At each point in the domain, draw an arrow representing the vector at that point. The arrow's direction shows the vector's direction, and its length represents magnitude. Regions with longer arrows have greater magnitude; regions with shorter arrows have lesser magnitude. The pattern of arrows reveals the overall structure of the field.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">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]?[reveal-answer q=\"872447946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"872447946\"][latex]12{\\bf{i}}-{\\bf{j}}[\/latex].[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Drawing a Vector Field<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">To visualize a vector field, draw representative vectors at their associated points. For [latex]\\mathbf{F}(x,y)[\/latex], evaluate the vector at point [latex](x,y)[\/latex] and draw it starting from that location. Choose enough points to reveal the pattern without creating clutter\u2014typically grid intersections from each quadrant work well.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Radial Fields<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Radial fields have vectors pointing directly toward or away from the origin. All vectors are perpendicular to circles centered at the origin, and magnitude depends only on distance from center. Points on the same circle have equal-magnitude vectors. These model gravitational fields (pointing inward) or energy sources (pointing outward).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Rotational Fields<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Rotational fields have vectors tangent to circles around the origin, rotating either clockwise or counterclockwise. Magnitude depends on distance from origin. To verify a field is rotational, check that [latex]\\mathbf{F}(a,b) \\cdot \\langle a,b \\rangle = 0[\/latex]\u2014the field vector is perpendicular to the radial direction. These model vortices and whirlpools.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Unit Vector Fields<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A unit vector field has [latex]||\\mathbf{F}(x,y)|| = 1[\/latex] everywhere\u2014all vectors have magnitude one. To create a unit vector field from [latex]\\mathbf{F}[\/latex], normalize by dividing: [latex]\\frac{\\mathbf{F}}{||\\mathbf{F}||}[\/latex]. This shows direction without magnitude, useful when only flow pattern matters, not speed.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Vector Fields in [latex]\\mathbb{R}^3[\/latex]<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Three-dimensional fields use three components: [latex]\\mathbf{F}(x,y,z) = \\langle P(x,y,z), Q(x,y,z), R(x,y,z) \\rangle[\/latex]. To visualize, first understand behavior in coordinate planes, then see how vectors change moving perpendicular to those planes. Choose representative points from different octants.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Draw the radial field [latex]\\small{{\\bf{F}}(x,y)=-\\frac{x}3{\\bf{i}}-\\frac{y}3{\\bf{j}}}[\/latex].[reveal-answer q=\"320974724\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"320974724\"][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]<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch vector field [latex]{\\bf{F}}(x,y)=\\langle-2y,2x\\rangle[\/latex]. Is the vector field radial, rotational, or neither?[reveal-answer q=\"458017451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458017451\"]Rotational.[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]<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"239547128\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"239547128\"][latex]\\sqrt{65}[\/latex] m\/sec.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Is vector field [latex]{\\bf{F}}(x,y)=\\langle-y,x\\rangle[\/latex] a unit vector field?[reveal-answer q=\"594329457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594329457\"]No.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch vector field [latex]{\\bf{G}}(x,y,z)=\\langle2,\\frac{z}2,1\\rangle[\/latex].[reveal-answer q=\"234184589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"234184589\"][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]<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"834187034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834187034\"][latex]1.49063\\times10^{-18},4.96876\\times10^{-19},9.93752\\times10^{-19}\\text{N}[\/latex][\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Gradient Fields<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A gradient field (also called a conservative field) arises when a vector field equals the gradient of some scalar function. These fields model physical systems where energy is conserved, such as gravitational and electrostatic forces. Understanding gradient fields reveals important connections between scalar functions and vector fields.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>What is a Gradient Field?<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Vector field [latex]\\mathbf{F}[\/latex] is a gradient field if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla f = \\mathbf{F}[\/latex]. The function [latex]f[\/latex] is called a potential function for [latex]\\mathbf{F}[\/latex]. For two variables, [latex]\\nabla f = \\langle f_x, f_y \\rangle[\/latex]. For three variables, [latex]\\nabla f = \\langle f_x, f_y, f_z \\rangle[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Relationship to Level Curves<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Gradient vectors are always perpendicular to level curves of the function. Where level curves are closely spaced (steep regions), gradient vectors have large magnitude. Where level curves are widely spaced (gradual regions), gradient vectors have small magnitude. This makes sense because the gradient points in the direction of greatest increase.