Analysis

A17

f = {Sin[t], Cos[t] Sin[t], Cos[t]}

g = x ((x^2) + (y^2) + (z^4))

G = 
      g /. {x -> f[[1]], y -> f[[2]], 
          z -> f[[3]]}

G = G // Simplify

Bestimmung der Ableitung direkt:

D[G, t]

Mehrdimensionale Kettenregel:

Grad[f_, x_] := Table[D[f, x[[k]]], {k, Length[x]}]

(Grad[g, {x, y, z}] /. {x -> f[[1]], y -> f[[2]], 
              z -> f[[3]]}) . D[f, t] // Simplify

A18

Kugelkoordinaten:

f = {r Cos[ϕ] Sin[θ], r Sin[ϕ] Sin[θ], 
        r Cos[θ]}

Solve[{x == r Cos[ϕ] Sin[θ], 
        y == r Sin[ϕ] Sin[θ], 
        z == r Cos[θ]}, {r, ϕ, θ}]

umkehr = {(Sqrt[(x^2) + (y^2) + (z^2)]), ArcTan[(y/x)], 
        ArcCos[(z/(Sqrt[(x^2) + (y^2) + (z^2)]))]}

jacobi = Transpose[Grad[umkehr, {x, y, z}]] // Simplify

Transpose[Grad[f, {r, ϕ, θ}]]

Simplify[(jacobi /. {x -> f[[1]], y -> f[[2]], 
              z -> f[[3]]}) . 
        Transpose[Grad[f, {r, ϕ, θ}]], r > 0]

A19

f = (x^3) - 3  (x^2) y + 3  x (y^2) + (y^3) - 3  x - 21  y

Plot3D[f, {x, -5, 5}, {y, -5, 5}]

erste Ableitungen

ableitung = Grad[f, {x, y}]

stationäre Punkte

stationär = Solve[ableitung == 0, {x, y}]

Test auf Sattelpunkt

HinreichendeBedingung[f_, {x_, y_}, x0_] := 
      Module[{testausdruck}, 
testausdruck = 
          D[f, {x, 2}] D[f, {y, 2}] - 
            D[D[f, x], y]^2; 
If[(testausdruck /. x0) < 
            0, Return[{{x, y} /. 
                x0, "Sattelpunkt"}]]; 
testausdruck = 
          D[f, {x, 2}]; 
If[(testausdruck /. x0) < 0, 
          Return[{{x, y} /. x0, "Maximum"}], 
          Return[{{x, y} /. 
                x0, "Minimum"}]]
]

Map[HinreichendeBedingung[f, {x, y}, #] &, 
      stationär]

Lineare Algebra

L17

A = {{1, -1}, {-2, 1}};  
    B = {{-3, 3}, {-1, 0}};

Inverse[A]

Inverse[B]

Inverse[A . B]

Inverse[A . B] . A

L18

A = {{1, 1, 0}, {2, -2, 3}, {1, 2, 3}}

Det[A]

Inverse[A]

Det[Inverse[A]]

L19

A = {{1, 2, 3}, {4, 1, 4}, {0, 2, 1}}

charpol = CharacteristicPolynomial[A, λ]

Factor[charpol]

Eigenvalues[A]