(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 16553, 495] NotebookOptionsPosition[ 15546, 461] NotebookOutlinePosition[ 15881, 476] CellTagsIndexPosition[ 15838, 473] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["3 Dimensions", "Section", CellChangeTimes->{{3.537019244235758*^9, 3.537019255280658*^9}}], Cell["hi, this is a cell. look at the blue thing on the right ->", "Text", CellChangeTimes->{{3.537027142135014*^9, 3.537027150507303*^9}, 3.537027233665954*^9}], Cell[TextData[{ "Here, we are working with the equation:", "\n", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "-", RowBox[{"2", SuperscriptBox["z", "2"]}]}], "=", "4."}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537019291063635*^9, 3.537019367464292*^9}, 3.537019812803794*^9}], "\n", "We shall first look at the individual contours, developed by substituting \ the desired number for the variable. For example, say one wanted to find the \ x = 3 contour. Just substitute 3 for the variable x to get or", "\n", Cell[BoxData[ RowBox[{ RowBox[{"9", "+", SuperscriptBox["y", "2"], "-", RowBox[{"2", SuperscriptBox["z", "2"]}]}], "=", RowBox[{ RowBox[{ RowBox[{"4", " ", "or"}], " ", "-", SuperscriptBox["y", "2"], "+", RowBox[{"2", SuperscriptBox["z", "2"]}]}], "=", "5"}]}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.53701946079873*^9, 3.537019467024252*^9}, { 3.537019505537512*^9, 3.537019588404785*^9}, 3.537019812803954*^9}], "\n", "This is a hyperboloid, which makes sense. The 3 - D shape, after all, is a \ hyperboloid of one sheet, which should have hyperbolas (or X' s) and circles \ as cross sections." }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537019261115552*^9, 3.537019284722372*^9}, { 3.537019812803672*^9, 3.537019831164973*^9}, {3.537020448905198*^9, 3.537020451466679*^9}}, FontSize->14], Cell[BoxData[{ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "-", RowBox[{"2", RowBox[{"z", "^", "2"}]}]}], "\[Equal]", "4"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}]}], "]"}], "\[IndentingNewLine]", RowBox[{"plotz1", "=", RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{ 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"]"}]}], "}"}]], "Input",\ CellChangeTimes->{{3.53701848484577*^9, 3.537018555700628*^9}, { 3.537018993337442*^9, 3.537019093344821*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", " ", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{"z", "\[Equal]", RowBox[{"Sin", "[", RowBox[{"x", " ", "y"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.536958220371166*^9, 3.536958261064823*^9}, { 3.536958301875165*^9, 3.536958313881206*^9}, {3.536958370973864*^9, 3.536958395378384*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["4 Dimensions", "Section", CellChangeTimes->{{3.537019221222994*^9, 3.537019236538792*^9}}], Cell[TextData[{ "The following are the contours of a 4 dimensional equation,\n", Cell[BoxData[ RowBox[{"w", "=", RowBox[{ RowBox[{"f", RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}]}], "=", RowBox[{ SuperscriptBox["x", "3"], "+", SuperscriptBox["y", "2"], "-", SuperscriptBox["z", "2"]}]}]}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537019996127287*^9, 3.537020020269704*^9}, 3.537020363409116*^9}], "\nObserve that there are 4 variables: ", StyleBox["x, y, z", "InlineFormula"], ", and w (or f). When we set w to a specific number, say a constant c, we \ will get an equation with only 3 variables:\n", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", SuperscriptBox["y", "2"], "-", SuperscriptBox["z", "2"]}], "=", RowBox[{"c", "."}]}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537020130750292*^9, 3.537020154880882*^9}, 3.537020363409272*^9}], "\nTo understand what this might look like, we can make a contour plot in 3 \ D and include the command \"Contours.