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Potential Functions<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]\\mathbf{F}[\/latex] is conservative with potential functions [latex]f[\/latex] and [latex]g[\/latex], then [latex]f = g + C[\/latex] for some constant [latex]C[\/latex]. This parallels how antiderivatives differ by constants in single-variable calculus. Potential functions are unique up to an additive constant.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Cross-Partial Property<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Conservative fields satisfy special equations called the cross-partial property. In [latex]\\mathbb{R}^2[\/latex], if [latex]\\mathbf{F} = \\langle P, Q \\rangle[\/latex] is conservative, then [latex]\\frac{\\partial P}{\\partial y} = \\frac{\\partial Q}{\\partial x}[\/latex]. In [latex]\\mathbb{R}^3[\/latex], three equations must hold: [latex]P_y = Q_x[\/latex], [latex]Q_z = R_y[\/latex], and [latex]R_x = P_z[\/latex]. These come from Clairaut's theorem: [latex]f_{xy} = f_{yx}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Important Limitation<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If a field fails the cross-partial property, it's definitely not conservative. However, satisfying the cross-partial property doesn't guarantee the field is conservative\u2014it's necessary but not sufficient. Most vector fields are not conservative since the cross-partial equations are restrictive.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use technology to plot the gradient vector field of\u00a0[latex]f(x,y)=\\sin{x}\\cos{y}[\/latex].[reveal-answer q=\"485726668\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485726668\"][caption id=\"attachment_3309\" align=\"aligncenter\" width=\"717\"]<img class=\"wp-image-3309 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162738\/6-1-tryitans4.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/> Figure 2. A virtual representation of the gradient vector field\u00a0[latex]f(x,y)=\\sin{x}\\cos{y}[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Is [latex]f(x,y,z)=x^2\\cos{(yz)}+y^2z^2[\/latex] a potential function for [latex]{\\bf{F}}(x,y,z)=\\langle2x\\cos{(yz)},-x^2z\\sin{(yz)}+2yz^2,y^2\\rangle[\/latex]?[reveal-answer q=\"764528702\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"764528702\"]No.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Verify that [latex]f(x, y)=x^{2}y^{2}+x[\/latex] is a potential function for velocity field [latex]{\\bf{v}}(x,y)=\\langle2xy^2+1,2x^2y\\rangle[\/latex].[reveal-answer q=\"324714857\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324714857\"][latex]\\nabla{f}=\\bf{v}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Show that vector field [latex]{\\bf{F}}(x,y)x=y{\\bf{i}}-x^2y{\\bf{j}}[\/latex] is not conservative.[reveal-answer q=\"736457208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"736457208\"][latex]\\sqrt{2}[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Is vector field [latex]{\\bf{G}}(x,y,z)=\\langle{y},x,xyz\\rangle[\/latex] conservative?[reveal-answer q=\"183428645\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183428645\"]No.[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize a vector field in a plane or in space<\/li>\n<li>Sketch a vector field from a given equation<\/li>\n<li>Identify a conservative field and its associated potential function<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Examples of Vector Fields<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A vector field assigns a vector to each point in space, describing phenomena where both magnitude and direction matter at every location. Think of a weather map showing wind velocity\u2014at each point, an arrow indicates wind direction and its length shows speed. Vector fields model gravitational forces, fluid flow, electromagnetic fields, and many other physical systems.<\/p>\n<p class=\"whitespace-normal break-words\">In two dimensions, a vector field [latex]\\mathbf{F}[\/latex] assigns a two-dimensional vector [latex]\\mathbf{F}(x,y)[\/latex] to each point <a class=\"underline\" href=\"x,y\">latex<\/a>[\/latex] in its domain. In three dimensions, it assigns [latex]\\mathbf{F}(x,y,z)[\/latex] to each point <a class=\"underline\" href=\"x,y,z\">latex<\/a>[\/latex]. You can think of a vector field as a function whose inputs are points and outputs are vectors.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Representing Vector Fields<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A vector field in [latex]\\mathbb{R}^2[\/latex] can be written in two equivalent forms:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Component form:<\/strong> [latex]\\mathbf{F}(x,y) = \\langle P(x,y), Q(x,y) \\rangle[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Standard unit vector form:<\/strong> [latex]\\mathbf{F}(x,y) = P(x,y)\\mathbf{i} + Q(x,y)\\mathbf{j}[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The functions [latex]P(x,y)[\/latex] and [latex]Q(x,y)[\/latex] are the component functions\u2014[latex]P[\/latex] gives the horizontal component and [latex]Q[\/latex] gives the vertical component. A vector field is continuous if both component functions are continuous.