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As the orientations of each part of the curve are different the areas \ could compesate if we compute it with a single line integral. Therefore we \ compute the area for both parts in a separate way. To use the Theorem of \ Green we introduce a vector field with constant curl. For instance\ \>", "Text", CellChangeTimes->{{3.7855054126471868`*^9, 3.785505614509632*^9}, { 3.7855240360664234`*^9, 3.7855240568135176`*^9}},ExpressionUUID->"5ac5572d-bf16-473f-8660-\ e1b267adeb78"], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x_", ",", "y_"}], "}"}], "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"-", "y"}], ",", "x"}], "}"}]}]], "Input", CellChangeTimes->{{3.785505616080741*^9, 3.7855056234904957`*^9}, 3.7855056616086884`*^9},ExpressionUUID->"e9990ff6-c62c-42d0-a09a-\ 826be21a15c1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Curl", "[", RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.785505626467803*^9, 3.785505648985948*^9}},ExpressionUUID->"63e00b5a-2fde-41d3-99ab-\ 65b7b8e5d1a0"], Cell[BoxData["2"], "Output", CellChangeTimes->{{3.7855056499663506`*^9, 3.7855056634696455`*^9}},ExpressionUUID->"75f8e4a1-9a8a-463e-9508-\ d2532751a1aa"] }, Open ]], Cell[TextData[{ "By means of Green\[CloseCurlyQuote]s Theorem we know that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Integral]", "D"], RowBox[{"curl", "(", "F", ")"}]}], "=", RowBox[{ SubscriptBox["\[Integral]", "r"], "F"}]}], TraditionalForm]], ExpressionUUID->"a1f92776-c578-4fbe-ab6b-0d4961f2b758"], " and as curl(F)=2 we finally have that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"area", "(", "D", ")"}], "=", RowBox[{ FractionBox["1", "2"], RowBox[{ SubscriptBox["\[Integral]", "r"], "F"}]}]}], TraditionalForm]], ExpressionUUID->"636378b4-549a-403d-b574-b749aa9aac52"], ". " }], "Text", CellChangeTimes->{{3.7855056746812315`*^9, 3.785505757979257*^9}, { 3.785524090923004*^9, 3.7855240932966857`*^9}},ExpressionUUID->"6b572d3a-1406-4efb-bdfa-\ c2bb3829e154"], Cell["Therefore, for the first part of the curve", "Text", CellChangeTimes->{{3.7855057618280573`*^9, 3.7855057908074784`*^9}},ExpressionUUID->"0f9071f8-fb18-49c0-a06a-\ e0e7b0de810b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"area1", "=", RowBox[{"N", "[", RowBox[{ FractionBox["1", "2"], RowBox[{ SubsuperscriptBox["\[Integral]", FractionBox[ RowBox[{"-", "\[Pi]"}], "2"], FractionBox["\[Pi]", "2"]], RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"r", "[", "t", "]"}], "]"}], ".", RowBox[{ RowBox[{"r", "'"}], "[", "t", "]"}]}], RowBox[{"\[DifferentialD]", "t"}]}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.7855057987839603`*^9, 3.7855058363794518`*^9}, { 3.7855058962752037`*^9, 3.7855058981491537`*^9}, {3.785505967056706*^9, 3.785505968624832*^9}},ExpressionUUID->"ad20dd14-cb0e-45a1-b5a7-\ 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3.7855241143323855`*^9},ExpressionUUID->"6e0e960a-ad11-4599-8898-\ 2a10b58bf109"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3", "Subsection", CellChangeTimes->{{3.785489790046318*^9, 3.785489814502342*^9}, { 3.7854986388362226`*^9, 3.7854986389061623`*^9}, {3.785506057429943*^9, 3.7855060575388794`*^9}},ExpressionUUID->"6a162753-74cb-4ff0-b619-\ dca64ce0dc51"], Cell["We have the vector field", "Text", CellChangeTimes->{{3.785506101832114*^9, 3.785506108328396*^9}},ExpressionUUID->"10820c3c-0b4e-457d-a6c2-\ 38520e8d277a"], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["z", "2"]}]], "+", RowBox[{"3", " ", "x"}], "-", RowBox[{"7", " ", "y", " ", "z"}]}], ",", RowBox[{ RowBox[{"x", " ", "z"}], "+", RowBox[{"2", " ", "x", " ", "y", " ", "z"}], "+", RowBox[{"Cos", "[", SuperscriptBox["z", "2"], "]"}]}], ",", RowBox[{ RowBox[{"y", " ", "z"}], "+", RowBox[{"Cos", "[", RowBox[{"2", " ", SuperscriptBox["y", "2"]}], "]"}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.7855061102632885`*^9, 3.785506120440469*^9}, { 3.7855063177404456`*^9, 3.7855063233622327`*^9}},ExpressionUUID->"fd97c8d7-a804-40a6-a728-\ 51e4c725190a"], Cell[TextData[{ "The surface ", Cell[BoxData[ InterpretationBox["\<\"S\[Congruent](\\!\\(\\*FractionBox[\\(\\!\\(\\*\ RowBox[{\\\"2\\\", \\\"+\\\", \\\"x\\\"}]\\)\\), \ \\(\\!\\(\\*RowBox[{\\\"1\\\"}]\\)\\)]\\)\\!\\(\\*SuperscriptBox[\\()\\), \ \\(2\\)]\\)+(\\!\\(\\*FractionBox[\\(\\!\\(\\*RowBox[{RowBox[{\\\"-\\\", \ \\\"2\\\"}], \\\"+\\\", \\\"y\\\"}]\\)\\), \\(\\!\\(\\*RowBox[{\\\"3\\\"}]\\)\ \\)]\\)\\!\\(\\*SuperscriptBox[\\()\\), \\(2\\)]\\)+(\\!\\(\\*FractionBox[\\(\ \\!\\(\\*RowBox[{\\\"3\\\", \\\"+\\\", \\\"z\\\"}]\\)\\), \ \\(\\!\\(\\*RowBox[{\\\"3\\\"}]\\)\\)]\\)\\!\\(\\*SuperscriptBox[\\()\\), \ \\(2\\)]\\)=1\"\>", StringForm[ "S\[Congruent](\!\(\*FractionBox[\(`1`\), \ \(`4`\)]\)\!\(\*SuperscriptBox[\()\), \(2\)]\)+(\!\(\*FractionBox[\(`2`\), \ \(`5`\)]\)\!\(\*SuperscriptBox[\()\), \(2\)]\)+(\!\(\*FractionBox[\(`3`\), \ \(`6`\)]\)\!\(\*SuperscriptBox[\()\), \(2\)]\)=1", 2 + $CellContext`x, -2 + $CellContext`y, 3 + $CellContext`z, 1, 3, 3], Editable->False]], "Print", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{3.785499517037701*^9, 3.7854995852236967`*^9, 3.785503383925978*^9, 3.785504129433913*^9, 3.7855042451749525`*^9, 3.785504586403664*^9, 3.7855046429253316`*^9, 3.7855048683359528`*^9, 3.7855051321918125`*^9},ExpressionUUID-> "3b3ac3a8-d37e-41e0-b852-d6fd6b390ae7"], " is an ellipsoid and therefore is a closed surface. \nThe general form of \ the ellipsoid centered at p=(p1,p2,p3) with radius r1, r2 and r3 for the axis \ x,y,z, resp., has general form " }], "Text", CellChangeTimes->{{3.7855063250192833`*^9, 3.7855063787056274`*^9}, { 3.7855069073816996`*^9, 3.7855071922742467`*^9}, {3.785507421035576*^9, 3.785507421766183*^9}},ExpressionUUID->"0d4ee411-68ed-407f-bb46-\ 2bafbb1c14a6"], Cell[BoxData[ RowBox[{ RowBox[{"S", "\[Congruent]", RowBox[{ SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"x", "-", "p1"}], "r1"], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"x", "-", "p2"}], "r2"], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"x", "-", "p3"}], "r3"], ")"}], "2"]}]}], "=", "1"}]], "Input", CellChangeTimes->{{3.7855072390970497`*^9, 3.7855073033851457`*^9}, { 3.7855073430683126`*^9, 3.7855073450182085`*^9}},ExpressionUUID->"ad2c5f8d-4b74-4202-b757-\ 111724eec3c7"], Cell["\<\ Therefore, we have an ellipsoid centered at p=(-2,2,-3) with radius 1, 3 and 3\ \>", "Text", CellChangeTimes->{{3.785507370889928*^9, 3.7855074352068615`*^9}},ExpressionUUID->"7cb4b9a1-b132-489d-a719-\ 