(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 61421, 1446] NotebookOptionsPosition[ 56208, 1340] NotebookOutlinePosition[ 57999, 1382] CellTagsIndexPosition[ 57920, 1377] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"0ba3833e-070c-4801-82af-cacc8780f61f"], Cell["MS-A0201 Diff. int. 2 (TFM)", "Title", CellChangeTimes->{{3.755527483885651*^9, 3.755527502028553*^9}},ExpressionUUID->"6388e47a-6b1f-498b-9257-\ 9a2787a9a01b"], Cell["Luento 1: Parametrisoidut k\[ADoubleDot]yr\[ADoubleDot]t ja \ kaarenpituus", "Subtitle", CellChangeTimes->{ 3.755527445583179*^9},ExpressionUUID->"0dec9ce5-4078-44d2-9a36-\ 3bd59c31ebf1"], Cell["Harri Hakula /Kev\[ADoubleDot]t 2023", "Subsubtitle", CellChangeTimes->{{3.755527633806271*^9, 3.7555276424907227`*^9}, { 3.7873214947370253`*^9, 3.787321494818349*^9}, 3.787321939540863*^9, 3.818996208282979*^9, {3.850837587474247*^9, 3.850837588462668*^9}, 3.882269013395791*^9},ExpressionUUID->"77a8381a-6add-4b11-99b3-\ 21687dda672a"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"5dc6cc66-657d-4069-bc3b-3aabaa3e0109"], Cell[CellGroupData[{ Cell["Yksinkertaisia esimerkkej\[ADoubleDot]: Tavallinen funktio", "Section", CellChangeTimes->{{3.755527760267074*^9, 3.755527767530978*^9}, { 3.755528518996209*^9, 3.755528528891755*^9}},ExpressionUUID->"53b37233-59d6-421a-a684-\ 08d68b2136e9"], Cell[TextData[{ StyleBox["r", FontSize->36, FontWeight->"Bold"], StyleBox[" (t) = t ", FontSize->36], StyleBox["i ", FontSize->36, FontWeight->"Bold"], StyleBox["+ sin (\[Pi] t)", FontSize->36], StyleBox[" j", FontSize->36, FontWeight->"Bold"], StyleBox[", t \[Element] [-1, 1]", FontSize->36] }], "Text", CellChangeTimes->{{3.755527784943656*^9, 3.755527805625931*^9}, { 3.755527886962586*^9, 3.755527935276455*^9}, {3.755528106019261*^9, 3.755528107923967*^9}, 3.850837687554183*^9}, FontSize->20, Background->GrayLevel[ 0.85],ExpressionUUID->"7afa3ded-7324-4a6f-a5c2-96853b37b9a8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"Pi", " ", "t"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.75552769383191*^9, 3.755527723889923*^9}}, CellLabel-> "In[929]:=",ExpressionUUID->"3c7cfb20-d2c7-4fad-9411-80b23cd7f8fb"], Cell[BoxData[ GraphicsBox[{{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[ 1.], FaceForm[Opacity[0.3]], LineBox[CompressedData[" 1:eJwtmXk4VH/0xxUlKZQWkYpEVBSyJaeUEC0KRSVEqZDstNmSItmSFKVEIZQU kmNJqSzZxZi5M5YsM3NF9ZWlfp/f8/vNP/O8nufemXvPOe/3ed/nyjmeO+A8 U0BAIG+GgMD/fv/fZwSFjy7SUMxIK91eMj/1zwSNQ8Kfd8bHHccQbxn9sV80 1rJ2ud2KO4dCg1r+/cM07pTcKh4UdwVFmtxGattpHC8xvGMYdwsXpndRd/No FDytVGIe+wAVTN5UaByncV75XU+fm3loGuMeeqqMj88vy8jLXEbMuP3rX4UD D+mbCeUiRl+RDrJR0V7ORR+rf6/efmvHa+tzZcS7h9BkMM7ahMtE9T0ysztD BvH3/FD56jk9+PmgcyrCAGaeaGgwXdCPPg0f1ZLY/WjxpWRu9vgAOi1XltVl 9ONfpZpxTcFBtDx9Y25nWz86J+mOvpk/iJqCe/pkavsx3OXu5FP5Qfyp1ZT8 4HU/Mu34S7TNB9E7lTErM7IfowbthwVSB9HT7UdX4eZ+nPjruldbfwjdRWUi mq/34QBzd/Ydx2H8/UWD+hrWh1MfuiaLzgzjpShznforfXhOY2NWs+cwRs2/ PFDj04e+O/ZI/QoZxmxxtkmZQx/enG4VHHg0jAOSmSLPdPvwmuTR01HsYXRY rhF5ebAX19ie2lNpw8UBhlnPhd5ebLyu8UzUgYseKU5bAli9uDtfWWmvCxeD ViQOe7X2onhbdEiZLxdTV/0xO13Ri6FuvuHm8VzsVMB5lnd70a9NyHD5Fy5a bjCLVjbtRY0/Dzsvb+Zh/j6TlyU7etFNXsndS5+H8zyN2swMerGIaZl/fAcP 37+GFe4avfjydo3OSgseamzTzH0p24sbnYNW67nycOFB2fotoz3YKRxY0P2Q h1/96fl77/Xgyy9HWusF+bj+HncT63YPehTL3CsU4WPEu0Erj5gerH0l2xov zsdtM3tT4q724LPX+g+0ZPiYF9mxof1cD7p6VRxbrM7H6NSKPfY7e/BYUKm0 lh0f97yPv+nJ46Ddx74vf/LI+bYmy+0HOMj6Vrwv8BUfNUemsvb0cHC/UaTW WBEfly8/WbP2GwdPhBetra/g45CXjiDrPQf/k8sMWtvMx/DV3f5m9zmYFGsn 4f6Tj4HFscK6dzj4yUZ6h/c4H9327UpUjONgZ+b4R88pPlpeyCuYEcFB7eS1 7w8L0bi6OZj/xouDaj8Xnq2RpLE8RNFJwYyDk5ytDzZvovGVVNfogl0cvHfO N7hck8bM3FvB/7Zx8O8iZsYOHRqjO8dTO7U4+L2+/I060HhM/cu3GHkOBl0f +lZgTuME+9y+6T9s9L3TdHL7SRr5fgrMoZ9s7Hift3T/aRrZ87+5dtBsrCkq +GvtSmONruGNgj42zrLe5LrHk8bE2EXVpxvZ2GAWkM+4RKPG9mK9tqds1Liz c1tYPI1rk69bdj9m42pG0iqRRBqXj9m496aS/8uWCgxPonH2k4m00QQ2tmtu TTyTQmPnHH0R8WA2RoTyHTsyaGxwmLd6yUU2uq08NbXqGY3vSxj6sn5slLhx ea5jNo3PXS95rHNj4+sY55A64hNBX8vajW3YeDPkQpNpEY0+yrd+7LVk4/MV RjOtS2g8E3Jc1HofG5cx4h/ZltJ4UFMAnIzYuF66TX1POY2KSdszgjaxcVJ1 IKz9I411x997F4uw8aDFzJtBrTRWFiVElwux8bzIkVuSxKdeL3B++vEfhR+f HJ9M7aDxYdUsRusvCld6/RJ70EWjt5LxjlE2hRoz/KR3smk8HbT02B8GhYEt RvbJHBrtvn33Feig8IvTjPmDPTQaR0ZkidVTuDxqRvK5fhql6U8L1pVQeFiM FSBHfFLcJHmdeiGF1VrlfD0ujUJpZ4x08ylc1mJYY86jkXdANMA4g8JQbeWn NjSNnOyuuL1pFPKGxHMtR2hsE8rJsbpPoTA/ZanJDzJPr82pE3EUxuZ/WiQ1 RmOhuOzEmZvkfNZ45k/CWS48Sc8ICo22jqZ8+kljgnS0cdBlCpVsAl8c/U3j dS87h2sBFKbMfVYv/R+Nl2tVL0R7U/ikkb2nkfCpyw25989QOOty9XmlPzRu 1A1L0HamcOnyzXtbCE+M6QQ2Hafwn9p+dX+yJ6pyecfdbAnjgKzkJI1Rpx8Z zbGi8LZ1s8QTwlYKh9Y93kfh1Znf56pO0biCJbrAYDeFfdbTormE8y19GF5A 4dqudPn4aRoDxFUqxfQo3PTxl/Y4YcPPzMxnmuR6WmoOWv6lUfRq/M2dahQO qQj5ZhJuARMvljKFOQExqT8Ip0xMHQ5UoDBz9HCd+j8aTxa+MFi8ksJac7OZ Zwlv9DipkL+M3G/Voa3JhP+oyMw1W0Sh7dLzl8sJV/Y10H1iFL6OiqruJhz5 MKw1SIRcHy9t4ShhyyO6b2WEKPwQnu08TXjLesZ2t2kW7l+fUfaPcH2a0fzH oyxU2xwr+4ew/dK8jo7vLPS95B4ySHg0SipdrJuFlLMh7yvhsJkh53Y2sbC5 WdIuj/AS/2G9wI8stNnEbQkj/JRnOTu/lIVybu0WFoT1TpQ19r1gYcBET8si wnUdSikymSw0a5S3ayD3b7831sXiPgul3TN5QYRHqyY0rsWycKLxWqgy4TBd p3+l4SyMDGGs/ETquySv7vPoBRb+uF5eeZzwUwXtxLXnWVjva+HKI/3RS37o YHeShXpH85Z7Eq4Vn7sh4QgLixqnmnmkv8eveo1/2s/CDWN2cfaERycYVf+M WNhmP33oM5mPMI9dtzZvYaEBzFyzjvDTI8sU09awMMvnZ2vjOPn9xpAfbdIs bMk2LFlKuHYXt3SeBAv5YsUZlmQef2zCg/4TTJSqnkwqIDlEb45z8L4GJj7P 8RqcNUqOv1RvfvU9E7kuUYuFiD7sxrSl3hYzUSfik9kfoqdQ5tw8xXQmmn1l ttcQvdW+ymdM+zPRzXGH39QAOV5F+qmGOxPnBW6ZfvmdxpEHoV6nTzBxeM6v W/ZEz4sjree27mFieb1K90OidzuHKe0ceSYG2qTJve6mcVVZS0eyFBODJVmb tzGI/0o/D7guxkQxKdGQyk4anVqOvT050Y1/69Or8oj/nDVGA7mmbpSfivi9 vpHG9el3WOI13VhX1dri30D8XMAj6O+7bpzdqsYvraPRs2RVZdezbvR1xuYN n2n03xBilBjcjWfPPfB9WUnqccO2P8yvG53P3tctJ3442a9+zcutG6XkYMuH MqLvhz01+2260X/MeaqE+OlVSSNz0U3d+K1OM8z4JY1xf4QPBlEMnJYfXMl7 QObbmhpzb2PgHqODoleIvy8uKEo4VsvA767/torcozHJ9XSbXhEDlcfDL8wm ++EB67PNr1sMVC+/75IeSWNO9U37M8BAjyK7e+4+NH6MlXS3TO3CzoZ3ptt3 0Kj+8lXGmfgulHHZrZlA9tv9JitWUEQXLohXVmJuodFDMmn/c88ufFeloHKI 7Eep28s1Zht34eTFkKpuBbIPktaMv6E70UbiOxiQfSv2QDtEZnsnvgma98Gy lI/+2FGySasTHwk2Td14w0cOK2DMeF0nfhb76Vj0ko9vVpU6eS/uxBi/b5Xj T/lo/2ibcd3gNxRVuD4ufpuPL5+YzrsS9w0LA8ou3z3LR+ucI4mcng7sSnRN WL2Qj1UHNPvWfevAX4t+qdrN4+PGP/M0feo7sEdrPCBmNh9FdpU1Chd34G5r hhJjgoelrFXzN0R34J5lqRISPTyUk+wP9dPpwEzPe9eHX/CQG+Bxft7Ndszj v9pwyZSHocZXzbW02vBTxQsjL3surmqt9dBZ34Z/d0fVqJL8V+q46LaefBuG qDwN6LHg4s9Lj7sNxNpwiUWWm84OLjq9qnA37m/FXVn29TFruLhT7m/M4cRW 9H+mITQ9MIyzJ31bA3+34I93Qau/knx6PdfFDl83Y0hwQlCd7RDW//OXHsxu