// Numbas version: exam_results_page_options
{"name": "Find curl and divergence of a vector field", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find curl and divergence of a vector field", "tags": [], "metadata": {"description": "<p>Curl and divergence of a vector field. &nbsp;Determine whether the vector field is irrotational or solenoidal.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p style=\"text-align: center;\"><iframe width=\"640\" height=\"360\" src=\"https://www.youtube.com/embed/HCkfEBO4eYo?rel=0&amp;showinfo=0\" frameborder=\"0\" allow=\"autoplay; encrypted-media\" allowfullscreen=\"allowfullscreen\"></iframe></p>\n<p>For the vector field</p>\n<p style=\"text-align: center;\">$\\boldsymbol{u}=\\pmatrix{\\simplify{{a1}*x^{p1}+{b1}*y^{p2}*z^{p3}},\\simplify{{c1}*y^{p4}+{d1}*x^{p5}*z^{p6}},\\simplify{{e1}*z^{p7}+{f1}*x^{p8}*y^{p9}}}$</p>\n<p>calculate $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ and $\\boldsymbol{\\nabla\\cdot u}$, and determine whether $\\boldsymbol{u}$ is irrotational or solenoidal, or both.</p>", "advice": "<p>The curl of a vector field $\\boldsymbol{u}=\\pmatrix{u_x,u_y,u_z}$ is given by</p>\n<p>\\[\\boldsymbol{\\nabla}\\times\\boldsymbol{u}=\\pmatrix{\\frac{\\partial u_z}{\\partial y}-\\frac{\\partial u_y}{\\partial z},\\frac{\\partial u_x}{\\partial z}-\\frac{\\partial u_z}{\\partial x},\\frac{\\partial u_y}{\\partial x}-\\frac{\\partial u_x}{\\partial y}}.\\]</p>\n<p>The divergence of the same vector field is given by</p>\n<p>\\[\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}.\\]</p>\n<h4>a)</h4>\n<p>By partial differentiation</p>\n<p>\\[\\boldsymbol{\\nabla \\times u}=\\pmatrix{\\simplify{{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}},\\simplify{{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}},\\simplify{{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}}}.\\]</p>\n<h4>b)</h4>\n<p>Again, by partial differentiation</p>\n<p>\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.\\]</p>\n<p>A vector field is irrotational if its curl is equal to the zero vector; a vector field is solenoidal if its divergence is equal to zero.</p>\n<h4>c)</h4>\n<p>Since $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ {irrequal} to the zero vector, the vector field {isirr}.</p>\n<h4>d)</h4>\n<p>Since $\\boldsymbol{\\nabla\\cdot u}$ {solequal} to zero, the vector field {issol}.</p>", "rulesets": {}, "extensions": [], "variables": {"solenoidal": {"name": "solenoidal", "group": "Ungrouped variables", "definition": "if(n=2,\"Yes\",\"No\")", "description": "", "templateType": "anything"}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "p7": {"name": "p7", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything"}, "p9": {"name": "p9", "group": "Ungrouped variables", "definition": "random(1..9 except p6)", "description": "", "templateType": "anything"}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything"}, "notsolenoidal": {"name": "notsolenoidal", "group": "Ungrouped variables", "definition": "if(n=2,\"No\",\"Yes\")", "description": "", "templateType": "anything"}, "p8": {"name": "p8", "group": "Ungrouped variables", "definition": "random(1..9 except p3)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything"}, "issol": {"name": "issol", "group": "Ungrouped variables", "definition": "if(n=2,\"is solenoidal\",\"is not solenoidal\")", "description": "", "templateType": "anything"}, "isirr": {"name": "isirr", "group": "Ungrouped variables", "definition": "if(n=1,\"is irrotational\",\"is not irrotational\")", "description": "", "templateType": "anything"}, "p4": {"name": "p4", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything"}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "random(1..9 except p2)", "description": "", "templateType": "anything"}, "solequal": {"name": "solequal", "group": "Ungrouped variables", "definition": "if(n=2,\"is equal\",\"is not equal\")", "description": "", "templateType": "anything"}, "irrotational": {"name": "irrotational", "group": "Ungrouped variables", "definition": "if(n=1,\"Yes\",\"No\")", "description": "", "templateType": "anything"}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything"}, "p6": {"name": "p6", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "irrequal": {"name": "irrequal", "group": "Ungrouped variables", "definition": "if(n=1,\"is equal\",\"is not equal\")", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything"}, "notirrotational": {"name": "notirrotational", "group": "Ungrouped variables", "definition": "if(n=1,\"No\",\"Yes\")", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "irrequal", "isirr", "b1", "d1", "issol", "e1", "irrotational", "a1", "c1", "solenoidal", "p2", "p3", "solequal", "p1", "p6", "p7", "p4", "p5", "notsolenoidal", "p8", "p9", "notirrotational", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\boldsymbol{\\nabla\\cdot u}=$ [[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\boldsymbol{\\nabla}\\times\\boldsymbol{u}=($[[0]]$,$[[1]]$,$[[2]]$)$.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Is the vector field $\\boldsymbol{u}$ irrotational? (An \"irrotational\" vector field is one with zero curl everywhere.)</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>{irrotational}</p>", "<p>{notirrotational}</p>"], "matrix": [1, 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Is the vector field $\\boldsymbol{u}$ solenoidal? 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