// Numbas version: exam_results_page_options
{"name": "Find surface of points in scalar field orthogonal to the z axis, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p4", "description": ""}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p1", "description": ""}, "p12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p12", "description": ""}, "p9": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p9", "description": ""}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,3,5,7)", "name": "p3", 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"p7", "p4", "p5", "p8", "p9", "p10", "p11", "p12"], "name": "Find surface of points in scalar field orthogonal to the z axis, ", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "marks": 1, "checkingType": "absdiff", "scripts": {}, "answerSimplification": "all", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}], "prompt": "<p>$f(x,y,z)=\\simplify{x^{p1}*y^{p2}*z^{p3}+x^{p4}*y^{p5}*z^{p6}+x^{p7}*y^{p8}*z^{p9}+x^{p10}*y^{p11}*z^{p12}}$.</p>\n      <p>$g(x,y,z)=$ [[0]].</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>For the following scalar field $f$, find all the points for which $\\boldsymbol{\\nabla}f$ is orthogonal to the $z$-axis. &nbsp;Enter your answer in the form of a surface $g(x,y,z)=0$.</p>", "tags": ["checked2015", "nabla", "scalar field", "surface"], "rulesets": {}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find all points for which the gradient of a scalar field is orthogonal to the $z$-axis.</p>\n<p>Should warn that multiplied terms need * to denote multiplication.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>A vector that is orthogonal to the $z$-axis has its $z$-component equal to zero. &nbsp;We need to find all the points, therefore, for which the $z$-component of $\\boldsymbol{\\nabla}f$ is zero, i.e. $(\\boldsymbol{\\nabla}f)_z=0$.</p>\n  <p>The $z$-component of $\\boldsymbol{\\nabla}f$ is</p>\n  <p>\\[(\\boldsymbol{\\nabla}f)_z=\\frac{\\partial f}{\\partial z}=\\simplify{{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}},\\]</p>\n  <p>and so the surface</p>\n  <p>\\[g(x,y,z)=\\simplify{{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}}=0\\]</p>\n  <p>defines the set of points for which $\\boldsymbol{\\nabla}f$ is orthogonal to the $z$-axis.</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/"}]}