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Physical Interpretation<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Vector fields naturally describe physical situations. In a gravitational field, each vector shows the force acting on a unit mass at that point\u2014larger vectors near massive objects indicate stronger gravitational pull. In a velocity field representing water flow, each vector shows the water&#8217;s velocity at that location\u2014longer vectors indicate faster flow, and the direction shows which way the water moves.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Visualizing Vector Fields<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">At each point in the domain, draw an arrow representing the vector at that point. The arrow&#8217;s direction shows the vector&#8217;s direction, and its length represents magnitude. Regions with longer arrows have greater magnitude; regions with shorter arrows have lesser magnitude. The pattern of arrows reveals the overall structure of the field.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q872447946\">Show Solution<\/button><\/p>\n<div id=\"q872447946\" class=\"hidden-answer\" style=\"display: none\">[latex]12{\\bf{i}}-{\\bf{j}}[\/latex].<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Drawing a Vector Field<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To visualize a vector field, draw representative vectors at their associated points. For [latex]\\mathbf{F}(x,y)[\/latex], evaluate the vector at point [latex](x,y)[\/latex] and draw it starting from that location. Choose enough points to reveal the pattern without creating clutter\u2014typically grid intersections from each quadrant work well.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Radial Fields<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Radial fields have vectors pointing directly toward or away from the origin. All vectors are perpendicular to circles centered at the origin, and magnitude depends only on distance from center. Points on the same circle have equal-magnitude vectors. These model gravitational fields (pointing inward) or energy sources (pointing outward).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Rotational Fields<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Rotational fields have vectors tangent to circles around the origin, rotating either clockwise or counterclockwise. Magnitude depends on distance from origin. To verify a field is rotational, check that [latex]\\mathbf{F}(a,b) \\cdot \\langle a,b \\rangle = 0[\/latex]\u2014the field vector is perpendicular to the radial direction. These model vortices and whirlpools.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Unit Vector Fields<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A unit vector field has [latex]||\\mathbf{F}(x,y)|| = 1[\/latex] everywhere\u2014all vectors have magnitude one. To create a unit vector field from [latex]\\mathbf{F}[\/latex], normalize by dividing: [latex]\\frac{\\mathbf{F}}{||\\mathbf{F}||}[\/latex]. This shows direction without magnitude, useful when only flow pattern matters, not speed.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Vector Fields in [latex]\\mathbb{R}^3[\/latex]<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Three-dimensional fields use three components: [latex]\\mathbf{F}(x,y,z) = \\langle P(x,y,z), Q(x,y,z), R(x,y,z) \\rangle[\/latex]. To visualize, first understand behavior in coordinate planes, then see how vectors change moving perpendicular to those planes. Choose representative points from different octants.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q320974724\">Show Solution<\/button><\/p>\n<div id=\"q320974724\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_3302\" aria-describedby=\"caption-attachment-3302\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q458017451\">Show Solution<\/button><\/p>\n<div id=\"q458017451\" class=\"hidden-answer\" style=\"display: none\">Rotational.<\/p>\n<figure id=\"attachment_3304\" aria-describedby=\"caption-attachment-3304\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q239547128\">Show Solution<\/button><\/p>\n<div id=\"q239547128\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sqrt{65}[\/latex] m\/sec.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q594329457\">Show Solution<\/button><\/p>\n<div id=\"q594329457\" class=\"hidden-answer\" style=\"display: none\">No.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q234184589\">Show Solution<\/button><\/p>\n<div id=\"q234184589\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_5236\" aria-describedby=\"caption-attachment-5236\" style=\"width: 388px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q834187034\">Show Solution<\/button><\/p>\n<div id=\"q834187034\" class=\"hidden-answer\" style=\"display: none\">[latex]1.49063\\times10^{-18},4.96876\\times10^{-19},9.93752\\times10^{-19}\\text{N}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Gradient Fields<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A gradient field (also called a conservative field) arises when a vector field equals the gradient of some scalar function. These fields model physical systems where energy is conserved, such as gravitational and electrostatic forces. Understanding gradient fields reveals important connections between scalar functions and vector fields.