854979806801"], Cell[TextData[{ "We need to compute the surface integral ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Integral]", "S"], "F"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "8a8ac2b4-dc0e-4035-9fd1-1009d7612037"], " but if we use Stokes\[CloseCurlyQuote] Theorem we know that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Integral]", "S"], "F"}], "=", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalOmega]"], RowBox[{"div", "(", "F", ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "4966a205-85bb-4e27-b083-8174430d58bf"], ", where \[CapitalOmega] is the interior domain of the considered surface. \ In principle it would be valid to compute any of these two integral. Of \ course we should choose the most simple one. In the formula of F we \ appreciate functions like ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]], " "}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "620c2620-41d9-4e34-aa1e-7493b0b0b66c"], " for which to compute the integral is difficult. On the other hand, if we \ compute the divergence of the vector field we have" }], "Text", CellChangeTimes->{{3.785506159517116*^9, 3.7855061975495844`*^9}, { 3.7855062597779875`*^9, 3.785506312230779*^9}, {3.785506436086787*^9, 3.78550654327947*^9}},ExpressionUUID->"93b5f711-9230-4d7d-9db9-\ 3f5d2325d86d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Div", "[", RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7855065446636925`*^9, 3.785506566225342*^9}},ExpressionUUID->"aae390da-c6cd-4510-a120-\ d273b9e0dd52"], Cell[BoxData[ RowBox[{"3", "+", "y", "+", RowBox[{"2", " ", "x", " ", "z"}]}]], "Output", CellChangeTimes->{{3.7855065614001017`*^9, 3.785506567233765*^9}},ExpressionUUID->"7a01be3a-a673-4613-94e5-\ 2b4617e0d94f"] }, Open ]], Cell["\<\ We can define the function divF to store the divergence in the following way\ \>", "Text", CellChangeTimes->{{3.785507746620239*^9, 3.7855077687586126`*^9}},ExpressionUUID->"be36e885-a151-45c5-bf9d-\ f271aa4242e4"], Cell[BoxData[ RowBox[{ RowBox[{"divF", "[", RowBox[{"{", RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], "]"}], ":=", RowBox[{"3", "+", "y", "+", RowBox[{"2", " ", "x", " ", "z"}]}]}]], "Input", CellChangeTimes->{{3.785507774130503*^9, 3.785507796525694*^9}},ExpressionUUID->"53c7cc2b-a9ad-45b6-af78-\ 810d74663aea"], Cell[TextData[{ "And we observe that the formula that we obtain is much more simple than the \ initial components that the field had. So we will compute ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Integral]", "\[CapitalOmega]"], RowBox[{"div", "(", "F", ")"}]}], TraditionalForm]],ExpressionUUID-> "af5e467a-526c-4c3e-a4b3-3164e7b592ef"], " to solve the problem. " }], "Text", CellChangeTimes->{{3.785506574773452*^9, 3.7855066302976894`*^9}},ExpressionUUID->"55d93578-3c62-444f-8b23-\ 1dbc85163746"], Cell[TextData[{ "We need a suitable change of variable to compute the volume integral over \ the ellipsoid. The change of variable is very similar to the spheric change \ but we have to adapt it to the shape of the ellipsoid. We consider the \ transformation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"T", ":", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"0", ",", RowBox[{"2", "\[Pi]"}]}], "]"}], "\[Times]", RowBox[{"[", RowBox[{"0", ",", "\[Pi]"}], "]"}], "\[Times]", RowBox[{"[", RowBox[{"0", ",", "1"}], "]"}]}], "-"}]}], "\[Rule]", SuperscriptBox["R", "3"]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "e5615c09-92a4-4419-857e-41b6393c8ceb"], " given by" }], "Text", CellChangeTimes->{{3.7855066550805225`*^9, 3.7855068040606775`*^9}},ExpressionUUID->"f4ac5e37-8a9e-4b70-aa49-\ fc9d9039fb4b"], Cell[BoxData[ RowBox[{ RowBox[{"T", "[", RowBox[{"\[Theta]_", ",", "\[Phi]_", ",", "r_"}], "]"}], ":=", RowBox[{ RowBox[{"r", RowBox[{"{", RowBox[{ RowBox[{"r1", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", RowBox[{"r2", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}], RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", RowBox[{"r3", " ", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "+", RowBox[{"{", RowBox[{"p1", ",", "p2", ",", "p3"}], "}"}]}]}]], "Input", CellChangeTimes->{{3.7855068077056303`*^9, 3.785506825651325*^9}, { 3.78550686888974*^9, 3.785506869872177*^9}},ExpressionUUID->"5f2bbcc6-ac85-4b05-b072-\ ce60554d5c33"], Cell["\<\ As we know the values for p=(-2,2,-3) and the radius r1=1, r2=3, r3=3, the \ final form for the transformation is\ \>", "Text", CellChangeTimes->{{3.785507459647869*^9, 3.7855075137951884`*^9}},ExpressionUUID->"116fa701-2ad2-48d7-a98f-\ 1099b00bc1e2"], Cell[BoxData[ RowBox[{ RowBox[{"T", "[", RowBox[{"\[Theta]_", ",", "\[Phi]_", ",", "r_"}], "]"}], ":=", RowBox[{ RowBox[{"r", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", RowBox[{"3", RowBox[{"Sin", "[", "\[Theta]", "]"}], RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", RowBox[{"3", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "+", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2", ",", RowBox[{"-", "3"}]}], "}"}]}]}]], "Input", CellChangeTimes->{{3.785507523503765*^9, 3.7855075690267205`*^9}},ExpressionUUID->"2ae6a3bc-3dba-4b09-bf77-\ 6a08fab9e131"], Cell["The Jacobian determinant for this transformation is", "Text", CellChangeTimes->{{3.7855075808169737`*^9, 3.7855075913779526`*^9}},ExpressionUUID->"e99f771b-64a3-4f98-ad8c-\ ae952aaadfa5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Det", "[", RowBox[{"Grad", "[", RowBox[{ RowBox[{"T", "[", RowBox[{"\[Theta]", ",", "\[Phi]", ",", "r"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "\[Theta]", ",", "\[Phi]"}], "}"}]}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.7855075938645115`*^9, 3.785507640254639*^9}, { 3.7855076748608046`*^9, 3.7855076788894997`*^9}},ExpressionUUID->"b6e37d92-c0e8-4841-8281-\ 618f69499c9a"], Cell[BoxData[ RowBox[{ RowBox[{"-", "9"}], " ", SuperscriptBox["r", "2"], " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}]], "Output", CellChangeTimes->{ 3.785507640953204*^9, {3.7855076722263117`*^9, 3.7855076796940417`*^9}},ExpressionUUID->"96ea0b49-7dcf-4a17-85d7-\ 0206fcf92655"] }, Open ]], Cell[TextData[{ "And we take this determinant with absolute value to apply the formula of \ change of variable for the integral ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Integral]", "\[CapitalOmega]"], RowBox[{"div", "(", "F", ")"}]}], TraditionalForm]],ExpressionUUID-> "ba272e07-17e9-4113-a49a-47991cd8a5b6"], ". We