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{"name": "Calculate divergence of vector fields", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a3"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c3"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "p1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,1))", "description": "", "name": "a2"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b3"}}, "ungrouped_variables": ["p1", "a1", "a3", "a2", "b1", "b3", "c3", "c1"], "name": "Calculate divergence of vector fields", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>$\\boldsymbol{u}=\\pmatrix{(\\simplify[std]{{a1}*x+{b1}*y+{c1}*z})(\\simplify[std]{{b1}*y-{c1}*z}),(\\simplify[std]{{a1}*y+{b1}*z+{c1}*x})(\\simplify[std]{{b1}*z-{c1}*x}),(\\simplify[std]{{a1}*z+{b1}*x+{c1}*y})(\\simplify[std]{{b1}*x-{c1}*y})}$.</p>\n<p>$\\boldsymbol{\\nabla\\cdot u}=$ [[0]].</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a1*b1-a1*c1}*x+{a1*b1-a1*c1}*y+{a1*b1-a1*c1}*z", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>$\\boldsymbol{u}=\\pmatrix{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\left(y^\\var{p1}+z^\\var{p1}\\right)\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\\left(z^\\var{p1}+x^\\var{p1}\\right)\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)}$.</p>\n<p>$\\boldsymbol{\\nabla\\cdot u}=$ [[0]].</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "0", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>$\\boldsymbol{u}=\\pmatrix{\\simplify{{a3}*x}+f_1(y,z),\\simplify{{b3}*y}+f_2(x,z),\\simplify{{c3}*z}+f_3(x,y)}$, for any general functions $f_1$, $f_2$, and $f_3$.</p>\n  <p>$\\boldsymbol{\\nabla\\cdot u}=$ [[0]].</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a3+b3+c3}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>For each of the following vector fields $\\boldsymbol{u}$, find the divergence $\\boldsymbol{\\nabla\\cdot u}$.</p>", "tags": ["checked2015"], "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Divergence of vector fields.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The divergence of a vector field $\\boldsymbol{u}=\\pmatrix{u_1,u_2,u_3}$ is given by</p>\n<p>\\[\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_1}{\\partial x}+\\frac{\\partial u_2}{\\partial y}+\\frac{\\partial u_3}{\\partial z}.\\]</p>\n<p>a)</p>\n<p>The variables $x$, $y$, and $z$ appear in a cyclical manner in each of the three components of $\\boldsymbol{u}$. &nbsp;Once you have calculated $\\frac{\\partial u_1}{\\partial x}$, you can use cyclic permutations to determine the other two derivatives. &nbsp;Hence</p>\n<p>\\[\\begin{align}\\frac{\\partial u_1}{\\partial x}&amp;=\\frac{\\partial}{\\partial x}\\left(\\simplify{({a1}*x+{b1}*y+{c1}*z)*({b1}*y-{c1}*z)}\\right)\\\\&amp;=\\simplify{{a1}*({b1}*y-{c1}*z)},\\end{align}\\]</p>\n<p>and so, cyclically permuting the variables,</p>\n<p>\\[\\frac{\\partial u_2}{\\partial y}=\\simplify{{a1}*({b1}*z-{c1}*x)}\\]</p>\n<p>and</p>\n<p>\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{a1}*({b1}*x-{c1}*y)}.\\]</p>\n<p>Finally, adding the components together gives the divergence</p>\n<p>\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify[std]{{a1}*({b1}*y-{c1}*z)+{a1}*({b1}*z-{c1}*x)+{a1}*({b1}*x-{c1}*y)}=\\simplify{{a1*b1-a1*c1}*x+{a1*b1-a1*c1}*y+{a1*b1-a1*c1}*z}.\\]</p>\n<p>&nbsp;</p>\n<p>b)</p>\n<p>As in part a) the variables $x$, $y$, and $z$ appear cyclically in each component of $\\boldsymbol{u}$, so we only need calculate one derivative explicitly, then use cyclic permutations to calculate the other two. &nbsp;Hence</p>\n<p>\\[\\begin{align}\\frac{\\partial u_1}{\\partial x}&amp;=\\frac{\\partial}{\\partial x}\\left\\{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)\\right\\}\\\\&amp;=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\end{align}\\]</p>\n<p>and by symmetry</p>\n<p>\\[\\frac{\\partial u_2}{\\partial y}=\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\\]</p>\n<p>and</p>\n<p>\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right).\\]</p>\n<p>Finally, adding the derivatives together gives the divergence</p>\n<p>\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)+\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right)+\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)=0.\\]</p>\n<p>&nbsp;</p>\n<p>c)</p>\n<p>First note that $f_1$ does not depend on $z$, $f_2$ does not depend on $y$, and $f_3$ does not depend on $z$. &nbsp;This makes the differentiation very straight forward, and hence</p>\n<p>\\[\\begin{align}\\boldsymbol{\\nabla\\cdot u}&amp;=\\frac{\\partial}{\\partial x}\\left(\\simplify{{a3}*x}+f_1(y,z)\\right)+\\frac{\\partial}{\\partial y}\\left(\\simplify{{b3}*y}+f_2(x,z)\\right)+\\frac{\\partial}{\\partial z}\\left(\\simplify{{c3}*z}+f_3(x,y)\\right)\\\\&amp;=\\simplify[all,!collectNumbers]{{a3}+{b3}+{c3}}\\\\&amp;=\\var{a3+b3+c3}.\\end{align}\\]</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}