xhXTvPnyFkO40OJ628K0ZlypVr/2nPEQ3h97tvdkZDMabF+5cExjCHN1hg3m 2zfjaKKDwKN5Q9hS6bbyiAjhHLV3qaWDuLLdk/p9pAmdOQIFSksHsfDfRUfV GY34JGyd5Io339FbNzHsyVgtiu7REjxPcuwX44S1Jf4fcXf4qhOh7mw8dshq wGZpJZYJpkhoXu5Cw2lJrFEowVsXf63+pNqC1QPB3KGwXFzEmK3t1fIRU85a nmeeSEDNG4WRAjPyMWDEUKrcJx5ub2c0ZUXlwuJWZ7xyOA98X7duSPergfhB t916lcXAc2tNM+prhhkP1X33Hq4E1aBnGzQ7u2Da9Gz2z8KP8Jjz/EJVIhuM j25U3vimFhwa1RWOvu2F1XWm7H0LGyFHW6hQ8dN38MiZP3x+eSMYVDVEfWr+ DqWRjT/jFRshPingyknmd7DebTOnQ68RTBsHsqLHvkPERxc1+xONcMFoXuMd 2QHgVly75PGqETJe/lgc6TEAr19XL4u1bgKdmfdzehYMgunD7Qeakpthcsu0 bfP2IUj4U2dUkd4M40XVnm93DwHzgK1ufm4zLAzcuTvl4BB4zvJaFV3ZDOnV Y2p7nYcg+Uw633SoGZ5QVvecIoZgWFM4skK3BW6I/1zlWDcENz/VVuV3tMB9 B8XnsQeGoUPe5s0DTgtQWkcrntsOg/zFvqxobgusm3hgWeU4DK9V/8a6CrRC b/KjkU7PYWDGq9mvXdsKQtsUPr2NGwY1u7ipB36toJJQYGXYOAyNo4c231rS Bq97JaasTLngGSETay3XBmsTvuTpW3Bh0QoWV3Z9G1xYs9JXxoYLh01Ppuds bwNv4apNFS5cYD7wlvzs2gZBc74pPb/GhSHz2B9CVW0wHHtmw6z3XJiZ+fl5 4Ll2iNT3mpWpxYN0/WgRw8B2iKI6RyO38sCoycJZ5Go7XOYV5Lvs5EHEdMfy pOR2uH98fqLIAR6IWX6PLKxuh6aZCk0DrjyQniF0ZkSmA95pprnJpPFA48hW xZM1HWBmf+buTkE+ZIjE5Xg3dYBVV1PX2zl8kCrqVw9ldMBE0uXadWJ8mFoU Aw9HOuDUJ8uCX1J8eF/POdwp9Q0EQw94blLlg7VhROTe09/giJ5SovEhPvgr N49oze2EvY2aHX2P+TDUruRntKgTPN2jvD2e8uFo+MXpgys6wef9lqSxHD5s 61kz97x6Jwye3Hq+p5APIin+q7NtO+HWeh+z0x/5kCyxwnpldifMYv8pmDfI h9L/XN4Km3eBmPyjfdmKNIRzfj7jW3eBzdqE37EqNOyrC0pqdegCK7Ub+edU aeA8SvJ57NcFzoJLbSS0aJizt0YNHnfBxdOWdv07abBKV0r3neiC1ewve3vt aVhxqyDu2CwGNMaUZGxxomEgAIJ3SjCgI0N5xo1TNFzcd8huoSIDquZVFou6 0/BoIlwq14IBf/UnVz4OpIG/vz+q7ykDbGxmTqyKo6Foi+eF2gIGpMH0/IEE GoIV/54uKGNA1syIJRl3aFg0tdg4uIUBwZfWzRROoUEv02iGrEA33BXcr3o0 kwahuEZaULQb6lyPNXY+o6Hu4jHm0OJumOyccNyfQ4P9AZ+3Reu6oVB56b5V L0g9ptN9LA91A11Xrn2whIb9Axudtjh2Q+45Jf3rpTQsay49IO/WDVpe75Vf l9GQ87RFbSSkG3y2KJb9rKSh6aDQUGRuN+QMl5QIfaFhZdYJu6pZTDBaZHz1 3DcaSp4H+f0QZ4LzjgW7xztpsHyRErNSmglSrt7SAQwaIoraKy+oMuGq8TWG I4uGkQ9mSpqHmPC+5FtXWy8NNz67bHN0IHzi8NjKfhoU6q/axJxlwhdehqTD dxpsWjGSe4UJxpS325dBGip7NEbSnzLh8+6oS9v4NBz9vl+k+SUTOmQ0m01p Gn4PucnPeMcE05mOamYjNKiMZloea2RC6tT+ic2jNLz/9d4tsosJLcF3XFaO 0WD3hx1e3McE1X/uDIGfNMQLLC9ePMGE+twfzdm/aFAV0m3aIcSCGEP1Qz6/ aagRth4+L8aCyFhVltZ/NEyJxcjWy7PggMS9qbRxGhIXPteaWs+C8/H5sSSw wcYln/epaLMgxL9WhUfYSVYoJNyMBebTfi5SkzT8XSV375UVC273lM57RDhJ weAV5zgLZhV7vlo9RYP62iN1EmdYoC4bcDyFcO06/34DbxbsyX0xX3yahpNq t/+5XmbBKhEB9CcsoPFS6l4EC2Q+mXp1Ek7Watj0KY4FNhG+Kpp/adisx939 330WZLLO9V4l/HWriNOaTBZcOymXRvI2nNmueOngCxZoF/jbL/hH5s1oR2Lw WxYU2jrLmxFONbHPy6tmQfVIR/9Fwjrml2q6G1jwQ/jj8wzCTfuS2aKdLIjd tMavhrDrwTcTur0s+DA9ZMghPPtQi6QLnwXhkwsW/CScZvtjfeI4uf+i22zy /AEr1IRXO86gIKrC6xV5/oDh/NtqAbMpaE7Lvk6eP6BIXUE/RpSC7B4Dh2HC V1+9NMmUoGC1xKYtrYQPaG23KltMwcq8a0vfEF5Z1ODQKk3BAePdv2MIc3Xt 3LkrKTCwC2l3JFz8lhsouIYCDf8tb9cTDt964Zq0CgUmUqFpfFKPgyiSsEmN gqPlLjeeEl61PemhiSYF4j+mfWwJ8yoVnx/XpQCVrZ2ECJfsLCz2NaAgbCDc KoPU39KkqSndhIK1Q43bmkj/VgkbWT7aQ8FVgfkP7Ahzq9+0PjhAwdiy4X09 pP9hO1I6ko9SEB4zJ6t1guh1poRtkgMFihkRB3cQXl4R0nX7JAXFahv2PyPz VGDgwozxoCDyYMAOOzJ/l6c7j0f7ULDwhIfeczKfu0v3sCMDyfU2qlz9SeaX o6vRGx5Ggf6v+0tcyXwv3Px38GISBfSrKtV/P2hgjnmcDUyhYJHKkk8ShLNe 9nD9HlFwY3NlkTTRk+HGT7RnDgVfVy++u5To7/y6hF8u5RQ4jN5yKiT61B+a 7XeymoKc2sj86wPEX58FjJ/4TOqtyztoTfT8UPH4pF0LBezXIo86iP6/yqnM sBqgICtH1SmY+MN96n7IAR4FMx43eEkxaXB5IC60f5QCvdpD3ZnETwRkf802 m6IgwDUY84j/qEmVz9suwYasFNvPJ5ppmGhTjzZYzAaJuaYF7xpp+HD7ibi+ NBtkahb/Ef9K9C4ZuVBbgQ0uDwfup9TSEC1mLbVBhw0NG9UsjlXTYFtXc1dl KxteVZWtD6yiQTFqi8xaQzY8WHrX5VYFDe9E5FasNmfDHEFhwcfvSP+EuKuX 2bPB6rDsV/dCGswmg9VmRbDJXmduufuY1OuwRJ/7TTZEjShXzkkjfl6YmtwR x4Zfixk251KJP517OysnhQ2LRE6Vy90l9er52XWggA3D1I4rMtHErz6finjI ZEP4TPQ54UtDqdLvrXN72dDqLe6x04v0PyxszGuQDW/OFEQs9yD7xyDNzvgn G0z+Rki9PE1D0MtOTb4IB9wGpr54HiX9Sd7D3rKZA1qWDdZm22jQ/K8r8Yke B2QPv89I1adhneUZc/FtHPidcEFhUIcGabFrbzi7ObDm44nnJzbRMB5cfvO6 PQd21opK0nI0FJ7W0GuP5EBY3qk19jOIX+oui/fkcKBN/eOTgmI+RMuPdF/5 zoHZ3cK3t5F9zhf9uPYmlwMbbEN/V+fzIZfpXZb5mwMKVMKikkw+bAj7OsiY 2wPNNz0Yxrf5sL7h2jZjjR6IXhd3b8KdDyrOv3kyYT3QLpimX72cDzf21uko X++BjO03ih2XknyhnR6qFd0DZQ9DG8YX8CFr7oFlFkk9oL9t6ouYMB+U85/v DM/pAVWjivWcER6snXS6N9LSA87bPCqUqniwJrbZpHpNL+w1ZmrqOPIgX+rD R4N1vTCxepmZ0xEe6D8s2lW0sRfuatj2X7PkgWVeys7sLb2Q/I1ZULiLByF1 p7bFWvSC1Iu5PdkqPGCKTGkfu9QLgj/zbIV+cOFOiILSr5ZeyCy95u3txwUR b59Za8L64GTh6UMNZ4chQcf8T9j1PnB6u9hq6YlhWDEtz+uN7oPaS/dfHiJ5 VT38a0v63T4oKJ5IrDAZhqNJ69IV8vpAc/MrU701w5BbyjZU6OwDs30doRWM IbCctTdo9cZ+WPIhc4XuriFITVSclGP0Q7TOK3NHwUEwbp4wFCI5fevA9gt1 x/uBkfw6dsx/EFzzjqaVyvXA1Yu31WveDcGBIr+t7++wgJUSOk9vahiebvfR rYnvAMXdNrwiKx60WXWYy0V/hSCJA6tkk0gfHZYXr7hfAQY1X7NLU/hwTjRX a4V1BUxd2ad9lOTAdI+Ms1ISFeBPm++9l8uHAErZYjq0HDzqjS8uq+bDplcN 5Uvay8A+amvH4lE+5LdMXkv0KoFtc5RjxMxpMHVLnU9lvIC/mCmTa0H2Wm32 Z9ayF1Dqp5i55xDJaSoXLTqj8kHn++qyKEcaOh4+/hrUlwsbP6zgzg0ge+Kx anfWxSxYFbbIRDiD6Ep6g0DsszRgbolvzsgme/DabcGFtx7C/bEFx3eRHGUq LvtbZjAVpE6I+14luSlWRmfryOVkkDCcmy5IfKei5umurVmxUP8nQu1xBw3O CQmfZg5FQ9QL4beGxMe4cf13g6/cgDlys5qCiU+K6vYddskKhg8docfkSG75 MHb0x5MrgRAWM3OwnOSQog2XZb2yzoOhSbC3PfH93jlP6n2yHOD/32/B/7/f 0v8fySjVmA== "]]}, Annotation[#, "Charting`Private`Tag$111928#1"]& ]}, Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImageSize->{640.66953125, Automatic}, Method->{"ScalingFunctions" -> None}, PlotRange->{{-1., 1.}, {-0.9999998782744886, 0.9999998592812047}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.05], Scaled[0.05]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{3.75552772539684*^9}, TextAlignment->Center,ExpressionUUID->"ec909fa3-b750-4edd-94a1-d1fd01b0c3e2"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"41a0421c-0c12-413b-b229-542e3537c546"], Cell[CellGroupData[{ Cell["Yksinkertaisia esimerkkej\[ADoubleDot]: Avaruusk\[ADoubleDot]yr\ \[ADoubleDot]", "Section", CellChangeTimes->{{3.755527760267074*^9, 3.755527767530978*^9}, { 3.7555285312927923`*^9, 3.755528537036922*^9}},ExpressionUUID->"1d019f27-d2a4-4e7e-8b18-\ 2055da7fec6d"], Cell[TextData[{ StyleBox["Helix: ", FontSize->36], StyleBox["r", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t) = a cos(t) ", FontSize->36], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a sin(t) ", FontSize->36], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b t", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["t \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0", FontSize->36, FontColor->GrayLevel[0]] }], "Text", CellChangeTimes->{{3.75552798669309*^9, 3.755528070823266*^9}, { 3.755528155314108*^9, 3.755528177738497*^9}}, FontSize->20, Background->GrayLevel[ 