have to recall that the formula of change of variable for the integral \ is ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Integral]", SuperscriptBox["\[CapitalOmega]", "*"]], RowBox[{ RowBox[{"dif", "(", "F", ")"}], RowBox[{"(", RowBox[{"T", "(", RowBox[{"\[Theta]", ",", "\[Phi]", ",", "r"}], ")"}], ")"}], RowBox[{"det", "(", RowBox[{ RowBox[{"T", "'"}], RowBox[{"(", RowBox[{"\[Theta]", ",", "\[Phi]", ",", "r"}], ")"}]}], ")"}], RowBox[{"\[DifferentialD]", "\[Theta]"}], RowBox[{"\[DifferentialD]", "\[Phi]"}], RowBox[{"\[DifferentialD]", "r"}]}]}], TraditionalForm]],ExpressionUUID-> "42394770-c2e9-4d3d-aa31-5950239b3ebc"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[CapitalOmega]", "*"], "=", RowBox[{ RowBox[{"[", RowBox[{"0", ",", RowBox[{"2", "\[Pi]"}]}], "]"}], "\[Times]", RowBox[{"[", RowBox[{"0", ",", "\[Pi]"}], "]"}], "\[Times]", RowBox[{"[", RowBox[{"0", ",", "1"}], "]"}]}]}], TraditionalForm]],ExpressionUUID-> "f14eec37-8fd1-41c0-8d00-5614da0d35ef"], ". Therefore" }], "Text", CellChangeTimes->{{3.785507690197036*^9, 3.7855077222242026`*^9}, { 3.7855078002795486`*^9, 3.785507822657266*^9}, {3.785507887496981*^9, 3.7855080025325975`*^9}, 3.7855242384090967`*^9},ExpressionUUID->"75d96179-c9c0-4155-939b-\ f2493c6fd31f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[{ SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[{ RowBox[{"divF", "[", RowBox[{"T", "[", RowBox[{"\[Theta]", ",", "\[Phi]", ",", "r"}], "]"}], "]"}], "9", " ", SuperscriptBox["r", "2"], " ", RowBox[{"Sin", "[", "\[Phi]", "]"}], RowBox[{"\[DifferentialD]", "r"}], RowBox[{"\[DifferentialD]", "\[Phi]"}], RowBox[{"\[DifferentialD]", "\[Theta]"}]}]}]}]}]], "Input", CellChangeTimes->{{3.7855079975024753`*^9, 3.7855080857478056`*^9}, { 3.785508241719965*^9, 3.7855082424125557`*^9}},ExpressionUUID->"4d3b5544-789c-4d5e-b79a-\ 82d77dbb1b19"], Cell[BoxData[ RowBox[{"204", " ", "\[Pi]"}]], "Output", CellChangeTimes->{3.785508088135425*^9, 3.7855082189499645`*^9},ExpressionUUID->"7e86c689-2288-4050-a06a-\ a25776bdd452"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.785508245373887*^9, 3.7855082468879833`*^9}},ExpressionUUID->"eb15464d-db4f-4f80-af42-\ 44ec2e3897d6"], Cell[BoxData["640.8849013323178`"], "Output", CellChangeTimes->{ 3.7855082473607135`*^9},ExpressionUUID->"53141f12-9865-4baa-88df-\ 7f858ff244b9"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Test 1 for serial number : 5", "Section", CellChangeTimes->{{3.7854897641951323`*^9, 3.7854897860316133`*^9}, { 3.7855139171004257`*^9, 3.7855139172273183`*^9}, {3.78551414960386*^9, 3.7855141497007875`*^9}},ExpressionUUID->"fc2ce443-601f-45ce-8aee-\ bd7ea136a536"], Cell[CellGroupData[{ Cell["Exercise 1", "Subsection", CellChangeTimes->{{3.785489790046318*^9, 3.785489814502342*^9}, { 3.7854986388362226`*^9, 3.7854986389061623`*^9}, {3.785506057429943*^9, 3.7855060575388794`*^9}, {3.785513928109089*^9, 3.785513928318968*^9}},ExpressionUUID->"c09638b6-2931-4f71-9d2d-\ 21408ab2705e"], Cell["\<\ In this case, we do not have two possible techniques since we have to compute \ the integral of the potential function over a 3-dimensional domain a we then \ need to compute the explicit expression for the potential. Therefore the only \ valid technique is what we called before Technique 1.