PK
aJ/G=c]J ]J mathcad/worksheet.xml ( 3. Praktikum. KompleksarvudDefineerime kompleksarvu reaalosa ja imaginaarosa.Moodustame kompleksarvulise muutuja (imaginaarosa sisestame kujul 1i).Kompleksarvulise muutuja "c" väärtus:Kaaskompleksarvu tähis on kriips kompleksarvu tähise peal, selle saab menüüst Math/Operators/Engineering:Kompleksarvu reaal- ja imaginaarosa:Kompleksarvu norm ehk moodul: ruutjuur reaalosa ruudu ja imaginaarosa ruudu summast, tähiseks püstkiipsud kompleksarvu tähise kõrval, mille saab | kiirklahviga (sama klahv, kus märgid >,< ) või menüüst Math/Operators/Algebra. Kompleksarvu argument (nurk komplekstasandil reaaltelje ja kompleksarvu kohavektori vahel). Kreeka tähti saab menüüst Math/Symbols.Muutujale omistatud funktsiooni väärtust võib küsida ka koos omistamisega:Kompleksarvu eksponentkuju:Kaaskompleksarv eksponentkujul:Kompleksarvu trigonomeetriline kuju:Kaaskompleksarv trigonomeetrilisel kujul:Kujutame kompleksarvu ja tema kaaskompleksi komplekstasandil.lineslines1113Võrrandilon kompleksarvude hulgas n erinevat lahenditKontrollime korral. Võrrandilahendamiseks kirjutame võrrandi veelkord alla, jätame aktiivseks, valime 'solve' menüüst Math/Symbolics/Keywords, vajutame 'Enter'-klahvi.Omistame selle vektori tähisele zFunktsiooni rakendamiseks vektorile või maatriksile komponenthaaval tuleb ta vektoriseerida, võttes menüüst Math/Operators/(Vector and Matrix) funktsioonile noole peale. Kujutame komponendid graafiliseltlineslines1111Graafikul on täiendavaid funktsioone sin(x) ja cos(x) kasutatud ühikringjoone moodustamiseks näitamaks punktide paiknemist ühikringjoonel. Graafikul punktide kujutamisel tuleb menüüs Plots/Styles/(Line Styles) teha valik 'none'. Lahendades samal viisil ka võrrandi saamelineslines1111Alternatiivsed kirjutised arvust -1 kuupjuure leidmiseks annavad erinevaid tulemusi:Nimelt eelistab Mathcad avaldise kompleksarvulistest väärtustest esitada vähima positiivse argumendiga väärtust.Kompleksarvulise funktsiooni graafilisel kujutamisel tuleb kujutada funktsiooni reaal- ja imaginaarosa eraldi. Mõistlik on seejuures kasutada kolmemõõtmelisi graafikuid. Esmalt defineerime funktsiooni argumendiks maatriksi, milles reaalosa muutub "vertikaalsuunas" (piki veerge) ja imaginaarosa horisontaalsuunas (piki ridu). Esitame järgnevalt 51x51 maatriksi M vasakpoolses ülanurgas asetseva 6x6 alammaatriksi Nüüd arvutame kompleksarvulise funktsiooni väärtused. Et teostada tehe "ruutjuur maatriksist", tuleb see tehe vektoriseerida, s.t. teostada juurimisoperatsioon igast maatriksi elemendist eraldi. Kanname funktsiooni reaalosa ja imaginaarosa kolmemõõtmeliste pindadena ("3P plot") graafikule:surface6661 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | surface6650.99910814302760165 | 0.01505776125471431 | -0.03944847730817358 | 0 | -0.014790680465670657 | 0.99986573287572234 | 0.0070535092157814625 | 0 | 0.039549390732347436 | -0.0064637486716850366 | 0.99919671018564182 | 0 | 0 | 0 | 0 | 1 | 111110Sama funktsioon "kontuurkaardina" (Contour Plot)111111Menüü Plots/Styles/Contour Values lisab kontuurjoontele arvväärtused.Kujutame graafiliselt logaritmfunktsiooni. Argumentmaatriksi reaalosale positiivse liitmiseta olekssurface666-0.99872292450582356 | -0.023867277579559275 | -0.044529463588458114 | 0 | -0.018486295348627439 | 0.99289345463406331 | -0.11756208840063435 | 0 | 0.047018899932036541 | -0.11658876792288149 | -0.99206667227732548 | 0 | 0 | 0 | 0 | 1 | surface6651 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 11119111110Kujutame graafikul ka kompleksset eksponentfunktsiooni.surface6650.25408624658106593 | -0.010747664122774968 | -0.9671218470359616 | 0 | 0.033285053667281857 | 0.99944310786167856 | -0.0023620648076061318 | 0 | 0.96660865116173678 | -0.031590534400246867 | 0.25430248452501381 | 0 | 0 | 0 | 0 | 1 | surface667-0.78371185212775973 | -0.042231590411495767 | -0.61968719980792919 | 0 | 0.086863413828669939 | 0.98042948556516818 | -0.17667136489099994 | 0 | 0.61502071523852431 | -0.19228758827791065 | -0.764705827898388 | 0 | 0 | 0 | 0 | 1 | 111110111111Sarnaseid tegevusi aitab sageli efektiivsemaks muuta esimesel korral tehtu kopeerimine, alla kleepimine ja redigeerimine. Kopeerida-kleepida saab korraga ka töölehe suuremat piirkonda, millele on hiirega raam ümber tõmmatud. Järgnevas ülesandes 1 tasub kompleksarvu tähisele (juhendis ülal c) omistada 1. kompleksarv, teha selle arvu jaoks tellitud tegevused, seejärel tulemused kopeerida-kleepida ja kui kleebitud osa alguses asendada 1. kompleksarv teisega, toimubki järgnev tegevus 2. kompleksarvuga. Kui ülesandes 2 on esimese funktsiooni jaoks kujutatav maatriks N moodustatud, tellitud 4 graafikut tehtud, värvitud ja sobivalt vormistatud ning paigutatud, tasub tulemusele raam ümber tõmmata, tulemus kopeerida, kleepida sobiv arv kordi ning kleebitud osade alguses asendada maatriksi N definitsioonis 1. funktsioon järgmistega.Ülesanne 1. Kujutada komplekstasandil arvudjaja nende kaaskompleksid. Leidke nende arvude trigonomeetriline ja eksponentkuju, samuti moodul ja argument.Ülesanne 2. Joonistage kompleksarvulise argumendi x korral reaalosa ja imaginaarosa graafikud piirkonnas(nii pinnana kui kontuurina) trigonomeetrilistelefunktsioonidele sin, cos, tan, cot, samuti funktsioonidele Katsuge seejuures vältida määramatust punktis PK
aJ/Gzb4
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aJ/Gʇ~ mathcad/header.xml ( PK
aJ/G鐎 mathcad/footer.xml ( PK
aJ/GU4 [Content_Types].xml ( PK
aJ/GMR R docProps/core.xml (
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aJ/Gj j docProps/app.xml ( 2.0.1.02.0.1.0PK
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aJ/Gk\# # mathcad/settings/calculation.xml ( PK
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