\" We set the number of contours \ (Automatic is also an option), and Mathematica will display several of the 3 \ D surfaces. A handy way to think about the fourth dimension (which humans can \ hardly conceptualize and certainly cannot picture) is to pretend the 4 th \ dimension is just time. So, over time, the 3 D graph is changing." }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537019978707047*^9, 3.537019991265362*^9}, 3.537020025747706*^9, {3.537020363408998*^9, 3.537020366574937*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "3"}], "+", RowBox[{"y", "^", "2"}], "-", RowBox[{"z", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"Contours", "\[Rule]", "5"}], ",", RowBox[{"Mesh", "\[Rule]", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.537019919286215*^9, 3.537019919286939*^9}, { 3.537019951627983*^9, 3.537019967304568*^9}, {3.537025781321375*^9, 3.53702578767531*^9}}], Cell[TextData[{ "For an example of something with which we are familiar, let us pull out the \ problem from earlier", "\n", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "-", RowBox[{"2", SuperscriptBox["z", "2"]}]}], "=", "4"}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.537020427023494*^9, 3.537020430927843*^9}, 3.537027302047488*^9}], "\n", "and change it to a 4 dimensional equation. Let", "\n", Cell[BoxData[ RowBox[{"t", "=", RowBox[{ RowBox[{"f", RowBox[{"(", RowBox[{"x", ",", "y", ",", "x"}], ")"}]}], "=", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "-", RowBox[{"2", SuperscriptBox["z", "2"]}]}]}]}]], "DisplayFormula", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.53702041155835*^9, 3.537020439368148*^9}, 3.537027302047778*^9}], "\n", "and see what it looks like when we graph it." }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.53702032691342*^9, 3.537020355276635*^9}, { 3.537027302047287*^9, 3.537027324577208*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "-", RowBox[{"2", RowBox[{"z", "^", "2"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"Contours", "\[Rule]", "5"}], ",", RowBox[{"Mesh", "\[Rule]", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.537020573590776*^9, 3.537020601430888*^9}}], Cell["\<\ One of these sheets should look awfully familiar. It is a bit buried, but the \ 3 D graph we plotted earlier is in there.\ \>", "Text", CellChangeTimes->{{3.537020623157565*^9, 3.537020672461548*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", RowBox[{"y", "^", "2"}]}], "-", RowBox[{"3", RowBox[{"z", "^", "2"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"Contours", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "Red"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "Blue"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "Yellow"}], "}"}]}], "}"}]}], ",", RowBox[{"Mesh", "\[Rule]", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.537020986384483*^9, 3.537021025207249*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", RowBox[{"y", "^", "2"}]}], "-", RowBox[{"3", RowBox[{"z", "^", "2"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"Contours", "\[Rule]", RowBox[{"{", RowBox[{"2", ",", "0"}], "}"}]}], ",", RowBox[{"Mesh", "\[Rule]", "None"}], ",", RowBox[{"ContourStyle", "->", RowBox[{"Opacity", "[", ".5", "]"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.537021064721613*^9, 3.537021099782421*^9}, { 3.537021147125819*^9, 3.537021149942681*^9}, {3.537021182627047*^9, 3.537021192758217*^9}, {3.5370212794101*^9, 3.537021362007316*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", RowBox[{"y", "^", "2"}]}], "-", RowBox[{"3", RowBox[{"z", "^", "2"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"Contours", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0"}], "}"}]}], ",", RowBox[{"Mesh", "\[Rule]", "None"}], ",", RowBox[{"ContourStyle", "->", RowBox[{"Opacity", "[", ".5", "]"}]}]}], "]"}]], "Input", CellChangeTimes->{3.537021393646223*^9}] }, Open ]] }, WindowSize->{1024, 664}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"8.0 for Linux x86 (32-bit) (November 7, 2010)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 97, 1, 72, "Section"], Cell[679, 25, 166, 2, 30, "Text"], Cell[848, 29, 1620, 42, 145, "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}], Cell[2471, 73, 2898, 77, 126, "Input"], Cell[5372, 152, 1577, 48, 50, "Input"], Cell[6952, 202, 1273, 37, 50, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[8262, 244, 97, 1, 72, "Section"], Cell[8362, 247, 1718, 38, 136, "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}], Cell[10083, 287, 714, 20, 30, "Input"], Cell[10800, 309, 1221, 36, 98, "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}], Cell[12024, 347, 638, 19, 30, "Input"], Cell[12665, 368, 211, 4, 30, "Text"], Cell[12879, 374, 919, 29, 50, "Input"], Cell[13801, 405, 935, 26, 50, "Input"], Cell[14739, 433, 791, 25, 50, "Input"] }, Open ]] } ] *) (* End of internal cache information *)