\ \>", "Text", CellChangeTimes->{{3.7855139303348527`*^9, 3.7855140127816896`*^9}},ExpressionUUID->"de752cf9-07f3-4946-ba18-\ 52aebe564b33"], Cell["The vector field is", "Text", CellChangeTimes->{{3.785514227076001*^9, 3.785514232772764*^9}},ExpressionUUID->"91736c67-50b7-4e7a-b173-\ 6e280a5fb4a6"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"P", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "x", " ", "y", " ", "z"}], "+", RowBox[{"y", " ", "z", RowBox[{"(", RowBox[{ RowBox[{"3", "z"}], "-", RowBox[{"2", "x"}]}], ")"}]}], "-", RowBox[{"2", SuperscriptBox["y", "2"]}], "+", RowBox[{"2", "y"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Q", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"-", "4"}], "x", " ", "y"}], "+", RowBox[{"x", " ", "z", RowBox[{"(", RowBox[{ RowBox[{"3", "z"}], "-", RowBox[{"2", "x"}]}], ")"}]}], "+", RowBox[{"2", "x"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"R", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], ":=", RowBox[{ RowBox[{"3", "x", " ", "y", " ", "z"}], "+", RowBox[{"x", " ", "y", RowBox[{"(", RowBox[{ RowBox[{"3", "z"}], "-", RowBox[{"2", "x"}]}], ")"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"P", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"Q", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"R", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "}"}]}], ";"}]}], "Input", CellChangeTimes->{{3.7855142357490654`*^9, 3.785514279564486*^9}, { 3.78551450730556*^9, 3.7855146184418163`*^9}},ExpressionUUID->"a706e28f-1827-4e8c-ade9-\ d5645286d3f1"], Cell["In this case, we start computing the integral for x: \ \[Phi]=\[Integral]P\[DifferentialD]x", "Text", CellChangeTimes->{{3.785514166180368*^9, 3.785514203145215*^9}},ExpressionUUID->"dd15c30a-c65e-48b6-ae15-\ 9405b58ec114"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Phi]", "=", RowBox[{ RowBox[{"\[Integral]", RowBox[{ RowBox[{"P", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], RowBox[{"\[DifferentialD]", "x"}]}]}], "+", RowBox[{"c", "[", RowBox[{"y", ",", "z"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.785514652528764*^9, 3.7855146794913397`*^9}},ExpressionUUID->"d99d51ae-8739-45a5-ac4b-\ 4aea8a094a2c"], Cell[BoxData[ RowBox[{ RowBox[{"y", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "x"}], "-", RowBox[{"2", " ", "x", " ", "y"}], "-", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", "z"}], "+", RowBox[{"3", " ", "x", " ", SuperscriptBox["z", "2"]}]}], ")"}]}], "+", RowBox[{"c", "[", RowBox[{"y", ",", "z"}], "]"}]}]], "Output", CellChangeTimes->{ 3.785514680111002*^9},ExpressionUUID->"1319856b-3f56-47cc-977c-\ bc81d0291820"] }, Open ]], Cell["We compute de derivative for y to compare with Q", "Text", CellChangeTimes->{{3.785514685582868*^9, 3.785514707443347*^9}, { 3.785514888026183*^9, 3.785514888488921*^9}},ExpressionUUID->"7819bf4a-c3a5-4136-88e4-\ 6c7c559b3f67"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], "\[Phi]"}], "\[Equal]", RowBox[{"Q", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.78551470991893*^9, 3.7855147361027136`*^9}, { 3.785514794751155*^9, 3.785514795133943*^9}},ExpressionUUID->"0382621c-e086-4ad1-8fae-\ 