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>What is a Gradient Field?<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Vector field [latex]\\mathbf{F}[\/latex] is a gradient field if there exists a scalar function [latex]f[\/latex] such that [latex]\\nabla f = \\mathbf{F}[\/latex]. The function [latex]f[\/latex] is called a potential function for [latex]\\mathbf{F}[\/latex]. For two variables, [latex]\\nabla f = \\langle f_x, f_y \\rangle[\/latex]. For three variables, [latex]\\nabla f = \\langle f_x, f_y, f_z \\rangle[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Relationship to Level Curves<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Gradient vectors are always perpendicular to level curves of the function. Where level curves are closely spaced (steep regions), gradient vectors have large magnitude. Where level curves are widely spaced (gradual regions), gradient vectors have small magnitude. This makes sense because the gradient points in the direction of greatest increase.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Potential Functions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If [latex]\\mathbf{F}[\/latex] is conservative with potential functions [latex]f[\/latex] and [latex]g[\/latex], then [latex]f = g + C[\/latex] for some constant [latex]C[\/latex]. This parallels how antiderivatives differ by constants in single-variable calculus. Potential functions are unique up to an additive constant.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Cross-Partial Property<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Conservative fields satisfy special equations called the cross-partial property. In [latex]\\mathbb{R}^2[\/latex], if [latex]\\mathbf{F} = \\langle P, Q \\rangle[\/latex] is conservative, then [latex]\\frac{\\partial P}{\\partial y} = \\frac{\\partial Q}{\\partial x}[\/latex]. In [latex]\\mathbb{R}^3[\/latex], three equations must hold: [latex]P_y = Q_x[\/latex], [latex]Q_z = R_y[\/latex], and [latex]R_x = P_z[\/latex]. These come from Clairaut&#8217;s theorem: [latex]f_{xy} = f_{yx}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Important Limitation<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If a field fails the cross-partial property, it&#8217;s definitely not conservative. However, satisfying the cross-partial property doesn&#8217;t guarantee the field is conservative\u2014it&#8217;s necessary but not sufficient. Most vector fields are not conservative since the cross-partial equations are restrictive.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use technology to plot the gradient vector field of\u00a0[latex]f(x,y)=\\sin{x}\\cos{y}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q485726668\">Show Solution<\/button><\/p>\n<div id=\"q485726668\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_3309\" aria-describedby=\"caption-attachment-3309\" style=\"width: 717px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3309 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/19162738\/6-1-tryitans4.jpeg\" alt=\"\" width=\"717\" height=\"497\" \/><figcaption id=\"caption-attachment-3309\" class=\"wp-caption-text\">Figure 2. A virtual representation of the gradient vector field\u00a0[latex]f(x,y)=\\sin{x}\\cos{y}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Is [latex]f(x,y,z)=x^2\\cos{(yz)}+y^2z^2[\/latex] a potential function for [latex]{\\bf{F}}(x,y,z)=\\langle2x\\cos{(yz)},-x^2z\\sin{(yz)}+2yz^2,y^2\\rangle[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q764528702\">Show Solution<\/button><\/p>\n<div id=\"q764528702\" class=\"hidden-answer\" style=\"display: none\">No.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Verify that [latex]f(x, y)=x^{2}y^{2}+x[\/latex] is a potential function for velocity field [latex]{\\bf{v}}(x,y)=\\langle2xy^2+1,2x^2y\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q324714857\">Show Solution<\/button><\/p>\n<div id=\"q324714857\" class=\"hidden-answer\" style=\"display: none\">[latex]\\nabla{f}=\\bf{v}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Show that vector field [latex]{\\bf{F}}(x,y)x=y{\\bf{i}}-x^2y{\\bf{j}}[\/latex] is not conservative.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q736457208\">Show Solution<\/button><\/p>\n<div id=\"q736457208\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sqrt{2}[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Is vector field [latex]{\\bf{G}}(x,y,z)=\\langle{y},x,xyz\\rangle[\/latex] conservative?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q183428645\">Show Solution<\/button><\/p>\n<div id=\"q183428645\" class=\"hidden-answer\" style=\"display: none\">No.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-744","chapter","type-chapter","status-publish","hentry"],"part":436,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/744","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/744\/revisions"}],"predecessor-version":[{"id":2219,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/744\/revisions\/2219"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/436"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/744\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=744"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=744"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=744"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}