Analysis

A11

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

?Log

Grad[Log[b, x], {x, b}]

Log[b, x] // Simplify

Grad[A*Cos[ω t + ϕ], {A, ω, t, ϕ}]

A12

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}]

zweite Ableitungen

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

Hesse[f, {x, y}]

A13

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

jacobi = Transpose[Grad[f, {r, ϕ, θ}]]

det = Det[jacobi]

Simplify[det]

Lineare Algebra

L11

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

Det[A]

Inverse[A]

L12

gleichungssystem = {x + 2  y + 3  z + 4  w == 1, 
        x + y - z - w == 1, 2  x + 3  y + 2  z + 3  w == 2}

Solve[gleichungssystem, {x, y, z}]

L13

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

Inverse[B]