748ccbba97c5"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["c", TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None], "[", RowBox[{"y", ",", "z"}], "]"}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{{3.7855147291157045`*^9, 3.7855147368013*^9}, 3.7855147972587175`*^9},ExpressionUUID->"fc1b2aed-d6b2-4916-8732-\ d74e098ece99"] }, Open ]], Cell["Therefore, c(y,z)=c(z) and ", "Text", CellChangeTimes->{{3.785514823049965*^9, 3.7855148497771196`*^9}},ExpressionUUID->"9b1ccfb0-cefe-4920-b8ed-\ d5bbb8a6ce16"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Phi]", "=", RowBox[{ RowBox[{"y", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "x"}], "-", RowBox[{"2", " ", "x", " ", "y"}], "-", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", "z"}], "+", RowBox[{"3", " ", "x", " ", SuperscriptBox["z", "2"]}]}], ")"}]}], "+", RowBox[{"c", "[", "z", "]"}]}]}]], "Input", CellChangeTimes->{{3.7855148639369645`*^9, 3.785514869007066*^9}},ExpressionUUID->"2f32ba38-c764-40c1-894b-\ 0c18828377ad"], Cell[BoxData[ RowBox[{ RowBox[{"y", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "x"}], "-", RowBox[{"2", " ", "x", " ", "y"}], "-", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", "z"}], "+", RowBox[{"3", " ", "x", " ", SuperscriptBox["z", "2"]}]}], ")"}]}], "+", RowBox[{"c", "[", "z", "]"}]}]], "Output", CellChangeTimes->{ 3.7855149252968626`*^9},ExpressionUUID->"d9e7778b-7602-4b9a-9f76-\ b6316693b9a2"] }, Open ]], Cell["\<\ And we finally compute the derivative for z to compare with R\ \>", "Text", CellChangeTimes->{{3.785514873928269*^9, 3.7855148940257535`*^9}},ExpressionUUID->"28c02f9d-c79f-4149-aac7-\ 4fc07fed5f85"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "z"], "\[Phi]"}], "\[Equal]", RowBox[{"R", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.7855148967332225`*^9, 3.785514916955654*^9}},ExpressionUUID->"ee847360-1b25-40de-8813-\ fe6257ba0340"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "[", "z", "]"}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{{3.785514917610259*^9, 3.7855149273257036`*^9}},ExpressionUUID->"6e633bc8-c6ef-4c21-a9c9-\ 241ecc77a89b"] }, Open ]], Cell["So we have that c[z]=c and ", "Text", CellChangeTimes->{{3.7855149329495087`*^9, 3.785514944664785*^9}},ExpressionUUID->"63a88e3b-b6b0-414d-aadb-\ be859c516479"], Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "=", RowBox[{ RowBox[{"y", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "x"}], "-", RowBox[{"2", " ", "x", " ", "y"}], "-", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", "z"}], "+", RowBox[{"3", " ", "x", " ", SuperscriptBox["z", "2"]}]}], ")"}]}], "+", "c"}]}], ";"}]], "Input", CellChangeTimes->{{3.785514957956184*^9, 3.7855149649452024`*^9}},ExpressionUUID->"ec0c4193-dd6d-4228-b764-\ c23f20bbf7be"], Cell["The potential at the origin is \[Phi](0,0,0)=1 so that", "Text", CellChangeTimes->{{3.7855149693447037`*^9, 3.7855149865957966`*^9}},ExpressionUUID->"2c8da746-48cc-4e24-ac59-\ ff0a41ee516b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"\[Phi]", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"y", "\[Rule]", "0"}], ",", RowBox[{"z", "\[Rule]", "0"}]}], "}"}]}], ")"}], "\[Equal]", "1"}]], "Input", CellChangeTimes->{{3.785514990133773*^9, 