Curl and divergence of a vector field. Determine whether the vector field is irrotational or solenoidal.
"}, "tags": ["checked2015", "MAS2104"], "variable_groups": [], "functions": {}, "name": "Find curl and divergence of a vector field", "parts": [{"type": "gapfill", "marks": 0, "prompt": "$\\boldsymbol{\\nabla}\\times\\boldsymbol{u}=($[[0]]$,$[[1]]$,$[[2]]$)$.
", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": true, "answersimplification": "all", "checkingaccuracy": 0.001, "marks": 1, "answer": "{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}", "expectedvariablenames": ["x", "y", "z"], "scripts": {}}, {"vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": true, "answersimplification": "all", "checkingaccuracy": 0.001, "marks": 1, "answer": "{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}", "expectedvariablenames": ["x", "y", "z"], "scripts": {}}, {"vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": true, "answersimplification": "all", "checkingaccuracy": 0.001, "marks": 1, "answer": "{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}", "expectedvariablenames": ["x", "y", "z"], "scripts": {}}], "scripts": {}}, {"type": "gapfill", "marks": 0, "prompt": "$\\boldsymbol{\\nabla\\cdot u}=$ [[0]].
", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "checkvariablenames": true, "answersimplification": "all", "checkingaccuracy": 0.001, "marks": 1, "answer": "{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}", "expectedvariablenames": ["x", "y", "z"], "scripts": {}}], "scripts": {}}, {"maxMarks": 0, "distractors": ["", ""], "prompt": "Is the vector field $\\boldsymbol{u}$ irrotational?
", "showCorrectAnswer": true, "displayType": "radiogroup", "minMarks": 0, "displayColumns": 0, "matrix": [1, 0], "shuffleChoices": false, "type": "1_n_2", "marks": 0, "choices": ["{irrotational}
", "{notirrotational}
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", "showCorrectAnswer": true, "displayType": "radiogroup", "minMarks": 0, "displayColumns": 0, "matrix": [1, 0], "shuffleChoices": false, "type": "1_n_2", "marks": 0, "choices": ["{solenoidal}
", "{notsolenoidal}
"], "scripts": {}}], "rulesets": {}, "type": "question", "showQuestionGroupNames": false, "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"], "statement": "For the vector field $\\boldsymbol{u}=\\pmatrix{\\simplify{{a1}*x^{p1}+{b1}*y^{p2}*z^{p3}},\\simplify{{c1}*y^{p4}+{d1}*x^{p5}*z^{p6}},\\simplify{{e1}*z^{p7}+{f1}*x^{p8}*y^{p9}}}$, calculate $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ and $\\boldsymbol{\\nabla\\cdot u}$, and determine whether $\\boldsymbol{u}$ is irrotational or solenoidal, or both.
", "advice": "The curl of a vector field $\\boldsymbol{u}=\\pmatrix{u_x,u_y,u_z}$ is given by
\n\\[\\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}}.\\]
\nThe divergence of the same vector field is given by
\n\\[\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}.\\]
\nBy straightforward partial differentiation
\n\\[\\boldsymbol{\\nabla\\cdot 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}}}.\\]
\nAgain, by partial differentiation
\n\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.\\]
\nA 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.
\nSince $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ {irrequal} to the zero vector, the vector field {isirr}.
\nSince $\\boldsymbol{\\nabla\\cdot u}$ {solequal} to zero, the vector field {issol}.
", "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}