// Numbas version: exam_results_page_options
{"name": "Use Green's theorem to convert line integral to double integral, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "b"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "f"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "d"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Use Green's theorem to convert line integral to double integral, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{(b+d)*f*g}", "maxValue": "{(b+d)*f*g}", "marks": 8}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$I=\\;\\;$[[0]]</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Green&rsquo;s theorem states that for a region R with boundary $\\Gamma$</p>\n  <p>\\[\\oint_{\\Gamma} \\left( u\\;dx+v\\;dy \\right)= \\int \\int_R\\left(\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}\\right)\\;dx\\;dy.\\]&nbsp;Use Green&rsquo;s theorem to find the value of:&nbsp;\\[I=\\oint_{\\Gamma} \\left( \\left(\\simplify[std]{{a}x^2-{b}y} \\right)\\;dx+\\left(\\simplify[std]{{c}y^2+{d}x}\\right)\\;dy\\right)\\]</p>\n  <p>where $\\Gamma$, mapped counter-clockwise, is the closed path, starting at $(0,0)$, around the boundary of a rectangle with vertices $(0,0),\\;(\\var{f},0),\\;(\\var{f},\\var{g}),\\;(0,\\var{g})$.</p>", "tags": ["Calculus", "calculus", "checked2015", "closed path", "differentiation", "Green's theorem", "green's theorem", "Greens theorem", "greens theorem", "integral over a closed path", "integral over a rectangle", "integral over a region", "line integral", "MAS2104", "partial differentiation", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>30/06/2012:</strong></p>\n            <p>Added tags. Could include Show steps on Green's theorem.</p>\n            <p><strong>19/07/2012:</strong></p>\n            <p>Added Show steps on Green's Theorem.</p>\n            <p>Added description.</p>\n            <p>Checked calculation.</p>\n            <p><strong>23/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>In the question and Steps added brackets so that Green's Theorem is valid.</p>\n            <p>Question appears to be working correctly.</p>\n            <p><strong>23/12/2012: (WHF)</strong></p>\n            <p>No Shoe steps on Green's theorem.</p>\n            <p>Checked calculations, OK. Added tested1 tag. Few minor typos - full stops added.</p>\n            <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>(Green&rsquo;s theorem). $\\Gamma$ a rectangle, find:&nbsp;$\\displaystyle \\oint_{\\Gamma} \\left(ax^2-by \\right)\\;dx+\\left(cy^2+px\\right)\\;dy$.</p>"}, "advice": "<p>First we identify the functions $u$ and $v$ and their required derivatives:</p>\n<p>\\[\\begin{eqnarray*} u &amp;=&amp; \\simplify[std]{{a}x^2-{b}y} &nbsp;\\Rightarrow \\frac{\\partial u}{\\partial y} = \\simplify[std]{-{b}}&nbsp;\\\\ v&amp;=&amp;\\simplify[std]{{c}y^2+{d}x} &nbsp;\\Rightarrow \\frac{\\partial v}{\\partial x} = \\simplify[std]{{d}}\\end{eqnarray*}\\]</p>\n<p>Hence: \\[\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}=\\simplify[std]{{d}-{-b}}=\\var{d+b}.\\]</p>\n<p><br/>So, using Green&rsquo;s Theorem, the integral $I$ becomes:</p>\n<p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int \\int_R\\left(\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}\\right)\\;dx\\;dy\\\\ &amp;=&amp;\\int \\int_R \\var{d+b}\\;dx\\;dy = \\var{d+b}\\int \\int_R dx\\;dy\\\\ &amp;=&amp;\\var{d+b}\\times \\textrm{Area of }R. \\end{eqnarray*}\\]</p>\n<p>Now the region $R$ is a rectangle of size $\\var{f}$ by $\\var{g}$. &nbsp;Hence:</p>\n<p>\\[\\begin{eqnarray*} I&amp;=&amp;\\var{b+d}\\times \\var{f} \\times \\var{g}\\\\ &amp;=&amp; \\var{(b+d)*f*g}. \\end{eqnarray*} \\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}