By Friedmar Schulz
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Extra resources for Analysis II: Funktionen mehrerer Variablen
Example text
N , xα ∶= xα1 1 ⋅ . . 1 Funktionen und Abbildungen 31 für x = (x1 , . . , xn ) ∈ Rn . αn xα1 1 ⋅ . . ⋅ xαnn . Im Fall k = 2 gilt zum Beispiel n n i,j =1 i=1 P (x) = ∑ aij xi xj + ∑ bi xi + c. (vi) σk ∶ Rn → R, σk (x) ∶= ∑ 1≤i1 <⋯ 3 Beispiele von Funktionen. (i) stante Funktion). (ii) f ∶ Rn → R, f (x) ∶= c, c ∈ R (kon- ϕi ∶ Rn → R, ϕi (x) ∶= xi (i-te Koordinatenfunktion) für i = 1, . . , n. 3: Wetterkarte mit Isobaren Wir schreiben auch ϕi = xi , das heißt, es gilt xi (x) = xi für x ∈ Rn . √ (iii) ∣ ∣ ∶ Rn → R, ∣x∣ ∶= x21 + . . + x2n (Absolutbetrag). (iv) ∶ Rn → R, (x) ∶= a ⋅ x + b (affine und für b = 0 lineare Funktion), dabei ist a = (a1 , . . , an ) ∈ Rn , b ∈ R und a ⋅ x + b = a1 x1 + . . + an bn + b. (v) P ∶ Rn → R, P (x) ∶= ∑ aα xα (Polynom), dabei ist α = (α1 , . Iv) ∶ Rn → R, (x) ∶= a ⋅ x + b (affine und für b = 0 lineare Funktion), dabei ist a = (a1 , . . , an ) ∈ Rn , b ∈ R und a ⋅ x + b = a1 x1 + . . + an bn + b. (v) P ∶ Rn → R, P (x) ∶= ∑ aα xα (Polynom), dabei ist α = (α1 , . . , αn ) ein ∣α∣≤k Multiindex mit α1 , . . , αn ∈ { 0, 1, . . αn ∈ R und wir setzen ∣α∣ ∶= α1 + . . + αn , xα ∶= xα1 1 ⋅ . . 1 Funktionen und Abbildungen 31 für x = (x1 , . . , xn ) ∈ Rn . αn xα1 1 ⋅ . . ⋅ xαnn . Im Fall k = 2 gilt zum Beispiel n n i,j =1 i=1 P (x) = ∑ aij xi xj + ∑ bi xi + c.