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$CellContext`y^2]}}, \ {{$CellContext`t$$, 5.32, "move point"}, 0, 6.283, ImageSize -> Tiny}, {{$CellContext`cylinder$$, True}, { False, True}}, {{$CellContext`axes$$, False}, { False, True}}, {{$CellContext`boxed$$, False}, { False, True}}, {{$CellContext`op$$, 0.705, "surface opacity"}, 0, 1, ImageSize -> Tiny}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{619., {228., 233.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->266249794] }, Open ]], Cell[CellGroupData[{ Cell["", "SnapshotsSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = False, $CellContext`i$$ = 3, $CellContext`op$$ = 0.29, $CellContext`t$$ = 1.04, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`i$$], 3, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, {{ Hold[$CellContext`t$$], 1.04, "move point"}, 0, 6.283}, {{ Hold[$CellContext`cylinder$$], False}, {False, True}}, {{ Hold[$CellContext`axes$$], False}, {False, True}}, {{ Hold[$CellContext`boxed$$], False}, {False, True}}, {{ Hold[$CellContext`op$$], 0.29, "surface opacity"}, 0, 1}}, Typeset`size$$ = {400., {198., 202.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`i$1522$$ = False, $CellContext`t$1523$$ = 0, $CellContext`cylinder$1524$$ = False, $CellContext`axes$1525$$ = False, $CellContext`boxed$1526$$ = False, $CellContext`op$1527$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = False, $CellContext`i$$ = 3, $CellContext`op$$ = 0.29, $CellContext`t$$ = 1.04}, "ControllerVariables" :> { Hold[$CellContext`i$$, $CellContext`i$1522$$, False], Hold[$CellContext`t$$, $CellContext`t$1523$$, 0], Hold[$CellContext`cylinder$$, $CellContext`cylinder$1524$$, False], Hold[$CellContext`axes$$, $CellContext`axes$1525$$, False], Hold[$CellContext`boxed$$, $CellContext`boxed$1526$$, False], Hold[$CellContext`op$$, $CellContext`op$1527$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> With[{$CellContext`f$ = Switch[$CellContext`i$$, 1, $CellContext`x + $CellContext`y, 2, $CellContext`x $CellContext`y, 3, $CellContext`x^2 - $CellContext`y^2]}, Show[ If[$CellContext`cylinder$$, ContourPlot3D[{$CellContext`x^2 + $CellContext`y^2 == 1, $CellContext`z == 0}, {$CellContext`x, -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -1.5, 1.5}, Mesh -> None, BoxRatios -> Automatic, BoundaryStyle -> None, ContourStyle -> { Directive[{ RGBColor[1, 0, 0], Opacity[0.2]}], Directive[{ RGBColor[0, 1, 0.25098], Opacity[0.2]}]}], {}], ContourPlot3D[{$CellContext`z == $CellContext`f$}, {$CellContext`x, \ -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -2, 2}, Mesh -> None, BoxRatios -> Automatic, ContourStyle -> { Directive[{ RGBColor[0, 0, 1], Opacity[$CellContext`op$$]}]}], Graphics3D[{ Opacity[1], PointSize[0.02], RGBColor[0.501961, 0.25098, 0], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}], RGBColor[0, 1, 1], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}], RGBColor[0.25098, 0, 0.501961], Line[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}]}], ParametricPlot3D[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}, {$CellContext`t$$, 0, 2 Pi}], Graphics3D[{ Text[ Style[ Row[{ Style["z", Italic], " = ", ToString[ N[ ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}], 3]]}], 14], {-1.2, -1.2, 2.5}], Text[ Style[ Row[{ Style["x", Italic], " = ", ToString[ N[ Cos[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.3}], Text[ Style[ Row[{ Style["y", Italic], " = ", ToString[ N[ Sin[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.1}]}], Axes -> $CellContext`axes$$, Boxed -> $CellContext`boxed$$, ImageSize -> {400, 400}]], "Specifications" :> {{{$CellContext`i$$, 3, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, \ {{$CellContext`t$$, 1.04, "move point"}, 0, 6.283, ImageSize -> Tiny}, {{$CellContext`cylinder$$, False}, { False, True}}, {{$CellContext`axes$$, False}, { False, True}}, {{$CellContext`boxed$$, False}, { False, True}}, {{$CellContext`op$$, 0.29, "surface opacity"}, 0, 1, ImageSize -> Tiny}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{619., {228., 233.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->328446442], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = False, $CellContext`i$$ = 2, $CellContext`op$$ = 0.705, $CellContext`t$$ = 5.32, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`i$$], 2, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, {{ Hold[$CellContext`t$$], 5.32, "move point"}, 0, 6.283}, {{ Hold[$CellContext`cylinder$$], False}, {False, True}}, {{ Hold[$CellContext`axes$$], False}, {False, True}}, {{ Hold[$CellContext`boxed$$], False}, {False, True}}, {{ Hold[$CellContext`op$$], 0.705, "surface opacity"}, 0, 1}}, Typeset`size$$ = {400., {198., 202.