0.85],ExpressionUUID->"9d8567dc-350d-4104-aed5-e461f39f1bcf"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"0.2", " ", "t"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{ "PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"LabelStyle", "\[Rule]", "16"}]}], "]"}]], "Input", CellChangeTimes->{{3.7555281623774557`*^9, 3.755528377504236*^9}}, CellLabel-> "In[938]:=",ExpressionUUID->"10d1dc12-bed9-4d22-951c-498bc5fb4faa"], Cell[BoxData[ Graphics3DBox[{{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[2], Line3DBox[CompressedData[" 1:eJxdm3k0Vf/3/ykhKtKkqIiQ4a0BSbRFhqSMqWRuVGhQqKSkFCoZMkQZSpnK kJlsU8g8XENm1zXe4chFEvqez1q/v37+uNZ6/HHOfe3XPs/Hfq2zrrjDNdML yzg4OFTIj+Xk/86lf+QfAecjdBuGlIMOxat1XlYUmcaCml1OkyTvP7ZDdHPI NQiTrZHiEZ7GPt7KjFKScx/WHakIfgD+W4pofeunUVVlWbMryZV/bj5MfH4K Xnxf4nPWTuP9HbI13CRv1l3ivBAcCDfnY+xerpnGXuLT27tLBNTs8VxuUxgC FyeCtl3kn8ZIIS7DxkUCrvF9Mnb7HA6WXT49GrzTeKxEUF1kgYAIcTbTOigG DhdcOs3knMaBascHQn8JSD07VBVcGgfKyWc2fl9i43WxqdiJPwR4iHcZXSh4 DzJvjlGi/7KxWUTRR+M3Afwzhh0NKZ9A8K6iseEMG594GtGdJwmwLj670utF Ggwc+K2XMsbGrSuO6DqxCGgwSaU3laUDZdf4Cp9hNp6Xzgs4xiBg9VSOd0tx BlRv7i63pLLRLMi3L3WUgE8bC4djc75C+p9i4Oth44UF+eH5HgKOWo4XGSXk wsN83/2OjWxM8DYYriwl4FiKyqXtXghiBzZKS+ew0Y9/eEfbNwLuxgrkWKmX wPpdvMNLmWxsUjO8W5FPgPv4zYnI/BLg2Twf357GxrWVN311MwgISnHiHPpa CsRc7zbfJDbuYZn6d74jQJbuFxb/qRwwL2HjcDQbq2xtzFVvEaB5KJvPOaAK bFWVud/7sNGVRyJzNS9Z57tWpQOajfB2v/nwNhM2ar30uBn7jwWR56u+hnxv hPpr1R8zj7Nxc/PTQoHfLLjWqSq1dmcTLH5Sv6xrwMZk9n8Cj4dZcEpqxcrL SU1gJbyT7qxNros231lRwoKPefIZ/2U0g+ifmckiZTayWkus06+zwJvCSugt aoXIgvAFyy1sNHqlH/Uohwmmazdx5JV1wGuN7nXhQ1P4bMsfh929dLLfsjKu evTDS20HLQG3KfS6+NXkaj0dHr/M9uVK7wchrVUbem9OoZZY9WP/b3Sw0WXj /dF+CNfMGU2+NoWWa91/335Lh9H9vzV55QYgRoPvha7jFMa8km9eZkWHY5KK MZ0RA5C+P7PjvtUUBgTklC6jTMAxEX7GqRuD0Cq3zJmhNYVG7XvurU8fh5tu 3oc2bBgC4fXx4T8EplD/0IffUaKjMDJ5w35CfATiaYMMn6RfWDD27Pw6gUF4 6nNvZkyIDrURz0r+Dkxiqo/nNv2pAWh9Uik7J0mHjntn3r/uncRuQ6kCx7YB WFv3oGRMhQ5DNrK+il2TuHrLnPCtqAG4zn+1yNmSDouSdcfOtU7irHzxyiSp ASjQX6riiqWDYoZAR833Sdzo1Wdg79oHuxPLFC12MiD4RxgzMnkSN3lVvzzy sQuOzAtFrBBmguX8hy2qtyYxj98v/9LlLoi4kmokKsWES723F5uvT2IUn+tm Z9kuuDtXKymkxATXEt2Bq86TmKvy+tX2Lz9BfEf52BsjJgT4jiXEXJzELio3 n1xWJ+wbL5be/oQJxULye3jOTKKcgM6yLYXtcCz4gNfwOBN2yH3Va9eYxNPt nr8Nc1rgQHHg3pVvWDB+tvTWLZ5JnHRNfRFxpwW+JPqf13zPAoMV1Zf1uSbx 0hvvhWb1FrAc+XTvTCoLUr40WIlyTqL33jVmAuXNwIz+oLynmAVdy/L8ns0Q aBW/+XpBQxMkbY8fvj3AAtpx5i6ZHgJpvYU8ayrrQWqrzD/D7QRMD526cimJ QIat0T+tZZXgriX9bOkJAesFFejD2gRGPY+TVnvyHYbkvU6cCyCgzUGt9YUG gWUHfWVFeb+DTH/vhqRXBIRn6xUq7ycw1rc/JX51BZzoEg3peEOA6FmHgCey BMLRdeurNpXB5tSVxfiZAMmP4XI71xK4NdlkMNOkCJ5cPs/MqyfzVX2Z0/k+ Ft4tPCN8/WQhBBy3jZRrJuD3SwHzVZ0svHEwxlXoTAE8M2vmfEAhoGBQVD2r mYUWP2lufHZ5ELZe+m9VFwEaT/ev4vrOwrntlW1UpyxQsEo6SB0mQK/FKfV9 CgslD+6qF+VOApWdT/OVZwk4dbmDQfVgYUqWSlnGwEdYei1xoZnM0VfHxruX 3WShSeZtY/nCD+AwZSOvT+ZuzX9/a3ZcZSGnhaF4z7VYUFncwlNF5rTGzLYk B2sWGl4WzJAPjYZPjgWbGslcd+/cHfHIgoXyb69LPtSPAP7DKocySQ+kF2o9 jTdi4QqB5z1p9oFQotFWvZL0yc5HFy9QD7OwwuSyTd0PX5hmXpXzIrndBQ/z ZQdZGBdbIf32nCd8EKuMrSf5G31/7R1KLJxnWsRwXroIRySeycyRnCIXvVdL gYUGX93a46ZOIdfgw7JFkq8W+CLuIEXW7atsS428O/IPJbn0kVxvCgUfbWeh OPKrOv73GB8YSyu8IfnDtuZ/ccIs1LrYWHSy8jmeZukt7SZ5Qd4Qq3QtC/30 DH0F04KRZ/sTajzpvemomd5BPhb+G3Z4NzAfhvTBR10EuV6FBzz1nFws/Jbv uidMNwpDWwRGN5H8ksPmIvFFJkrliM83sOIwKHPk8J95ArpkNN7YE0z8V8gV WvM4GQ/w60pyzhBQkv3YYrCViQOcZ3YtT05F7wDjYSk2AfMRYTqc9UwUVHBI 3tL4BWNuWVVJ/yJAyTNRSbySiY5vnx5xeZ6BhU5dvwtIv33SqhOyz2Mia4Wj 2r62bJz863hSmkpAYOO6xoFoJi5feSVIZjQHn7YZX5bqJ/c9c2cxRxgTp3qk 7i7/k4uJx0585SC9xxW2/7NYIBPzc+sFL4gW4Ez+IQ69NnLfrc4G2Hkz8db2 yR9Z9sVo6SI8sqWS9PB4vN7ABSY22t00zr9ZjrxVRcyhWNJjy/eW9P/HxNYj 3K8cabXIESYxI3GQgCoV44y+EgYuGv9p2nK5Djc1KY7m7iP33TQlf1sBA7mE OO6k0uvwdbCJ5C55ApxcuMtsvjLwgtDKbJepehwPio57L0qA4Meilr4EBrqu 0qbZLDXigvV/b//8ZYHlhl3Tff4MdFf7eDWXpwUXcpS/7cliAZPNub/fnIEm O8+hwsp2VNRbFiXCz4L1GV+L+sfoWO0rXwXFvWjWNlAVcZIBdnKbqgcE6Tjf IH3i10wvcvZzpo/rMqCntXcqlI+OJ+yYbU8U+lBgzya17aoMOOX5YetRLjre sa0WPBXdh5GF95xVtzDAqH6Pa8bvCTRQCDwd6tGPmdOeY5KkDzWuGW571DeB jfvUuhNeD+C2hB5gW9NB+Kv3LYnUCXw2R+UTU6diLRYdaT02AU1qDLGLehNY VPlUJMZ0GMONktzte0ZB06D0Dv3+OLKEOB94dE/gajrrwM8Lg7DqaSDjd/Uo mjaszC9mTOBPuoDD2mODkGhYqRlRNoqcZu2HJhYnkL7+6jnV3YOgI7QYqlo0 it5mueOT2+n4zOacldn8ANx/e0XjTtooLp8xGylwoOPAibUbSl4MAPFV5+X8 61Fs5WsW2zdExw/hSv7C5/qhaeCv4qLDKOp/FbPRa2TgqcMb5my7u+G12mXX ZYsjGNLklTtpzMJ9/sInv3hSYDtLc37VnhFc1Fp1m2rJwkozleCbhylgzii5 VC87gppvP2vnnmfhGt6ShxLcFHg2odn2QnIE04la1koy59xljpobBrbC1Ihm +hrhEZyMoHqavmVh57IHvCWxLVDZr3lR8N8wts4Mt1+msvD++EcP+5ImcG7W bFlXP4whe+Z3u1oQaNv6X4b62A8oytJM3uI4jHFpMVy5VgQGmz7d0uj8A6Yy SzZ1OwyTdQhn9zgQmBowq3uUXQ3SGZpPoqyGET3lVzW6EOjeY1o0u1gFrz5r 2okaD2PWpjb/Y08IdGl1pWkJVcL5j5obt+0fRin5u/viUkmfWGvUcO8vg1UR mo/EVwyjV2X4Qa4MAve0RZgYFpaCZlgJMbhEQ86iz2uOZhMYwqU99RBK4Xao pnX8HA1f/45Y4fGN5E6Bp/L1SqD/laaqBIOGbTfDaq/UEWizZHdw9EsRfPXT ZEm20rD17d6SWBqB4RdtdOhjWWB5T9NSJo6G/30p/mA2SmAiRXE0susr8PB9 b7B7Q8MVIsVF1HECf2a3HN5blwlZEUe1I0NoyBopu/eaRaBInFm/ZFo6rMox lefzpeGyPQ6E4SyBzHO7Bi+MJQJOnl+iX6HhEoMpPfibwAtLm0Tiaj6C04Px m5Lnadgrault8ofA3mtLpWWpH0B4jcuolTUNtRt8X1X+JVC3gt5edj0Wbsi5 N9Ub0XDoxhPt2gXyvgJRfIYJ0bC14O8R7qM0zGgaWJu6SGCNSJrW1OUIqNF/ mH9Ii4aZy4YXOP8RmJNVF0jtDwTJi37xaUo0vCa3Rtid5JZfUOHTxafQNL16 05gCDYWLQo9Vk5y5VnHd+677cN8nOEBMmobLw7eGz5A8Ul0qLLXREWSFNnGc EaNh19jt+SWSWxivHVwnb4ztsVG3gjaTdZYJuU0juUaHSmb34i18pCg2/kOI hoXCIbwfSb7Xf/2143OPULH4g/WyVeR+VRR+1Sa597r6wvAbAdhjuKtFbQUN X/rcc/+2RJDzwst39jNB+Kz7s67r0hAuTl82X0dyymkv1aE9Yah8ZW9hyu8h VF+mfUyHrIPU/ZqJYy5vkDqXo0ibHMLbC97+GmQ9F+RSamxd4/DgxhLhk0ND qCFgc/MBOWc1Prxwt20iCZnGLRMq1UM4PFutmcMmkKfru/h13lR8029he610 CCW9SpV+/CJwddwdc/rOL6jn0t36qWAIQ+NkwnyYBApWN7LUd2dgbACtSPjz EMKz/U8MhggUUxLy73LMRtPK2ZfzQUNoqXi1VqKaQHbt+y/6nd8w96DIvuKz Q0g9zji+rYLAZhMFmefDxXi+JubTrPkQtnxwkJtBAssVr86mTiEKnpEUVTwx hErZy/lUcwlUWx7N0OQuRUc3hRWxmkPYN5FAFfpAPl8phr1y28tRJAM6vHcO Ydj5wjFxd3Jemzh3rN2wCr2lzt/TIajYmz7z1vgGgTddiif2/KjCjsPmQlNj VFyefS7K7iqBTaaHjrvpVKOC9ZGkd1QqNtXI+0raEmj9UcYtV+MHdoZIdvxu o+KLh4e+q+sQeDBQbuMahVpU5Bzem1xERf0b9ZFZ/AQ+/OjUFs/diL3dFybW +FPR+eFn1RlvFm6ePSu21qQVVYMunemUpGI0/y6P3A1MpD6V0jFL70Li75X9 Vy0HcfquTLYlDxNjyhQjw1hdWPTJti3QbBAVTfJieuYYuNNtclmhfDf6m5m7 ZhkOYn6gWolnNwM7pS7NJyR2o2SKRtrioUF8e39u5e4YBh7uzstaeNeDlpaC 0q8kBrHamnfjTTEGst11P1970off87I3ZjMGsC+H89II6T11fq+95SKDGHWb Y3bJawBP6rt/2Px4DF+ee5jOeEXD4D45x516/ch1J8S504yKDkV2AcYHJzB8 7TZe7/FunNMvubBejYpH/704cuj4BGazrxdO9XTjoJA05ZAYFSfKR74I2U4g pa3c5XxTN7oJ7XC3ZQyi27UQ02uPJlDwjWObXm43jhZNBrX4DKK+18O/M1UT 6L8jO37Nk27cc6B+T1jaAI5yDpjInqCj5z5DjWixbuT5vnPry9le/C0gLVWi