3.7855150289435883`*^9}},ExpressionUUID->"1def4342-2504-4153-a5d6-\ 1f411b4232ed"], Cell[BoxData[ RowBox[{"c", "\[Equal]", "1"}]], "Output", CellChangeTimes->{{3.785515003242447*^9, 3.785515029963006*^9}},ExpressionUUID->"a13b181a-0839-4540-b227-\ d0e1eb01463f"] }, Open ]], Cell["And the final formula for the potential is", "Text", CellChangeTimes->{{3.7855150363323774`*^9, 3.785515043631426*^9}},ExpressionUUID->"427402ba-7b24-4406-bd84-\ e2a280c4566e"], Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "=", RowBox[{ RowBox[{"y", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "x"}], "-", RowBox[{"2", " ", "x", " ", "y"}], "-", RowBox[{"2", " ", SuperscriptBox["x", "2"], " ", "z"}], "+", RowBox[{"3", " ", "x", " ", SuperscriptBox["z", "2"]}]}], ")"}]}], "+", "1"}]}], ";"}]], "Input", CellChangeTimes->{{3.7855150578732986`*^9, 3.785515059327483*^9}},ExpressionUUID->"c2400b79-0409-49a1-9d76-\ 6b6531995575"], Cell[TextData[{ "We compute now the integral over the domain ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"[", RowBox[{"0", ",", "1"}], "]"}], "3"], TraditionalForm]],ExpressionUUID-> "a6d92c42-b4ab-4b5a-be9e-97e89d27f64e"], ". If the formula for \[Phi] is very complex, we use the instruction \ NIntegrate to obtain a numeric approximation for the integral" }], "Text", CellChangeTimes->{{3.7855150633231792`*^9, 3.785515117441223*^9}, 3.7855243602279243`*^9},ExpressionUUID->"5a98d35a-503c-4eba-9c86-\ 4d374390fdd6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NIntegrate", "[", RowBox[{"\[Phi]", ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.785515119206197*^9, 3.785515140945756*^9}},ExpressionUUID->"d15686bd-2bac-40b3-b58c-\ 2f8f8ba13b6c"], Cell[BoxData["1.25`"], "Output", CellChangeTimes->{ 3.785515141494444*^9},ExpressionUUID->"82faaea8-ca2f-47bb-977e-\ 184cec8ece17"] }, Open ]], Cell["\<\ Anyway, in this case we could have used also the usual definite integral \ since the formula for \[Phi] is not specially difficult.\ \>", "Text", 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In this case, as the formula for the field is simple, both of them are \ easily computable so we have two valid options." }], "Text", CellChangeTimes->{{3.7855162802309785`*^9, 3.785516304915857*^9}, { 3.7855163386245737`*^9, 3.785516395934179*^9}, {3.785524426863798*^9, 3.7855244480736666`*^9}},ExpressionUUID->"03cd65ea-715e-4764-aab8-\ 3c7413255707"], Cell[CellGroupData[{ Cell["Option 1", "Subsubsection", CellChangeTimes->{{3.785516397463315*^9, 3.785516422634907*^9}},ExpressionUUID->"4089e8ee-b59b-45e7-af15-\ 20eeaa07cdca"], Cell["We frist remove the value for variables that we used before:", "Text", CellChangeTimes->{{3.7855165628285065`*^9, 3.7855165757301435`*^9}},ExpressionUUID->"267e67ee-df56-48b1-8632-\ a8785d886c32"], Cell[BoxData[ RowBox[{"\[Phi]", "=."}]], "Input", CellChangeTimes->{{3.785516578815378*^9, 3.785516582461274*^9}},ExpressionUUID->"d435c944-7035-42c0-8929-\ 08d93c830c82"], Cell[TextData[{ "We compute ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Integral]", "S"], "F"}], TraditionalForm]], ExpressionUUID->"930f5b3a-4c54-4331-8cad-32eaa0e63596"], ". We then need a parameterization for the ellipsoid surface. 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