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`i$1605$$ = False, $CellContext`t$1606$$ = 0, $CellContext`cylinder$1607$$ = False, $CellContext`axes$1608$$ = False, $CellContext`boxed$1609$$ = False, $CellContext`op$1610$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = False, $CellContext`i$$ = 2, $CellContext`op$$ = 0.705, $CellContext`t$$ = 5.32}, "ControllerVariables" :> { Hold[$CellContext`i$$, $CellContext`i$1605$$, False], Hold[$CellContext`t$$, $CellContext`t$1606$$, 0], Hold[$CellContext`cylinder$$, $CellContext`cylinder$1607$$, False], Hold[$CellContext`axes$$, $CellContext`axes$1608$$, False], Hold[$CellContext`boxed$$, $CellContext`boxed$1609$$, False], Hold[$CellContext`op$$, $CellContext`op$1610$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> With[{$CellContext`f$ = Switch[$CellContext`i$$, 1, $CellContext`x + $CellContext`y, 2, $CellContext`x $CellContext`y, 3, $CellContext`x^2 - $CellContext`y^2]}, Show[ If[$CellContext`cylinder$$, ContourPlot3D[{$CellContext`x^2 + $CellContext`y^2 == 1, $CellContext`z == 0}, {$CellContext`x, -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -1.5, 1.5}, Mesh -> None, BoxRatios -> Automatic, BoundaryStyle -> None, ContourStyle -> { Directive[{ RGBColor[1, 0, 0], Opacity[0.2]}], Directive[{ RGBColor[0, 1, 0.25098], Opacity[0.2]}]}], {}], ContourPlot3D[{$CellContext`z == $CellContext`f$}, {$CellContext`x, \ -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -2, 2}, Mesh -> None, BoxRatios -> Automatic, ContourStyle -> { Directive[{ RGBColor[0, 0, 1], Opacity[$CellContext`op$$]}]}], Graphics3D[{ Opacity[1], PointSize[0.02], RGBColor[0.501961, 0.25098, 0], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}], RGBColor[0, 1, 1], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}], RGBColor[0.25098, 0, 0.501961], Line[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}]}], ParametricPlot3D[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}, {$CellContext`t$$, 0, 2 Pi}], Graphics3D[{ Text[ Style[ Row[{ Style["z", Italic], " = ", ToString[ N[ ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}], 3]]}], 14], {-1.2, -1.2, 2.5}], Text[ Style[ Row[{ Style["x", Italic], " = ", ToString[ N[ Cos[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.3}], Text[ Style[ Row[{ Style["y", Italic], " = ", ToString[ N[ Sin[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.1}]}], Axes -> $CellContext`axes$$, Boxed -> $CellContext`boxed$$, ImageSize -> {400, 400}]], "Specifications" :> {{{$CellContext`i$$, 2, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, \ {{$CellContext`t$$, 5.32, "move point"}, 0, 6.283, ImageSize -> Tiny}, {{$CellContext`cylinder$$, False}, { False, True}}, {{$CellContext`axes$$, False}, { False, True}}, {{$CellContext`boxed$$, False}, { False, True}}, {{$CellContext`op$$, 0.705, "surface opacity"}, 0, 1, ImageSize -> Tiny}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{619., {228., 233.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->497258914], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = True, $CellContext`i$$ = 1, $CellContext`op$$ = 0.705, $CellContext`t$$ = 5.32, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`i$$], 1, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, {{ Hold[$CellContext`t$$], 5.32, "move point"}, 0, 6.283}, {{ Hold[$CellContext`cylinder$$], True}, {False, True}}, {{ Hold[$CellContext`axes$$], False}, {False, True}}, {{ Hold[$CellContext`boxed$$], False}, {False, True}}, {{ Hold[$CellContext`op$$], 0.705, "surface opacity"}, 0, 1}}, Typeset`size$$ = {400., {198., 202.