y0C7k3dv5p7qQo3lzIsaLe2YzCeVdH+Yibsi23uY3zvRsGOlXEhwO/qVDwW+ +sXE/dF73xnldaLcLl2HRtN27MloC/Im5zydmJe2GcmdWJTEMlhoacNcj6sZ i+tZaJegS70V2IlmMWOTii0UNKqyYVlpsTA8I2f07+lO/C9QTf9QXQtut9Ox SA1jIVdN2NRKRge2ckcoTtvX461H0aGHdhLYP2/BKyXUgftVYmRO99ahm2xb 3KI8ec5ZzPzxlKsDVQ19bkedrkNWguqFSCUC5zkEAsZn29FxFdf5shO1SDuo zLhInn828FSt/tzdjmtexfHHH/iBzMn8dWfI/jVct3+dUkI7hixuzJDn+450 haNCIa8IzJfbtE1LtR1TUrRuKijn49i3F9a7OwkMterYZ2PThtdeKxE7VfNQ 7/nP9MPdBE578KklmrRhEucGf+6Duag+IfpmXx+B5q81NKeOtKHYvBTHY81s 1Ajgl8gn82B9w3tDX7k29Cz7BQEGGbi6q7PCgMyPELh+4cscBUM81yz1G6Yj rfX0fTuCwKmz76/O0Sn4+9st06PvvmD6uj+5FmQOmXq039Dqp6Dpf0dtl39K xucuCym0aQKFMtS92r9TMKlgR9Gx3HisSHlm5ztP+nzHyvClYAo2neKolS2M RYewlPkYMhenDqm/1felYGaMedbU/bdY2NzF94b0jOnZa++D71AwUnox6ufz 11j6O95PkcxdodC2NCk7CoZ37laTCw3CLq7U2RqS30znzbluRiGf90srOlQC UCAprl2XzPWWuoNFBboUDAoxKlHX8cGMopW73v/PA+MuZVxqFEyT/DW97ZMb eij7MftJHrQivvqEAgVnd781yFtzFvulaBLkeQcnxdsaIsQoSLQGzAaDA9iV LdbOk9z4EG8bdR0Fv7t2rFRdfgeen+oc7iR5uuXBbnkeCqYH8//WdX0Mq81f 3wgnuaC7y6DbfCu2BYcU955/Dj7at6+pkPx6SNxoCbMVLU6v15aXCwadxo7+ bHJdTWkUJt9gK+5jG9k5i76Bo+xDWyxIz7waU5t/V9WKptcKjoyvfwcDLu+G 3ci6TXK5cIwXtCJjZQi+544DEUMVKQ+yzsbicdz7vrRiiNqLj/VLH2DbVhF+ UdL/gpY866pCWzEuJ/bsZlYqaK3v8h2fIq9f2ypH2Ldi8pisgvO5HHBZ/O97 9ACBgV+crGGhBdm3/CRi7XLhiwXl7fdeAomamHN+RAuWHbdZVWSdB7u0Z1c2 dhFoNNri2EptwRJz8cbkUwVQuuSV6EMhcI3YAbdLP1pw0M2lRd+gGF6IrT99 q5LAF0ErAgPDWlBlbuO99vXlYH5CJ9ON9MnzW+9K+ne3oN9JpcCF6HK4Zqtv rhVDYHdGF2WzRAtuOdPP5N9ZAbZ61mfGIwmUIzaOm21owSa59vU0pe8QJ52S 8OMlOac4Bq6t/tOMFhu2tV02qwLRvJilVjfSt7YPHdLLmlHZWHVALaAW6qiV 8xs0SW8cdeDyNm9GpTWHRfonm0HMboHDP5eFt7ZJ6e/waMLuV3mTZ7m6oCoh MyRInYl5r87G/PCqxwNP/+iv0+qCzw/FZMrJ81gBc06++Fw97qzjN/v8oAsK xupkKWIkNwgryNSvxycvKxyfzXfB+msH8kO5mPiNq7ktSqgeP0hkSrayuiH5 fXZOUA0DSzx0V7l8rEMtebB80NwL+RfYz/2MGVhjt+eOUEMtvmzQVk8rHgB5 hZbsiSN07NrNY3Z2aw0m/Y2LCdhNg6LDexuzWsdwoSmTm1lYie7bzS7Hxo9D UshfFUnpITxTEPZzr3MJji/+mOPNHYddbsrdWfxD2BswvarTpAR37O2TN68d h5npGyWKk6RnrE0176uUoLywiUQwexxyLApraXlUpHCIngj8h5i8Zag6SWsC XAyunDU6SsXfz/nG8yQRLdHXQal7ApKjtlprXRxEtYTRbauufcNH5dNHxWfo EK6uo7JSoh9L2uL8M5cXoI7RLUWJaiYo5KmZ1Dp2YKPKRod/8l8xwNBEo6yV CQlzc1vN/utAMfivZ+JPJi7rE4/S7meCzqhkZ/lUO97U07Vor8zEc78ZwswZ Jvy4tMBpf68dN5x2M/hsl4ni+s1DP8VZMGi8/vOrgDa0vNO+92xoBk4szz2+ 9hYLbOzjv/751IrV7go8Lr/TcDbKxSSTk4CkHVue7E1oxCnv/wx6Q5MwxXbx +tOVBLR7u9ld3NWIocE2h7L+S8JjDgqpGmsJeH+6cP+9zw24//3LvQHVibiK 2W1xXIyAsOUeVUez69GzgrXlwMInlCoV+TKnTkDozajJp+W1yMOTRg899xGt aupO3LlOgF51CHO6qwpFniu+MNz7HmMvvFo1hwS0drjXz/xC9Je3j3WPicQo b++0pnLy/N/9i/vOn2Kk2Y1lyHRE4PVEJ8K/igAbg9l3fRzFqPH6WvnPNRHY VMGn+qGBgLy9/+LMBYpwcvH+yEGvMIzSzQ107CZgk4/v4gq5PDzZ+EaewzoE ReyWik70ETD2dGFfzp5cTFu+41BGaDB+zxx13jBIwJ8oCzVj1RzkVU0ycqgL wvmSbcZ6IwR8eyVyQ0cnCwtjc29WHHyF+cOTLeoEAUfLx9+fsElHLVrijbXK /rhho/FPhV8EuPHW0xuqv+CMYW6o64dnqPVX24SDTYBEaJ/oECUVE7O/57at e4ocSmsCj84SYKtqGlPASMQ1T6mLkVM+SKtGk5LfBCxmeXz2nvuIJcSk2IL9 I9SKKHog/IeAw9fVy/ZyJaDr6X/aNs0PUUDulbDzXwJe7xK+aykSh1lmqut1 7O5h2n55b8cFAmqvHshOOv4ON+88abkSPFBv3TtjvUUCZoVk7niff4MPZm/E 1m+9jaMv1vpyLxHQY5q9CPfCcLjq5UjQwg0s/tYo9onkn6wyOcaDgvH13lNn UnOcsfLfIbFd/wi4ObbyIYfJC7RrezWsdOsS2sxRfZ6TPD5Xjy3k9QTlPWpu fNtjj4UhVseaSf56yfCLuMU9nFTmXQjoO4m63FfcfpP8Dv/KGG67S7j696yM 2lct7Kkb4frf+yztR1X0PRlGIPjbL+Kl5S6gp9bOjpL89kcjDsNoV3g9dfC2 t50ReGf9Z5hJcjqaPVSM84asr7LxispWsFtNe4UNyetltkrI3PKDNbndRwSs zwO158wOOrku3ReqKo/fvyJ9Ep/nG3MFnvkMvD9DcpfZXddpz0OhUFVp47/r 10Gw72l7F1lPoQcH5S/aRsPk63k/t69uUFa+0Y9J1j9TyXqzpmQs+Ff0amdS 7kCqwFT84DwBG0vjU74Jx4Mku2SROeMJ4gdSBV3mCBAZUFrmw/kR5BPL8/ZG e0M275Xfm6YI6OWV3bas5TPYrfCXT+57CiWXa7hVJgm4ITUN/d/T4HP0vZX6 1/ygIDD1mTKLAONjdksNPekwv895ZPifP8S4FtE7x0ne9e6SVVMmvHYwjhUX fwk/NbquZvQT0LBqsSY8JweGbfn96/YEg9GqE29SvhPwTunXqhKvYtj8rXlR +1YEKL0ZNnhZSsBz5/7t9s8Rau7k8Qcfj4ThxT61498IMB3hPkUYlsBdlZjN A1JvYPi3HsM2i4Bh4bOgbFkKPWlOyp4/o0BAnS35M5aAq/byI82u5fA2nsc5 61AMLKc7/NruStZNf06nJr4KBCu2uQRzvYcDH+x1FRdZ8Oz6ROkbeiOEJpn3 LmtMgqFrGZvPJDHBmPNB2U6uTjihNtD7K+Mr9PAfzrj4lgk8Rpa3Og51wt7H i8c/0L4Cd14CnApigmZBvt2NO52woWFLscWmLDi1YHF90IMJYTwGy61ZndBj f/JdoWcWtF676hqlS+atSKX86o6fcNmvxuaJXjb4+z5npvcw4FKfgEHM+254 1JnVL9ybAy13JduK2XRIb/hXzCnTD7lufoPAWwCW36b+JTaNQ5kjPW1ijAqv zkRO6SqWwEu9qci/A0NgJUh8CVs3DvrbMn+9WF0FkZ33BZ9UDoFmu7T9XZlx mHvnl+grUwW8jn4hcylDkNxgLKWlQXpvu73tA+0qWFW4Wfmd2xA0HTjx4v7F ceATF6y/fqcKDDKVd47yDcHcoUqjONKHdRIuiWbDVaBq3fTPT5EKfYP+on7m E3Bil6zt5m/VcPzgM58JxwEYefNt8xEPOpgrxdd9cKoBy/R9HNLmP0FM40Cx 1Dkm2BoEfyqsrQfqqjvjp7f8BK+Rrkf+15hwJaKQMkSrh68EPfTmQCcMhjzr ar7HBPcRGueqpXoQPiQsZOPUCblB2ev4Q5nw4tF+K6vdDXCdo6JaxKcDph1F LkuVM+FbQY/gYmgDhGqd8z+X2gYMPqGaWREWbJWTvnvIphGKdipWfCXIeSab nZWdywJihc1HO48maNzjTj/2tRkutLsvuJWwYCZ1qmj6cROo8E3Nt7g1w3Hz lKAtP1jw1+xp67OgJmAoSRd9WWyCM6f7fHi7WMATn86RkdwEmPbhv/d8TcAT SHzwmmfB9kPLz3L2NoG728z9i9p1UP1HSbJemYATbokC8YeboT5lzFf0eQV5 vy7r8tcEfB6Z8hjia4FP0yKezisq4EdKD0U2ioByJy/vp8ItMJ17O+uTVzkI zFEvupDPQdfUSn85qRYQ7f5l2nW9DJpjDlb4JRPA808syvVwC4zXF/AdNS+B 4t8VlYLkc+awyejbco8WcH+nHZGgmAf9Lh98z/7Pb/opnJLDLaBb2//j3/Zc eLPy14FHpN8UGlT4qqda4Ibu4TwQzIFvXlWFj0m/aZuXCTlxtIL8Rubb25Nf wWTo5dato2Su2P2UyBIh+RJ3co5tGqDHr+VRZN7UefDoHjFtBU/YeJkPPgOv RAclhswn6lLwiTHbVri/Q4tQ35YCHGdcrB9OEzD3eNup586tsPfPzHL9no+Q mG72gkJ6bGeQ8mXKs1ZIXLFq1vVLNIj6u6j8IHP0YdI5//PYCrQj/10JvBoB QR+VglNIL4UrEsG89a1QneS3I1AmFDqGxSIdyTz+nH03KrWrFW5rHGMQNYHw 0uOswXIy1ysOcn8wHmuFdFE9j7dHnoJwq3KCB8m7SoNSp2daISZw+D97z/tg /UU26QfJJ/W2Zkcsp4Cqnd1gbd9luMDbd3KW5DwNid/U11JgRard1xdupoiS W1OWSL7VXKlyYBsF8r85Pv+UfBsLn/clUkmu1IUNj+UpsMrK7tmPsz54W0DL KJ7kBnbHOmTUKMD7zvLELmoAZpS+eqfxv/dZI+39dXoUaLSgZCkpBePTRfXw LHJd7k4OY9dPUsDq+58Nmu5hOGhPPcBD8pdTzMn15yiQG+rH3lPwBuubdZ4q k/VJ8LjzJ+86BXLynrutWHyHb1xPPgSynkVLXMusvSjw4OQg7/bS99ilyHF3 hJwXxvhE1yVEUiC+53yxITUB63p07vuQ/vn36pPI0U8U+Fl3L9J+eSKG9T7f s0DOIxs27ZNkZlHgy+Wz6/frpKJm3IGsh+T8oi1hoKzcRNZh8/K8j+cycXV3 vVX7BAFvD3qc+s7dBtuDmm48GSzA03uWReS0kH5wokQNu7aB2PRid0FfIfpq R4obNZJ9+/OZbdzDNjgTcz2D0l2E1mandlfXEtCtoyFh/aINqH2WWRltxTjW Z00/W0HA+e0fkykf22Bd+uq++XOlKNv6R5OP9MztFvf88p9tsOGbwMdh0++4 7migiO8LAiJURTrioB0oN8p4JrfXY3gGz41tu0ivcjkI2fB3QLuY0M0FhXbk /Nt1/iQXCxzfEs/a4zvhT/rYs/uX2zGp+bGP2x8mLESs3WKa1gkiR2SMx+Pb sanPXeweiwmBoftS6gs74eju/LUPN3XgriCRf/91MiE7wL2+gtIJ0Wefmn74 14FuP6/f1UpmwrK7/9Z+5f4JWgmc0eG1P/G4scWWHUeZEG0hEBV45Se8C9W5 m2rciy4/2hY33mRAvYDiZ/295Dkt+7bvIicVfR9otlS4TsDuRy7NBaXdMMCs 1NYyHMMlHmHr6hYabJdLCI+S7Ic3LQ2+yh8Z2DsieaTtUQ8g8oVmSw3CjHqL yKVsBv5I+6txQ6cHJgWk5QMVBoFr5FO9ewUDg8NHKAyeHthhp11xWWkQbB1H tVSoDJR8wjHt87wbfDg8p0W0BqG/sjZypygTD5+7nX0gtAv0DzPNH1kPgvds zkf0Y+Jj74MpnO86oamscYNR6CAs7lUNdDvGQj+faufFaApQv4eGj3FSIT9E