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The fact that grad rho is orthogonal to the columns of JPsi amounts \ to the matrix equation grad rho( Psi(s,t) ) JPsi(s,t) = 0 (which I have \ mathematica verify). This fact follow from differentiating the identity rho( \ Psi(s,t) ) = 0 and applying the chain rule.\ \>", "Text", CellChangeTimes->{{3.595677007394209*^9, 3.595677083783329*^9}, { 3.5956773547517853`*^9, 3.595677358559964*^9}, {3.595690438732148*^9, 3.59569056744285*^9}, {3.5956906493702374`*^9, 3.595690670050164*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{"gradrho", "[", RowBox[{"{", RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"rho", "[", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}], "}"}]}], "]"}]}], "]"}], "\n", RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{"JPsi", "[", RowBox[{"{", RowBox[{"s_", ",", "t_"}], "}"}], "]"}], " ", "=", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"Psi", "[", RowBox[{"{", RowBox[{"s", ",", "t"}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"{", RowBox[{"s", ",", "t"}], "}"}], "}"}]}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gradrho", "[", RowBox[{"Psi", "[", RowBox[{"{", RowBox[{"s", ",", "t"}], "}"}], "]"}], "]"}], ".", RowBox[{"JPsi", "[", RowBox[{"{", RowBox[{"s", ",", "t"}], "}"}], "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.595675449347249*^9, 3.5956755338078833`*^9}, { 3.595675646001073*^9, 3.595675685602762*^9}, {3.5956757605168123`*^9, 3.595675791047093*^9}, {3.5956761526275253`*^9, 3.595676152896862*^9}, 3.5956769848770514`*^9, {3.5956773431823683`*^9, 3.59567736452353*^9}, { 3.595690619942637*^9, 3.595690641541168*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{ RowBox[{ RowBox[{"-", "32"}], " ", "x"}], "+", RowBox[{"4", " ", "x", " ", RowBox[{"(", RowBox[{"3", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], ")"}]}]}]}, { RowBox[{ RowBox[{ RowBox[{"-", "32"}], " ", "y"}], "+", RowBox[{"4", " ", "y", " ", RowBox[{"(", RowBox[{"3", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], ")"}]}]}]}, { RowBox[{"4", " ", "z", " ", RowBox[{"(", RowBox[{"3", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], ")"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.5956773119658012`*^9, 3.595677366040339*^9, 3.595690410383586*^9, 3.5956905724025383`*^9, {3.595690617555984*^9, 3.595690642141073*^9}, 3.595690673246393*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"2", "+", RowBox[{"Cos", "[", "t", "]"}]}], ")"}]}], " ", RowBox[{"Sin", "[", "s", "]"}]}], RowBox[{ RowBox[{"-", RowBox[{"Cos", "[", "s", "]"}]}], " ", RowBox[{"Sin", "[", "t", "]"}]}]}, { RowBox[{ RowBox[{"Cos", "[", "s", "]"}], " ", RowBox[{"(", RowBox[{"2", "+", RowBox[{"Cos", "[", "t", "]"}]}], ")"}]}], RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "s", "]"}]}], " ", RowBox[{"Sin", "[", "t", "]"}]}]}, {"0", RowBox[{"Cos", "[", "t", "]"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.5956773119658012`*^9, 3.595677366040339*^9, 3.595690410383586*^9, 3.5956905724025383`*^9, {3.595690617555984*^9, 3.595690642141073*^9}, 3.595690673249296*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]], "Output", CellChangeTimes->{ 3.5956773119658012`*^9, 3.595677366040339*^9, 3.595690410383586*^9, 3.5956905724025383`*^9, {3.595690617555984*^9, 3.595690642141073*^9}, 3.5956906732507887`*^9}] }, Open ]], Cell[TextData[{ "To illustrate, let me draw the tangent plane and normal lines for the \ surface at the point p = Psi[Pi/4,Pi/4]. \n\nFirst a warning: as I defined \ them in class, the ", StyleBox["tangent and normal spaces", FontSlant->"Italic"], " to a hypersurface at a given point are vector subspaces of R^n; in \ particular, they pass through the origin. The", StyleBox[" tangent plane and normal line ", FontSlant->"Italic"], "are drawn here so that they instead pass through the given point p on the \ torus--i.e. what I really draw is the tangent space translated by p. One \ generally does things this way to facilitate an easy visual comparison of the \ surface with the tangent space.\n\nNote that the parametrization of the \ (translated) tangent plane is just given by the linear approximation of Psi \ (plot2). Alternatively, the tangent plane is the level set of the linear \ approximation of the defining function rho (plot3). 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