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For the particular case of n=3:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Curl", "[", RowBox[{ RowBox[{"LeavingKansas", "[", RowBox[{"x", ",", "y", ",", "z", ",", "3"}], "]"}], ",", RowBox[{"Cartesian", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "]"}], "//", "Simplify"}]], "Input", CellTags->"mmtag:13:Curl[]"], Cell["Define a new vector function for the curl for general n", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Glenda", "[", RowBox[{"x_", ",", "y_", ",", "z_", ",", "n_"}], "]"}], ":=", RowBox[{"Simplify", "[", RowBox[{"Curl", "[", RowBox[{ RowBox[{"LeavingKansas", "[", RowBox[{"x", ",", "y", ",", "z", ",", "n"}], "]"}], ",", RowBox[{"Cartesian", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "]"}], "]"}]}]], "Input"], Cell["\<\ Demonstrate the assertion that the curl has a fairly simple form and is \ sphericaly symmetric for n=1\ \>", "Text"], Cell[BoxData[ RowBox[{"Glenda", "[", RowBox[{"x", ",", "y", ",", "z", ",", "n"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Glenda", "[", RowBox[{"x", ",", "y", ",", "z", ",", "1"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{ "The", " ", "above", " ", "is", " ", "a", " ", "vector", " ", "field", " ", "that", " ", "points", " ", "radially", " ", "from", " ", "the", " ", "origin"}], "Text"], StyleBox[",", "Text"], StyleBox[" ", "Text"], RowBox[{ StyleBox[ RowBox[{ "with", " ", "a", " ", "magnitude", " ", "that", " ", "falls", " ", "off", " ", "like", " ", RowBox[{"1", "/", SuperscriptBox["r", "2"]}]}], "Text"]}]}]], "Text"], Cell["\<\ Visualize the curl for n=1, it will be necessary to \"zoom\" in to see the \ field.\ \>", "Text", CellChangeTimes->{{3.3991080248751783`*^9, 3.399108042076261*^9}}], Cell[BoxData[ RowBox[{"VectorFieldPlot3D", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{"Glenda", "[", RowBox[{"x", ",", "y", ",", "z", ",", "1"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "0.5"}], ",", "0.5"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "0.5"}], ",", "0.5"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "0.25"}], ",", "0.25"}], "}"}], ",", RowBox[{"VectorHeads", "\[Rule]", "True"}], ",", RowBox[{"ColorFunction", "\[Rule]", RowBox[{"(", RowBox[{ RowBox[{"Hue", "[", RowBox[{"#1", " ", "0.66`"}], "]"}], "&"}], ")"}]}], ",", RowBox[{"PlotPoints", "\[Rule]", "21"}]}], "]"}]], "Input", CellChangeTimes->{{3.3991073581007423`*^9, 3.399107382094822*^9}, { 3.399107762708005*^9, 3.399107766800351*^9}, {3.3991078004322863`*^9, 3.399107817865197*^9}, {3.3991079059780703`*^9, 3.399107925055943*^9}, 3.3993860110930023`*^9}], Cell["\<\ Demonstrate that the divergence of the curl vanishes for the above function \ independent of n\ \>", "Text", CellTags->"mmtag:13:divergence_of_curl__example"], Cell[BoxData[ RowBox[{"DivCurl", "=", RowBox[{"Div", "[", RowBox[{ RowBox[{"Glenda", "[", RowBox[{"x", ",", "y", ",", "z", ",", "n"}], "]"}], ",", RowBox[{"Cartesian", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"Simplify", "[", "DivCurl", "]"}]], "Input"] }, Closed]] }, CellGrouping->Manual, WindowSize->{1057, 724}, WindowMargins->{{259, Automatic}, {Automatic, 127}}, WindowTitle->"Lecture 13 MIT 3.016 (Fall 2007) \[Copyright] W. 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