Differentiabilitet af vektorfunktionMan kan ikke se p\303\245 2D-banekurve for en vektorfunktion f(t)=(x(t),y(t)) om den er differentiabel.NB: Man skal unders\303\270ge om hver af koordinatfunktionerne x(t) og y(t) er differentiabel.Eksempel p\303\245 en differentiabel vektorfunktion af 2 variableLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=Definer f\303\270lgende koordinat-funktioner: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 og 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Banekurven i planen kn\303\246kker LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JlEjOTBGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctRiM2Jy1JI21pR0YkNidRIm9GJ0YyLyUnaXRhbGljR0Y0L0Y2USxib2xkLWl0YWxpY0YnRjhGMkZARkJGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn i (0,0):NB: Her er kun 2 kordinater: x og y.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 funktionen f(t)=(x(t),y(t)) er en differentiabel funktion, da koordinatfunktionerne x(t) og y(t) er differentiable: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Koordinatfunktionernes afledede x'(t) og y'(t) er kontinuerte funktioner af 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og 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i rummet er differentiabel:NB: Her er 3 kordinater: t, x og y.(Rot\303\251r grafen og se fra forskellige vinkler. Grafen er simpelthen glat - ogs\303\245 i LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1EidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyMDQsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSI9RidGMkY1L0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JlEjMC5GJ0YyRjVGPkYyRjVGPg==)(Pr\303\270v ogs\303\245 at dreje koordinatsystemet, s\303\245 t-aksen g\303\245r ud af sk\303\246rmen samt x-aksen til h\303\270jre og y-aksen opad. S\303\245 fremkommer banekurven i x-y-planen.)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Eksempel p\303\245 en ikke-differentiabel vektorfunktion af 2 variableLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiDefiner f\303\270lgende koordinat-funktioner: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 og 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PkkieUc2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjI5JCIiISwkRjAjIiIiIiIjMUYxRjAsJEYwRjVGJEYkRiQ=Banekurven i planen er en ret linje:NB: Her er kun 2 kordinater: x og y.LUklcGxvdEc2IjYnNyUtSSJ4R0YkNiNJInRHRiQtSSJ5R0YkRikvRio7ISIlIiIiL0kmY29sb3JHRiRJJHJlZEdGJC9JKnRoaWNrbmVzc0dGJCIiJC9JJ2xhYmVsc0dGJDckRihGLC9JJ2xlZ2VuZEdGJDcjUTRiYW5la3VydmVufml+cGxhbmVuRiQ=Men koordinatfunktionerne x(t) og y(t) er ikke differentiable:(grafen kn\303\246kker i 0)LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYoLUkieEdGJzYjSSJ0R0YnL0YsOyEiIyIiIi9JJmNvbG9yR0YnSSVibHVlR0YnL0kqdGhpY2tuZXNzR0YnIiIkL0knbGFiZWxzR0YnNyRGLEYqL0knbGVnZW5kR0YnNyNRJXgodClGJw==LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYoLUkieUdGJzYjSSJ0R0YnL0YsOyEiIyIiIi9JJmNvbG9yR0YnSSVibHVlR0YnL0kqdGhpY2tuZXNzR0YnIiIkL0knbGFiZWxzR0YnNyRGLEYqL0knbGVnZW5kR0YnNyNRJXkodClGJw==Banekurven i rummet er ikke-differentiabel:NB: Her er 3 kordinater: t, x og y.(Rot\303\251r grafen og se fra forskellige vinkler. Grafen kn\303\246kker i LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2J1EidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyMDQsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYwUSI9RidGL0YyRjVGNy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2J1EjMC5GJ0YvRjJGNUY3Ri9GMkY1Rjc=)(Pr\303\270v ogs\303\245 at dreje koordinatsystemet, s\303\245 t-aksen g\303\245r ud af sk\303\246rmen samt x-aksen til h\303\270jre og y-aksen opad. S\303\245 fremkommer banekurven i x-y-planen.)LUkrc3BhY2VjdXJ2ZUc2IjYoNyUtSSJ4R0YkNiNJInRHRiQtSSJ5R0YkRilGKi9GKjshIiMiIiMvSSp0aGlja25lc3NHRiQiIiQvSSZjb2xvckdGJEkmYnJvd25HRiQvSSVheGVzR0YkSSZCT1hFREdGJC9JJ2xhYmVsc0dGJDclRihGLEYq