{"id":22652,"date":"2023-02-28T03:04:05","date_gmt":"2023-02-28T03:04:05","guid":{"rendered":"https:\/\/www.booksofall.com\/?post_type=product&#038;p=22652"},"modified":"2023-02-28T03:04:05","modified_gmt":"2023-02-28T03:04:05","slug":"clp-3-multivariable-calculus","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/pt\/clp-3-multivariable-calculus\/","title":{"rendered":"CLP-3 Multivariable Calculus"},"content":{"rendered":"<h2>Chapter 1 &#8211; Vectors and Geometry in Two and Three Dimensions<\/h2>\n<p>Before we get started doing calculus in two and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Three-dimensional_space\">three dimensions<\/a> we need to brush up on some basic <a href=\"https:\/\/en.wikipedia.org\/wiki\/Geometry\">geometry<\/a>, that we will use a lot. We are already familiar with the <a href=\"https:\/\/byjus.com\/maths\/cartesian-plane\/\">Cartesian plane<\/a>, but we\u2019ll start from the beginning.<\/p>\n<p><b>1.1 Points\u00a0<\/b><\/p>\n<p>Each point in two dimensions may be labeled by two coordinates1(<i>x, y<\/i>) which specify the position of the point in some units with respect to some axes as in the figure below.<\/p>\n<p id=\"QHNsWDj\"><img loading=\"lazy\" decoding=\"async\" width=\"162\" height=\"157\" class=\"alignnone size-full wp-image-22662 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6b767ca90.png\" alt=\"\" \/><\/p>\n<p>The set of all points in <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/2-dimensional\">two dimensions<\/a> is denoted2 R2. Observe that<\/p>\n<ul>\n<li>the distance from the point (<i>x, y<\/i>) to the\u00a0<i>x<\/i>-axis is |<i>y<\/i>|<\/li>\n<li>if <i>y &gt;\u00a0<\/i>0, then (<i>x, y<\/i>) is above the\u00a0<i>x<\/i>-axis and if\u00a0<i>y &lt;\u00a0<\/i>0, then (<i>x, y<\/i>) is below the\u00a0<i>x<\/i>-axis<\/li>\n<li>the distance from the point (<i>x, y<\/i>) to the\u00a0<i>y<\/i>-axis is |<i>x<\/i>|<\/li>\n<li>if <i>x &gt;\u00a0<\/i>0, then (<i>x, y<\/i>) is to the right of the\u00a0<i>y<\/i>-axis and if\u00a0<i>x &lt;\u00a0<\/i>0, then (<i>x, y<\/i>) is to the left of the\u00a0<i>y<\/i>-axis<\/li>\n<li>the distance from the point (<i>x, y<\/i>) to the origin (0<i>,\u00a0<\/i>0) is \u221a\u00a0<i>x<\/i>2 +\u00a0<i>y<\/i>2<\/li>\n<\/ul>\n<div>\n<p>Similarly, each point in three dimensions may be labeled by three coordinates (<i>x, y, z<\/i>), as in the two figures below.<\/p>\n<p id=\"yUorrPN\"><img loading=\"lazy\" decoding=\"async\" width=\"463\" height=\"195\" class=\"alignnone size-full wp-image-22663 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6c1635724.png\" alt=\"\" \/><\/p>\n<p>The set of all points in three dimensions is denoted R3. The plane that contains, for example, the\u00a0<i>x<\/i>&#8211; and\u00a0<i>y<\/i>-axes is called the\u00a0<i>xy<\/i>-plane.<\/p>\n<ul>\n<li>The <i>xy<\/i>-plane is the set of all points (<i>x, y, z<\/i>) that satisfy\u00a0<i>z\u00a0<\/i>= 0.<\/li>\n<li>The <i>xz<\/i>-plane is the set of all points (<i>x, y, z<\/i>) that satisfy\u00a0<i>y\u00a0<\/i>= 0.<\/li>\n<li>The <i>yz<\/i>-plane is the set of all points (<i>x, y, z<\/i>) that satisfy\u00a0<i>x\u00a0<\/i>= 0.