VsnZgoVxKXRFYSsKrNuYoVjGSwXuG8rCh+1ZOLHqWIyICAW0L9ZXRQlQIZ53 u9NDNxbKBsel9Ua0wkdu7rkTW6nQJlH562wMeX2ZsFKLkBa4oud+OluVCpkl 6508J1iomNtpZkP6gF19ZvOja1Q42c4vIXKRwLuVuZbJGdWwvG5blEgvFcbW /51RcSJQn/JWXFipGgICGwW9h6hgNzd5Yd9NAqv6/Aqdc6pgndlD35FxKtw/ vXNFw30CA0/H2bQUVIJE1+CNjFkqCJ79F+sXQuCS4dYdgWUVoD32QV9PcAii z/+XtyGfQJOauNEjDSXgwyU3e1N7CMpDG1tUvxHYPJU7fMWoBPiqu692Hh2C O7YvDqiXEnhqj81ezjiE4IDngxrGQ5D36nb3UDWBipuqVeY/fIN4IVYdr/UQ yC368ua0Eyh14VbPq5R8KBPLeP+/+aD7+lH5cwwCN3oo8ClmZcAyjf0mNUlD EL95rbMfi0CljWI/xrLSwZ9jrFIxfQhOaH+tDpwkkH5cOl3iahoIVUSov84h +URgtfo0gS5X71RcdU6BHcfmpe3KhyBHZjB/0zyB8+KrxMuvvwetM9+WZnqG YOQhJebQXwL9jo1dO3cjDmpEXW6dHRqC4aHJEJ0Fsm5WXAm0oXdgOrBtomR8 CJKEk+PZiwQ6lrdEVw2HgcOlh20Bs0OgYBlZ+GaJwLac3bWhI8EwIbvn2K+F IUh5atEv/o/ALy/X7nnX8AJusgZLLJbTQDLBcrUfyV97jK6YaXwC8xnBKkUr aUCPP6TTRHI71vkdTvH34NFt7VRxQRoI2BT6kucO7L1go5Fw9BLwHZgWf7qR Bm9jk5r/977G9EjmZoNzJhi88CGcIUqDOvFlUiMkDxdYNHO1uIVbSk6uNpWg AZdvr89nkp9P9B/7t+sRxvtw++TuokGJzS26Ock/m7ksy5LxR1m93DnR3TSw kt1u3UOuSzbp0+LHdUGYyXfZ5ZEKDar2ZNoFkHW4efN13UqhSCwL+nHmuDYN Jrjp7EyybuJ5K99vWvsWDU7ebcw8SoMjHD9DCsg6GxneUeV6FostwnI6wsY0 SPpAWXT8Q/ZD9GavUt8PSI15vnvIigYDBnpvj7IJDBlZ4lfxSUXOL0Y8d9xo oPeYqnvvF9m3UjKxqz2+4KRq7rliTxrp7wc6LwlyH1M7/pxTSceB8m0ly31o 0BildfQ0ncCrXZwpLNlMxJ9Mj5eBNFhLYSZ4DpLXZ7tqyK3LQc8VzyfeJ9LA lMeJwUv2bT61Uf5a/zf8bVVd39BFg/TNqW56XgRO8b0uOZZeiQz+w6GSMAz6 RwJeJnkQWDh38HWaXBV2hyVOOuoMk7l+9RqTfB733+eWW/xYhTXigsfTjg3D Z2/j+C2XCBRtDSsxe1uNifv7V6idHga9+r/O/CcI3C7l9nHwWQ1ePHff3ejG MMxq/lthuJFAaoXN4c1nG3CgIN/y7vthUBHYdOnuKxbGdD/e7jDdgu2Oe8Wb eEbgveLppxa+LPSWsL5npNyKx8+/dnu7ZgTCTz6XF/RkIXrGSUq7teJ3m7na KxtGoPGFvZLoJRZKPg1QC/3ditlmxW4rJEZATMDgWcRBFkYn7nG3/kPBUA2D OrVDI8D30lBJoY+JsTYbPzjNtaPxWgf3hFsjcEnmo5AUHxP5505seErtwpq8 oPq7gyMwkMGVU8dBxwNBTFqa3CAiz687OwtHoWa40fDi7DB+qHnEN68zinb3 6vbfNBmHI1M7755b7MV0i/hbk0cYuKnZR+nbSjpUaeowPpX2opRQTPtXcwa2 3VbQ8xKkQ4e3llrtk160HZ5MPnWegaFbOs7AJjq8KU6+lba6Fz1X8V3Q9GGg 4HnZB6WSdDjVSOTMiPTg8TPcmqHIQN7fTT++Ax3i/lr5ZO3tQv4Gy9X79zJx XnSbbcNtOqxVNfvSZdCO3yP26y0sMbH3cp5f/yAd/LYaX/Dib8ez+z3HvvCR dVrtEB07Rgf/5dSXK+vacM7k6HrNjSy0zORPsyfo0GMRbN5h2IZXTm7pXJRn YcdfG8rQAh2k30ZfPX2cgvmbhDKGT7Ow8eWK7WObGGDxxk9G0aAFKV/spK2S WViSZZo1eZwBKv/SHx2/UodSyWcHw/cR+IGD2cdZwIDKVOs5e6IWV3zmEnqq RmCesFzjiRIGWG1UOXv2Vi2WiWcvnTpMYP1uR4yqZMA74Rvfl+7XYP6x3l/v yL6bsxuOUW5lgKxXrfqGV9XoduXlixKyT4+X9to5MhggE2oUMJ5WgfvvHHA+ HkrgjHcDtWkbEx51ZzEe76vA46o8vicjCeR7s6p1604mVLivK1iVW46Zd+aX Dr0jUCzToPyKHBOoEifvDX0rw5M3zy9lfyLQgFr5nkuVCby9vA0FNSWY2GKQ vY/0z1stPK9qwoRpT5X9M2X5GMK37/YV0ht6y9NHYnyYcHy3ULdsdR7+I2TD fH8SaCXCamf4kee2yKt39etzUWqj0E+fHgKv75OvOvCKCev0jnhARzZSnJXz xakERp9L/NQazYSHj2dX75/IwEkUiJwgc2KyPPYyTw5539kHslWsdByuJLwE SP+s6Ok7bV7EhL0PaYUvRtIQdDtvbiD9s2Va9GhcGRPaq5Uz9syloEs57/0K MreOSEbuOtjIhOGDCcHaWz6gT6gD/QuZcxGPgyZcxphAHOEODt4Rj9bmimt7 SC99jm7sKiTnNaOOZUuNsrEYwfNWdYLMy4qs1bW8M0yo//nkzkG3COzgdXiW ROYuk+aXEs/JAmenoyXnfEMxq4UnzZLM6WWLVVEEDwu2VWfv8Ah7hcf86K0M km/awP1cfQ0LOl8fHpoz9MOXfg9mbcm8l1c44um3ngVTDf1/x5y8kaVhvjGL 5Jo6j5zat7DgZ7WJw6vNrlixTXjfOMlPWpdYSYiz4FXLojH1hwHGZYYb/iP5 ldtLhtelWRAh9W2zSsNlePTymwN5TsGHL9Q1vimwwHX6s81FaU/oPxl8u+F/ 3ku4q8CnxAKme9q+WvYTSClQf/KU5Mnf8raeUmPB9KGXv+pbX4D0fc7g//mz pG129QdNFsw+MvEr+BEGYg4+H+bIOkyscGVpHGeBQlam8p01UXC6kNYkQfKn L1I7hMxZUFqYs3qzWQzsvSrrL0t6aeeGkZJRSxboyzpfnGa+B3vPhU815L7Y SZ4JCbrMgmX2z/bF3k6F2tNcBSfJfWzTAtVhHxaI6r6V1gz/AkdffbltTfrn Vo2HeIE/CzgO7Im1sUoH3tZPtZqkfwRNM/kCg1hwR/h3aPz9TDBQMDF+M0H2 uf3OXtUYFmx+pzUrgTlwxGQ9X2Q/ua4H/I+eF7BgR/1Rf93GXFgKTaLdJ/vW mkfnqn0pC8ouCCwe78+D+miLG4fIPl946WWuUs2C1nSRT9z/CsB2lJVm2kJ6 5t0vqYE2cr+sHvy3qI5ACxnaV11GYGpRR+2+Xyx49Lxef66zHKaIeCd6NDkX /Hm/oVuagOG3ivxjthUwjVsfvAgn5yzv3qW0/8hz2C6jsJKRCthRpRHMH0xg C++mscfKBNSr6sfumv4OJ2PSW6N8CVwt7F/wnzYBjzvKCmtWV0OrusL91S4E PlK5bvvIhgDbGhPV8oN10DAzXXxQhUBnV/WkXa8J+E9zuHwvtMAGh5u/l1JY GO92ZZs65yQAV1LpMc8WcOL+LNMZz0LRW83eTlyTMOL06pxpfgusLGliBEey MPyG6nA0zyQ8ueoSJ72vFV4+/G76mfRegBN36uLqSTi8f+EyfScF3ArevHOz ZaGrQ/wB3DIJTg7dPu952kGtX4F6jp+FR050mWspTcKROOcNygU/QUhELmza kImjOw0C9C9NQsR9I7fgSwMgQxvxTr5DR7m2XXPm9ZOgpWQyRwkcAanTmY5n M0YwTqws8crrXxDNDpj/mkuHo6kJW8v/9qOYdbXHA5cpmFygvHGvooPIFipn f0U/Fqe1FCTdmIJwQcE1Yh106M5rb+5+0Y9nl/UutN6agjXme3Zvn6WDrqnh qjvb+zHs06+HMvem4MmLmiLKXgaMcn9t0zvch/xTW/yan07B2JFBUcFPDEi6 +dfp4oMe/O3rHCkRNwV/07UKLt4l8y/ehVNutBMb04UKqlqnwIi6c1OSLxNu 3FVqaEjtRKflWxd+tU8B/dyDoIZgJnxZMHO3vNmJfBbSh0S7pkC3fldaGXk+ XLfmtrbsYgce+Xuw7PrAFFwdf6d/4icTtpdI8mWu7cACnQs1m5lToEZESozv ZoFfuERLqEobJnTldl3lZkPHpnvdtB8ssC/P/AQ3mtGTy/qvwAE2rC5XlXFt ZcGeE4l8qgrNePKX0J0IdTZUG+JAdw8LjtuL3BcZb0KFvupZMU02yOm4XlIn WND0Iavng10T9ucqTe3RY0OOTsRrVSECmN1O2q/fNqDm1VXjZifZoPtP2Hq9 CQHfT2xqsjCqwRUthe3hN9hwXz3xypHTBPA3Ur7/rviBfcU3zMVus6Hk/kat E7YE6C9dMbmj9gNzU6RbEj3Y8PmikzzhRADDK7pebGc1Xn4SUp//gA2XLh3g u+BLQOFWj651f75jjerViu4XbOCuz7HekknApL1o/rboUgyM2ZK5PZkNArso jkM5BHQuy/YUFizFy8+bFBM/s2Hbave6J4UEaHMYui76lKDmHd/PuzPY4HVz yMuigoDa7V5HN3UU4y/TqcTDeWxwkSpfIU0h4F/JQGS4awGac9fEnKtkw/sb insLOghw3fI795xlPsqzH2xl/GADo1X4hVQ3AZJarUzBw3m4YkA56lY9GyJ7 Br75DxKweLRpYd2aHMzJjwt7QmGD76718Q8ZBHCf5t9H/ZiBws53XnyiskFs D6Ojh0V+/1VXI3uep+OvM4r8u0fYcH7fajGhXwRYhvslPBn+gjW6w8/yxtlg KPO6h2eGgIBeBfV0z2S8J2by+MckG8INZAyrZwm4nH6oZJ1+Ipqv5llmOs0G rLOqsJ8joH/7/eQz6z6i/HzRg67fbNCvWFW1/S8Bfj8uuoYnxWFvq8w9+hIb 9jnvPnFkgcyVuwkXd42/w5ySvjnXZdMgNajTcXiRAA7bhv1LXm8w8HOo28KK abi4StheeIkA6i/eosz1YXjpjcH045XT8PhC3EQVyUM1NU4aJgeh5lOOm6tX T0Mwi9/V+H/va06P1cQ+8EfhWznEa8Fp2K2S8SeD5GppcW+fJXvjpJ2T87b1 02B+eosXg+S79GQaspNv4I/jO+gfN03D//e7p+L/97sn+D8MPiEa "]]}, Annotation[#, "Charting`Private`Tag$117684#1"]& ]}, Axes->True, DisplayFunction->Identity, FaceGridsStyle->Automatic, ImageSize->{722.9978668393392, 929.