{"id":216,"date":"2019-08-28T19:56:09","date_gmt":"2019-08-28T19:56:09","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=216"},"modified":"2023-06-12T16:15:50","modified_gmt":"2023-06-12T16:15:50","slug":"parametric-equations","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/parametric-equations\/","title":{"rendered":"Parametric Equations"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Parametric Equations &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nThink of a curve being traced out over time, sometimes doubling back\non itself or crossing itself.  Such a curve cannot be described by a\nfunction $y=f(x)$.  Instead, we will describe our position along the\ncurve at time $t$ by\n\\begin{eqnarray*}\n\tx&amp;=&amp;x(t)\\\\\n\ty&amp;=&amp;y(t).\n\\end{eqnarray*}\nThen $x$ and $y$ are related to each other through their dependence on\nthe <b>parameter<\/b> $t$. \n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n<applet codebase=\"..\/..\/java\" code=\"Parametric.class\" width=\"270\" height=\"200\" align=\"right\"><\/applet>\nSuppose we trace out a curve according to\n\\begin{eqnarray*}\n\tx&amp;=&amp;t^2-4t\\\\\n\ty&amp;=&amp;3t\n\\end{eqnarray*}\nwhere $t\\geq 0$.  The arrow on the curve indicates the direction of\nincreasing time or <b>orientation<\/b> of the curve.  Drag the box\nalong the curve and notice how $x$ and $y$ vary with $t$.\n\n<br><br><br>\n<\/p>\n\n\n\n<p>\nThe parameter does not always represent time: \n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n<applet codebase=\"..\/..\/java\" code=\"Parametric2.class\" width=\"270\" height=\"200\" align=\"right\"><\/applet>\nConsider the parametric equation\n\\begin{eqnarray*}\n\tx&amp;=&amp;3\\cos\\theta\\\\\n\ty&amp;=&amp;3\\sin\\theta.\n\\end{eqnarray*}\nHere, the parameter $\\theta$ represents the polar angle of the position on a\ncircle of radius $3$ centered at the origin and oriented\ncounterclockwise.\n\n<br><br><br><br><br>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Differentiating Parametric Equations<\/h4>\n\n\n\n<p>\n\nLet $x=x(t)$ and $y=y(t)$.\nSuppose for the moment that we are able to re-write this as $y(t)=f(x(t))$.\nThen $\\displaystyle \\frac{dy}{dt}=\\frac{dy}{dx}\\cdot \\frac{dx}{dt}$ by\nthe Chain Rule.  Solving for $\\displaystyle \\frac{dy}{dx}$ and\nassuming $\\displaystyle \\frac{dx}{dt}\\neq 0$,\n\\[\\frac{dy}{dx}=\\frac{~\\frac{dy}{dt}~}{~\\frac{dx}{dt}~}\\]\na formula that holds in general.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nIf $x=t^2-3$ and $y=t^8$, then $\\displaystyle \\frac{dx}{dt}=2t$ and\n$\\displaystyle \\frac{dy}{dt}=8t^7$.  So\n\\begin{eqnarray*}\n\t\\frac{dy}{dx}&amp;=&amp;\\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}}=\\frac{8t^7}{2t}=4t^6\\\\\n\t\\frac{d^2y}{dx^2}&amp;=&amp;\\frac{d}{dx}\\left[\\frac{dy}{dx}\\right]=\n\t\\frac{~\\frac{d[\\frac{dy}{dx}] \\strut}{dt}~}{~\\frac{dx}{dt}~}=\\frac{24t^5}{2t}=12t^4.\n\\end{eqnarray*} \n\n<br><br>\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> It is often possible to re-write the parametric equations without the parameter.  In the second example, $\\displaystyle \\frac{x}{3}=\\cos\\theta$, $\\displaystyle \\frac{y}{3}=\\sin\\theta$. Since $\\cos^2 \\theta +\\sin^2 \\theta =1$, $\\displaystyle \\left(\\frac{x}{3}\\right)^2+\\left(\\frac{y}{3}\\right)^2=1$.  Then $x^2+y^2=9$, which is the equation of a circle as expected.  When you do eliminate the parameter, always check that you have not introduced extraneous portions of the curve. <br><br> <\/li><li> Every curve has infinitely many parametrizations, amounting to different scales for the parameter.  For example, \\begin{eqnarray*} x&amp;=&amp;3\\cos 2\\theta\\\\ y&amp;=&amp;3\\sin 2\\theta \\end{eqnarray*} traces out the circle from the second example twice as &#8220;quickly,&#8221; completing a full revolution in $\\pi$ rather than $2\\pi$ units of $\\theta$. <br><br> <\/li><li> Every equation $y=f(x)$ may be re-written in parametric form by letting $x=t$, $y=f(t)$.  <\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p>\n\nA curve in the $xy$-plane may be described by a pair of parametric equations\n$$ x = x(t) $$\n$$ y = y(t) $$\nwhere $x$ and $y$ are related through their dependence on $t$. This is particularly useful when\nneither $x$ nor $y$ is a function of the other.\n\n<\/p>\n\n\n\n<p> The derivative of $y$ with respect to $x$, in terms of the parameter $t$, is given by $$ \\frac{dy}{dx} = \\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}}. $$  <br> <\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1110\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Parametric Equations &#8211; HMC Calculus Tutorial Think of a curve being traced out over time, sometimes doubling back on itself or crossing itself. Such a curve cannot be described by a function $y=f(x)$. Instead, we will describe our position along the curve at time $t$ by \\begin{eqnarray*} x&amp;=&amp;x(t)\\\\ y&amp;=&amp;y(t). \\end{eqnarray*} Then $x$ and $y$ are&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-216","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/216","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=216"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/216\/revisions"}],"predecessor-version":[{"id":1254,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/216\/revisions\/1254"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=216"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}