<\/li>\n<\/ul>\n<p>More generally,<\/p>\n<ul>\n<li>The set of all points (<i>x, y, z<\/i>) that obey\u00a0<i>z\u00a0<\/i>=\u00a0<i>c\u00a0<\/i>is a plane that is parallel to the\u00a0<i>xy<\/i>-plane and is a distance |<i>c<\/i>| from it. If\u00a0<i>c &gt;\u00a0<\/i>0, the plane\u00a0<i>z\u00a0<\/i>=\u00a0<i>c\u00a0<\/i>is above the\u00a0<i>xy<\/i>-plane. If\u00a0<i>c &lt;\u00a0<\/i>0, the plane\u00a0<i>z\u00a0<\/i>=\u00a0<i>c\u00a0<\/i>is below the\u00a0<i>xy<\/i>-plane. We say that the plane\u00a0<i>z\u00a0<\/i>=\u00a0<i>c\u00a0<\/i>is a signed distance\u00a0<i>c\u00a0<\/i>from the\u00a0<i>xy<\/i>-plane.<\/li>\n<li>The set of all points (<i>x, y, z<\/i>) that obey\u00a0<i>y\u00a0<\/i>=\u00a0<i>b\u00a0<\/i>is a plane that is parallel to the\u00a0<i>xz<\/i>-plane and is a signed distance\u00a0<i>b\u00a0<\/i>from it.<\/li>\n<li>The set of all points (<i>x, y, z<\/i>) that obey\u00a0<i>x\u00a0<\/i>=\u00a0<i>a\u00a0<\/i>is a plane that is parallel to the\u00a0<i>yz<\/i>-plane and is a signed distance\u00a0<i>a\u00a0<\/i>from it.<\/li>\n<\/ul>\n<p id=\"WlvvizS\"><img loading=\"lazy\" decoding=\"async\" width=\"545\" height=\"195\" class=\"alignnone size-full wp-image-22665 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6c59d39fe.png\" alt=\"\" \/><\/p>\n<ul>\n<li>Observe that our 2d distances extend quite easily to 3d.<\/li>\n<li>the distance from the point (<i>x, y, z<\/i>) to the\u00a0<i>xy<\/i>-plane is |<i>z<\/i>|<\/li>\n<li>the distance from the point (<i>x, y, z<\/i>) to the\u00a0<i>xz<\/i>-plane is |<i>y<\/i>|<\/li>\n<li>the distance from the point (<i>x, y, z<\/i>) to the\u00a0<i>yz<\/i>-plane is |<i>x<\/i>|<\/li>\n<li>the distance from the point (<i>x, y, z<\/i>) to the origin (0<i>,\u00a0<\/i>0<i>,\u00a0<\/i>0) is <img loading=\"lazy\" decoding=\"async\" width=\"103\" height=\"32\" class=\"alignnone size-full wp-image-22666 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6c6f8e28b.png\" alt=\"\" \/><\/li>\n<\/ul>\n<p><span style=\"font-size: 1rem;\">To see that the distance from the point (<\/span><i style=\"font-size: 1rem;\">x, y, z<\/i><span style=\"font-size: 1rem;\">) to the origin (0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0) is indeed\u221a\u00a0<\/span><i style=\"font-size: 1rem;\">x<\/i><span style=\"font-size: 1rem;\">2 +\u00a0<\/span><i style=\"font-size: 1rem;\">y<\/i><span style=\"font-size: 1rem;\">2 +\u00a0<\/span><i style=\"font-size: 1rem;\">z<\/i><span style=\"font-size: 1rem;\">2, <\/span><\/p>\n<ul>\n<li><span style=\"font-size: 1rem;\">apply <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem\">Pythagoras<\/a> to the right-angled triangle with vertices (0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0), (<\/span><i style=\"font-size: 1rem;\">x,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0) <\/span><span style=\"font-size: 1rem;\">and (<\/span><i style=\"font-size: 1rem;\">x, y,\u00a0<\/i><span style=\"font-size: 1rem;\">0) to see that the distance from (0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0) to (<\/span><i style=\"font-size: 1rem;\">x, y,\u00a0<\/i><span style=\"font-size: 1rem;\">0) is \u221a\u00a0<\/span><i style=\"font-size: 1rem;\">x<\/i><span style=\"font-size: 1rem;\">2 +\u00a0<\/span><i style=\"font-size: 1rem;\">y<\/i><span style=\"font-size: 1rem;\">2 <\/span><span style=\"font-size: 1rem;\">and then<\/span><\/li>\n<li><span style=\"font-size: 1rem;\">apply Pythagoras to the right-angled