}, LabelStyle->16, Method->{}, PlotLabel->FormBox[ "\"a=1, b=1/5, t \[Element] [-2\[Pi],2\[Pi]]\"", TraditionalForm], PlotRange->{{-0.999999512844876, 0.9999999999999671}, {-0.9999996658276197, 0.9999993650500513}, {-1.2566370101446087`, 1.2566370101446087`}}, PlotRangePadding->{ Scaled[0.02], Scaled[0.02], Scaled[0.02]}, Ticks->{Automatic, Automatic, Automatic}, ViewPoint->{1.2890308397716308`, -2.5536044577951023`, 1.8076237902967052`}, ViewVertical->{0.02274983435442942, -0.08538715610997255, 0.9975245361862535}]], "Output", CellChangeTimes->{{3.755528208271995*^9, 3.755528239764348*^9}, 3.755528301193284*^9, {3.755528335606996*^9, 3.755528342727436*^9}, { 3.755528373002487*^9, 3.75552837874992*^9}}, TextAlignment->Center,ExpressionUUID->"86ee88a1-e2b8-4480-92a5-c89811fc8116"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"1fe59a13-79a6-4a12-93d8-84994256f8e4"], Cell[CellGroupData[{ Cell["Yksinkertaisia esimerkkej\[ADoubleDot]: Avaruusk\[ADoubleDot]yr\ \[ADoubleDot]", "Section", CellChangeTimes->{{3.755527760267074*^9, 3.755527767530978*^9}, { 3.7555285312927923`*^9, 3.755528537036922*^9}},ExpressionUUID->"8ebf104b-89bb-41e9-a719-\ b75e05684c0f"], Cell[TextData[{ StyleBox["Helix: ", FontSize->36], StyleBox["r", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t) = a cos(t) ", FontSize->36], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a sin(t) ", FontSize->36], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b t", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["t \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0", FontSize->36, FontColor->GrayLevel[0]] }], "Text", CellChangeTimes->{{3.75552798669309*^9, 3.755528070823266*^9}, { 3.755528155314108*^9, 3.755528177738497*^9}}, FontSize->20, Background->GrayLevel[ 0.85],ExpressionUUID->"d383d4da-46f2-4c06-8c14-b99dbda7ff37"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"b", " ", "t"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "10", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "0.1", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7555281623774557`*^9, 3.755528377504236*^9}, { 3.75552857419386*^9, 3.75552866912462*^9}, {3.755528745393684*^9, 3.755528765651289*^9}, {3.75552880247812*^9, 3.755528813006901*^9}}, CellLabel-> "In[941]:=",ExpressionUUID->"e4166f5c-bb8a-4ec4-ace9-594c357c7502"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 8, $CellContext`b$$ = 0.275, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], 1, 10, 1}, { Hold[$CellContext`b$$], 0.1, 1}}, Typeset`size$$ = {518., {179., 187.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$121185$$ = 0, $CellContext`b$121186$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 0.1}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$121185$$, 0], Hold[$CellContext`b$$, $CellContext`b$121186$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot3D[{$CellContext`a$$ Cos[$CellContext`t], $CellContext`a$$ Sin[$CellContext`t], $CellContext`b$$ $CellContext`t}, \ {$CellContext`t, (-2) Pi, 2 Pi}], "Specifications" :> {{$CellContext`a$$, 1, 10, 1}, {$CellContext`b$$, 0.1, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{566., {246., 255.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.755528803897852*^9, 3.755528818520871*^9}, { 3.7555290459036913`*^9, 3.755529047746291*^9}}, TextAlignment->Center,ExpressionUUID->"4c626f5a-41ac-4079-ab8d-6d25274ae7b2"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"a243125f-d587-46b8-b229-6718a95221ac"], Cell[CellGroupData[{ Cell["Avaruusk\[ADoubleDot]yr\[ADoubleDot]n tangenttivektori", "Section", CellChangeTimes->{{3.755527760267074*^9, 3.755527767530978*^9}, { 3.7555285312927923`*^9, 3.755528537036922*^9}, {3.755581526320593*^9, 3.7555815389356327`*^9}},ExpressionUUID->"377b3bca-ec2d-411e-9f03-\ 71291b0eb2a2"], Cell[TextData[{ StyleBox["Helix: \n", FontSize->36], StyleBox["r", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t) = a cos(t) ", FontSize->36], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a sin(t) ", FontSize->36], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b t", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["t \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0\n", FontSize->36, FontColor->GrayLevel[0]], StyleBox["r\[CloseCurlyQuote]", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(c) = -a sin(c) ", FontSize->36, FontColor->GrayLevel[0]], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a cos(c) ", FontSize->36, FontColor->GrayLevel[0]], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["c \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0", FontSize->36, FontColor->GrayLevel[0]] }], "Text", CellChangeTimes->{{3.75552798669309*^9, 3.755528070823266*^9}, { 3.755528155314108*^9, 3.755528177738497*^9}, {3.755581544081231*^9, 3.7555815798836317`*^9}, {3.7555823063591547`*^9, 3.75558231521772*^9}}, FontSize->20, Background->GrayLevel[ 0.85],ExpressionUUID->"f50e3d37-0ddf-4aa9-862d-ca1b903a56e2"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"ParametricPlot3D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"b", " ", "t"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Thick", ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "c", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "c", "]"}]}], ",", RowBox[{"b", " ", "c"}]}], "}"}], ",", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "c", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "c", "]"}]}], ",", RowBox[{"b", " ", "c"}]}], "}"}], "+", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "a"}], " ", RowBox[{"Sin", "[", "c", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Cos", "[", "c", "]"}]}], ",", "b"}], "}"}]}]}], "}"}], "]"}]}], "}"}], "]"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}]}], "\[IndentingNewLine]", "]"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "10", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "0.1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"c", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7555281623774557`*^9, 3.755528377504236*^9}, { 3.75552857419386*^9, 3.75552866912462*^9}, {3.755528745393684*^9, 3.755528765651289*^9}, {3.75552880247812*^9, 3.755528813006901*^9}, { 3.755581627616954*^9, 3.755581682398143*^9}, {3.755581752968676*^9, 3.7555818895661*^9}, {3.755581929951229*^9, 3.755581931870739*^9}, { 3.755582038210887*^9, 3.755582038687866*^9}, {3.755582084392789*^9, 3.7555821327734947`*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"79406004-50d9-4a40-84b2-b9f9efef5658"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 0.30200000000000005`, $CellContext`c$$ = -2.764601535159018, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], 1, 10, 1}, { Hold[$CellContext`b$$], 0.1, 1}, { Hold[$CellContext`c$$], (-2) Pi, 2 Pi}}, Typeset`size$$ = { 518., {263., 270.