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`i$1688$$ = False, $CellContext`t$1689$$ = 0, $CellContext`cylinder$1690$$ = False, $CellContext`axes$1691$$ = False, $CellContext`boxed$1692$$ = False, $CellContext`op$1693$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`axes$$ = False, $CellContext`boxed$$ = False, $CellContext`cylinder$$ = True, $CellContext`i$$ = 1, $CellContext`op$$ = 0.705, $CellContext`t$$ = 5.32}, "ControllerVariables" :> { Hold[$CellContext`i$$, $CellContext`i$1688$$, False], Hold[$CellContext`t$$, $CellContext`t$1689$$, 0], Hold[$CellContext`cylinder$$, $CellContext`cylinder$1690$$, False], Hold[$CellContext`axes$$, $CellContext`axes$1691$$, False], Hold[$CellContext`boxed$$, $CellContext`boxed$1692$$, False], Hold[$CellContext`op$$, $CellContext`op$1693$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> With[{$CellContext`f$ = Switch[$CellContext`i$$, 1, $CellContext`x + $CellContext`y, 2, $CellContext`x $CellContext`y, 3, $CellContext`x^2 - $CellContext`y^2]}, Show[ If[$CellContext`cylinder$$, ContourPlot3D[{$CellContext`x^2 + $CellContext`y^2 == 1, $CellContext`z == 0}, {$CellContext`x, -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -1.5, 1.5}, Mesh -> None, BoxRatios -> Automatic, BoundaryStyle -> None, ContourStyle -> { Directive[{ RGBColor[1, 0, 0], Opacity[0.2]}], Directive[{ RGBColor[0, 1, 0.25098], Opacity[0.2]}]}], {}], ContourPlot3D[{$CellContext`z == $CellContext`f$}, {$CellContext`x, \ -1.5, 1.5}, {$CellContext`y, -1.5, 1.5}, {$CellContext`z, -2, 2}, Mesh -> None, BoxRatios -> Automatic, ContourStyle -> { Directive[{ RGBColor[0, 0, 1], Opacity[$CellContext`op$$]}]}], Graphics3D[{ Opacity[1], PointSize[0.02], RGBColor[0.501961, 0.25098, 0], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}], RGBColor[0, 1, 1], Point[{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}], RGBColor[0.25098, 0, 0.501961], Line[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}]}], ParametricPlot3D[{{ Cos[$CellContext`t$$], Sin[$CellContext`t$$], 0}, { Cos[$CellContext`t$$], Sin[$CellContext`t$$], ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}]}}, {$CellContext`t$$, 0, 2 Pi}], Graphics3D[{ Text[ Style[ Row[{ Style["z", Italic], " = ", ToString[ N[ ReplaceAll[$CellContext`f$, {$CellContext`x -> Cos[$CellContext`t$$], $CellContext`y -> Sin[$CellContext`t$$]}], 3]]}], 14], {-1.2, -1.2, 2.5}], Text[ Style[ Row[{ Style["x", Italic], " = ", ToString[ N[ Cos[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.3}], Text[ Style[ Row[{ Style["y", Italic], " = ", ToString[ N[ Sin[$CellContext`t$$], 3]]}], 14], {-1.2, -1.2, 2.1}]}], Axes -> $CellContext`axes$$, Boxed -> $CellContext`boxed$$, ImageSize -> {400, 400}]], "Specifications" :> {{{$CellContext`i$$, 1, "choose function"}, { 1 -> TraditionalForm[$CellContext`x + $CellContext`y], 2 -> TraditionalForm[$CellContext`x $CellContext`y], 3 -> TraditionalForm[$CellContext`x^2 - $CellContext`y^2]}}, \ {{$CellContext`t$$, 5.32, "move point"}, 0, 6.283, ImageSize -> Tiny}, {{$CellContext`cylinder$$, True}, { False, True}}, {{$CellContext`axes$$, False}, { False, True}}, {{$CellContext`boxed$$, False}, { False, True}}, {{$CellContext`op$$, 0.705, "surface opacity"}, 0, 1, ImageSize -> Tiny}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{619., {228., 233.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->297548900] }, Open ]], Cell[CellGroupData[{ Cell["", "DetailsSection"], Cell[TextData[{ "To find the extrema of a function ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"f", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]], "InlineMath"], ", subject to the constraint that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}], "=", "0"}], TraditionalForm]], "InlineMath"], ", form an auxiliary function ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{ RowBox[{"f", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}], "+", RowBox[{"\[Lambda]g", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}]}]}], TraditionalForm]], "InlineMath"], ". The coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]], "InlineMath"], " of the extrema satisfy three equations: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}], "=", "0"}], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["w", "x"], "=", "0"}], TraditionalForm]], "InlineMath"], ", and ", Cell[BoxData[ FormBox[ SubscriptBox["w", "y"], TraditionalForm]], "InlineMath"], "=0. The function ", Cell[BoxData[ FormBox["w", TraditionalForm]], "InlineMath"], " is referred to as the Lagrangian, and ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]], "InlineMath"], " as the Lagrange multiplier. One observes that the first of the three \ equations is simply ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["w", "\[Lambda]"], "=", "0"}], TraditionalForm]], "InlineMath"], ", and that we really have three variables ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y", ",", "\[Lambda]"}], ")"}], TraditionalForm]], "InlineMath"], ", so the system we solve is not overdetermined. 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