triangle with vertices (0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0), (<\/span><i style=\"font-size: 1rem;\">x, y,\u00a0<\/i><span style=\"font-size: 1rem;\">0) <\/span><span style=\"font-size: 1rem;\">and (<\/span><i style=\"font-size: 1rem;\">x, y, z<\/i><span style=\"font-size: 1rem;\">) to see that the distance from (0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0<\/span><i style=\"font-size: 1rem;\">,\u00a0<\/i><span style=\"font-size: 1rem;\">0) to (<\/span><i style=\"font-size: 1rem;\">x, y, z<\/i><span style=\"font-size: 1rem;\">) is <\/span><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-22668 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6cbcd6586.png\" alt=\"\" width=\"140\" height=\"29\" \/><img loading=\"lazy\" decoding=\"async\" width=\"113\" height=\"26\" class=\"alignnone size-full wp-image-22669 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6cc28d3aa.png\" alt=\"\" \/><img loading=\"lazy\" decoding=\"async\" width=\"227\" height=\"207\" class=\"alignnone size-full wp-image-22670 \" style=\"font-size: 1rem;\" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6cd6c13ae.png\" alt=\"\" \/><\/li>\n<\/ul>\n<\/div>\n<p>More generally, the distance from the point (<i>x, y, z<\/i>) to the point (<i>x<\/i>\u2032<i>, y<\/i>\u2032<i>, z<\/i>\u2032) is <img loading=\"lazy\" decoding=\"async\" width=\"235\" height=\"30\" class=\"alignnone size-full wp-image-22671 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6d0d1bebd.png\" alt=\"\" \/>.<\/p>\n<p>Notice that this gives us the equation for a sphere quite directly. All the points on a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Sphere\">sphere<\/a> are equidistant from the centre of the sphere. So, for example, the equation of the sphere centered on (1<i>,\u00a0<\/i>2<i>,\u00a0<\/i>3) with radius 4, that is, the set of all points (<i>x, y, z<\/i>) whose distance from (1<i>,\u00a0<\/i>2<i>,\u00a0<\/i>3) is 4, is<\/p>\n<p id=\"mITOzIO\"><img loading=\"lazy\" decoding=\"async\" width=\"252\" height=\"29\" class=\"alignnone size-full wp-image-22675 \" src=\"https:\/\/www.booksofall.com\/wp-content\/uploads\/2023\/02\/img_63fd6d20d3014.png\" alt=\"\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p><iframe style=\"width: 100%; height: 750px; border: none;\" src=\"https:\/\/online.visual-paradigm.com\/share\/book\/clp-3-multivariable-calculus-19mdv4wisv?p=1\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"featured_media":22677,"template":"","meta":{"_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learning mathematics can enhance problem-solving skills and improve logical reasoning, which can be valuable in many areas of life. Learn more here!"},"product_brand":[],"product_cat":[365],"product_tag":[],"class_list":{"0":"post-22652","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-mathematics-subjects","8":"first","9":"instock","10":"shipping-taxable","11":"product-type-simple"},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>CLP-3 Multivariable Calculus - BooksOfAll Portuguese<\/title>\n<meta name=\"description\" content=\"Learning mathematics can enhance problem-solving skills and improve logical reasoning, which can be valuable in many areas of life. 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