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$9632$$ = 0, $CellContext`b$9633$$ = 0, $CellContext`c$9634$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 0.1, $CellContext`c$$ = (-2) Pi}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$9632$$, 0], Hold[$CellContext`b$$, $CellContext`b$9633$$, 0], Hold[$CellContext`c$$, $CellContext`c$9634$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Show[ ParametricPlot3D[{$CellContext`a$$ Cos[$CellContext`t], $CellContext`a$$ Sin[$CellContext`t], $CellContext`b$$ $CellContext`t}, \ {$CellContext`t, (-2) Pi, 2 Pi}], Graphics3D[{Thick, Arrow[{{$CellContext`a$$ Cos[$CellContext`c$$], $CellContext`a$$ Sin[$CellContext`c$$], $CellContext`b$$ $CellContext`c$$}, \ {$CellContext`a$$ Cos[$CellContext`c$$], $CellContext`a$$ Sin[$CellContext`c$$], $CellContext`b$$ $CellContext`c$$} + \ {(-$CellContext`a$$) Sin[$CellContext`c$$], $CellContext`a$$ Cos[$CellContext`c$$], $CellContext`b$$}}]}], PlotRange -> All], "Specifications" :> {{$CellContext`a$$, 1, 10, 1}, {$CellContext`b$$, 0.1, 1}, {$CellContext`c$$, (-2) Pi, 2 Pi}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{566., {344., 352.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.755528803897852*^9, 3.755528818520871*^9}, { 3.7555290459036913`*^9, 3.755529047746291*^9}, 3.755581683570608*^9, 3.7555818907900743`*^9, 3.755581933628839*^9, 3.7555820401634407`*^9, 3.755582095289888*^9, 3.755582133971101*^9}, TextAlignment->Center,ExpressionUUID->"a5ea9552-6763-4e92-9dd0-0eaf759955f3"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"d4404ae5-8ed9-4fdf-93ef-0d8c976dd1ea"], Cell[CellGroupData[{ Cell["Avaruusk\[ADoubleDot]yr\[ADoubleDot]n pituus", "Section", CellChangeTimes->{{3.755527760267074*^9, 3.755527767530978*^9}, { 3.7555285312927923`*^9, 3.755528537036922*^9}, {3.755581526320593*^9, 3.7555815389356327`*^9}, {3.755583338926352*^9, 3.755583340134562*^9}},ExpressionUUID->"8c4b390e-1a0e-487e-8e57-\ 071c605011ca"], Cell[TextData[{ StyleBox["Helix: \n", FontSize->36], StyleBox["r", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t) = a cos(t) ", FontSize->36], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a sin(t) ", FontSize->36], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b t", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["t \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0\n", FontSize->36, FontColor->GrayLevel[0]], StyleBox["r\[CloseCurlyQuote]", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t) = -a sin(t) ", FontSize->36, FontColor->GrayLevel[0]], StyleBox["i", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" + a cos(t) ", FontSize->36, FontColor->GrayLevel[0]], StyleBox["j", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], StyleBox["+ ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["b", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" k, ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["t \[Element] ", FontSize->36, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["I, a,b > 0\n||", FontSize->36, FontColor->GrayLevel[0]], StyleBox["r\[CloseCurlyQuote]", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["(t)||=", FontSize->36, FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], " ", "+", " ", SuperscriptBox["b", "2"]}]], TraditionalForm]], FontSize->36,ExpressionUUID->"732c158f-4b24-4561-b923-940d9aa3fc25"], StyleBox["\nKaarenpituus l(c) = ", FontSize->36, FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ RowBox[{"-", "2"}], "\[Pi]"}], "c"], RowBox[{ StyleBox["||", FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[ RowBox[{"r", "'"}], FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], FontColor->GrayLevel[0]], StyleBox["||", FontColor->GrayLevel[0]], RowBox[{"\[DifferentialD]", "t"}]}]}], TraditionalForm]], FontSize->36,ExpressionUUID->"b060b5b1-fa45-40cf-8492-3e766e50c5bb"] }], "Text", CellChangeTimes->{{3.75552798669309*^9, 3.755528070823266*^9}, { 3.755528155314108*^9, 3.755528177738497*^9}, {3.755581544081231*^9, 3.7555815798836317`*^9}, {3.7555823063591547`*^9, 3.75558231521772*^9}, { 3.755583376663636*^9, 3.755583755600296*^9}}, FontSize->20, Background->GrayLevel[ 0.85],ExpressionUUID->"92198362-af19-4674-99d8-261e756cf3bc"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"ParametricPlot3D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"b", " ", "t"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Thin"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"a", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"b", " ", "t"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", "c"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Thick", ",", "Red"}], "}"}]}]}], "]"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"\"\\"", "<>", " ", RowBox[{"ToString", "@", RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"a", "^", "2"}], "+", RowBox[{"b", "^", "2"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{ RowBox[{"-", "2"}], "Pi"}], ",", "c"}], "}"}]}], "]"}]}]}]}], ",", RowBox[{"LabelStyle", "\[Rule]", "16"}]}], "\[IndentingNewLine]", "]"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "10", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "0.1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"c", ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "Pi"}], "+", ".1"}], ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7555281623774557`*^9, 3.755528377504236*^9}, { 3.75552857419386*^9, 3.75552866912462*^9}, {3.755528745393684*^9, 3.755528765651289*^9}, {3.75552880247812*^9, 3.755528813006901*^9}, { 3.755581627616954*^9, 3.755581682398143*^9}, {3.755581752968676*^9, 3.7555818895661*^9}, {3.755581929951229*^9, 3.755581931870739*^9}, { 3.755582038210887*^9, 3.755582038687866*^9}, {3.755582084392789*^9, 3.7555821327734947`*^9}, {3.755583788970901*^9, 3.755583804275379*^9}, { 3.7555838343652782`*^9, 3.755583848206956*^9}, {3.755583878717855*^9, 3.755583981552272*^9}, {3.75558404767799*^9, 3.755584051167013*^9}}, CellLabel->"In[20]:=",ExpressionUUID->"4df7abfe-8eae-4a86-a404-8eaa5cd8fd3f"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 0.1, $CellContext`c$$ = -6.183185307179587, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], 1, 10, 1}, { Hold[$CellContext`b$$], 0.1, 1}, { Hold[$CellContext`c$$], -6.183185307179587, 2 Pi}}, Typeset`size$$ = { 518., {261., 268.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$22433$$ = 0, $CellContext`b$22434$$ = 0, $CellContext`c$22435$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 0.1, $CellContext`c$$ = -6.183185307179587}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$22433$$, 0], Hold[$CellContext`b$$, $CellContext`b$22434$$, 0], Hold[$CellContext`c$$, $CellContext`c$22435$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Show[ ParametricPlot3D[{$CellContext`a$$ Cos[$CellContext`t], $CellContext`a$$ Sin[$CellContext`t], $CellContext`b$$ $CellContext`t}, \ {$CellContext`t, (-2) Pi, 2 Pi}, PlotStyle -> Thin], ParametricPlot3D[{$CellContext`a$$ Cos[$CellContext`t], $CellContext`a$$ Sin[$CellContext`t], $CellContext`b$$ $CellContext`t}, \ {$CellContext`t, (-2) Pi, $CellContext`c$$}, PlotStyle -> {Thick, Red}], PlotRange -> All, PlotLabel -> "l(c) = " <> ToString[ NIntegrate[ Sqrt[$CellContext`a$$^2 + $CellContext`b$$^2], {$CellContext`t, \ (-2) Pi, $CellContext`c$$}]], LabelStyle -> 16], "Specifications" :> {{$CellContext`a$$, 1, 10, 1}, {$CellContext`b$$, 0.1, 1}, {$CellContext`c$$, -6.183185307179587, 2 Pi}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{566., {342., 350.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.755583952486363*^9, 3.755583983035172*^9, 3.755584053364648*^9}, TextAlignment->Center,ExpressionUUID->"785e1317-4dc3-4b28-b94b-03aa1d8d67ea"] }, {2}]] }, Open ]] }, Open ]] }, ScreenStyleEnvironment->"SlideShow", WindowSize->{Full, Full}, WindowMargins->{{47, Automatic}, {Automatic, 31}}, TaggingRules->{ "PresenterSettings" -> { "ShowToolbar" -> True, "ShowNavigationBar" -> False, "SlideBreakStyles" -> {}, "ShowSlideBreaks" -> True, "Theme" -> { "ThemeName" -> "Default", "FontSetName" -> "Default", "ColorSetName" -> "Default"}, "UserModifications" -> {}, "Dialogs" -> {}}, "SlideshowSettings" -> {"Toolbar" -> True}}, FrontEndVersion->"13.2 for Mac OS X ARM (64-bit) (November 18, 2022)", StyleDefinitions->FrontEnd`FileName[{"PresenterTools"}, "Default.nb", CharacterEncoding -> "UTF-8"], ExpressionUUID->"c99e07f7-0a11-4745-acb3-595d634a05f6" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "SlideShowHeader"->{ Cell[580, 22, 136, 2, 2, "SlideShowNavigationBar",ExpressionUUID->"0ba3833e-070c-4801-82af-cacc8780f61f", CellTags->"SlideShowHeader"], Cell[1485, 47, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"5dc6cc66-657d-4069-bc3b-3aabaa3e0109", CellTags->"SlideShowHeader"], Cell[12843, 278, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"41a0421c-0c12-413b-b229-542e3537c546", CellTags->"SlideShowHeader"], Cell[33813, 686, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"1fe59a13-79a6-4a12-93d8-84994256f8e4", CellTags->"SlideShowHeader"], Cell[38477, 825, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"a243125f-d587-46b8-b229-6718a95221ac", CellTags->"SlideShowHeader"], Cell[46691, 1061, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"d4404ae5-8ed9-4fdf-93ef-0d8c976dd1ea", CellTags->"SlideShowHeader"]} } *) (*CellTagsIndex CellTagsIndex->{ {"SlideShowHeader", 57036, 1361} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 136, 2, 2, "SlideShowNavigationBar",ExpressionUUID->"0ba3833e-070c-4801-82af-cacc8780f61f", CellTags->"SlideShowHeader"], Cell[719, 26, 170, 3, 222, "Title",ExpressionUUID->"6388e47a-6b1f-498b-9257-9a2787a9a01b"], Cell[892, 31, 196, 4, 99, "Subtitle",ExpressionUUID->"0dec9ce5-4078-44d2-9a36-3bd59c31ebf1"], Cell[1091, 37, 357, 5, 65, "Subsubtitle",ExpressionUUID->"77a8381a-6add-4b11-99b3-21687dda672a"] }, Open ]], Cell[CellGroupData[{ Cell[1485, 47, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"5dc6cc66-657d-4069-bc3b-3aabaa3e0109", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[1632, 53, 252, 4, 121, "Section",ExpressionUUID->"53b37233-59d6-421a-a684-08d68b2136e9"], Cell[1887, 59, 620, 22, 82, "Text",ExpressionUUID->"7afa3ded-7324-4a6f-a5c2-96853b37b9a8"], Cell[CellGroupData[{ Cell[2532, 85, 428, 12, 30, "Input",ExpressionUUID->"3c7cfb20-d2c7-4fad-9411-80b23cd7f8fb"], Cell[2963, 99, 9822, 172, 664, "Output",ExpressionUUID->"ec909fa3-b750-4edd-94a1-d1fd01b0c3e2"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[12843, 278, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"41a0421c-0c12-413b-b229-542e3537c546", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[12990, 284, 274, 5, 121, "Section",ExpressionUUID->"1d019f27-d2a4-4e7e-8b18-2055da7fec6d"], Cell[13267, 291, 1113, 47, 82, "Text",ExpressionUUID->"9d8567dc-350d-4104-aed5-e461f39f1bcf"], Cell[CellGroupData[{ Cell[14405, 342, 676, 19, 30, "Input",ExpressionUUID->"10d1dc12-bed9-4d22-951c-498bc5fb4faa"], Cell[15084, 363, 18671, 316, 952, "Output",ExpressionUUID->"86ee88a1-e2b8-4480-92a5-c89811fc8116"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[33813, 686, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"1fe59a13-79a6-4a12-93d8-84994256f8e4", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[33960, 692, 274, 5, 121, "Section",ExpressionUUID->"8ebf104b-89bb-41e9-a719-b75e05684c0f"], Cell[34237, 699, 1113, 47, 82, "Text",ExpressionUUID->"d383d4da-46f2-4c06-8c14-b99dbda7ff37"], Cell[CellGroupData[{ Cell[35375, 750, 942, 25, 30, "Input",ExpressionUUID->"e4166f5c-bb8a-4ec4-ace9-594c357c7502"], Cell[36320, 777, 2099, 41, 526, "Output",ExpressionUUID->"4c626f5a-41ac-4079-ab8d-6d25274ae7b2"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[38477, 825, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"a243125f-d587-46b8-b229-6718a95221ac", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[38624, 831, 298, 4, 121, "Section",ExpressionUUID->"377b3bca-ec2d-411e-9f03-71291b0eb2a2"], Cell[38925, 837, 2118, 89, 180, "Text",ExpressionUUID->"f50e3d37-0ddf-4aa9-862d-ca1b903a56e2"], Cell[CellGroupData[{ Cell[41068, 930, 2581, 66, 115, "Input",ExpressionUUID->"79406004-50d9-4a40-84b2-b9f9efef5658"], Cell[43652, 998, 2981, 56, 721, "Output",ExpressionUUID->"a5ea9552-6763-4e92-9dd0-0eaf759955f3"] }, {2}]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[46691, 1061, 122, 2, 1, "SlideShowNavigationBar",ExpressionUUID->"d4404ae5-8ed9-4fdf-93ef-0d8c976dd1ea", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[46838, 1067, 337, 5, 121, "Section",ExpressionUUID->"8c4b390e-1a0e-487e-8e57-071c605011ca"], Cell[47178, 1074, 3197, 127, 290, "Text",ExpressionUUID->"92198362-af19-4674-99d8-261e756cf3bc"], Cell[CellGroupData[{ Cell[50400, 1205, 2925, 73, 157, "Input",ExpressionUUID->"4df7abfe-8eae-4a86-a404-8eaa5cd8fd3f"], Cell[53328, 1280, 2843, 55, 717, "Output",ExpressionUUID->"785e1317-4dc3-4b28-b94b-03aa1d8d67ea"] }, {2}]] }, Open ]] }, Open ]] } ] *)