多変数の関数反復偏微分 3次の偏導関数、順序、指数関数、三角関数(正弦と余弦と逆正接)、累乗、合成関数の微分

続解析入門 (原書第2版)
楽天ブックス
Yahoo!ショッピング
au PAY マーケット
原著: Calculus of Several Variables

学習環境

続解析入門 (原書第2版) (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第3章(多変数の関数)、4(反復偏微分)の練習問題11、12、13、14、15、16、17、18.の解答を求めてみる。

D_{1} D_{2} D_{3} x y z

= D_{1} D_{2} x y

= D_{1} x

= 1

D_{3} D_{2} D_{1} x y z

= D_{3} D_{2} y z

= D_{3} z

= 1

D_{1} D_{2} D_{3} x^{2} y z = 2 x

D_{3} D_{2} D_{1} x^{2} y z = 2 x

D_{1} D_{2} D_{3} e^{x y z}

= D_{1} D_{2} x y e^{x y z}

= D_{1} (x e^{x y z} + x^{2} y z e^{x y z})

= D_{1} e^{x y z} (x + x^{2} y z)

= y z e^{x y z} (x + x^{2} y z) + e^{x y z} (1 + 2 x y z)

= e^{x y z} (x^{2} y^{2} z^{2} + 3 x y z + 1)

D_{3} D_{2} D_{1} e^{x y z}

= D_{3} D_{2} y z e^{x y z}

= D_{3} (z e^{x y z} + x y z^{2} e^{x y z})

= D_{3} e^{x y z} (z + x y z^{2})

= x y e^{x y z} (z + x y z^{2}) + e^{x y z} (1 + 2 x y z)

= (x^{2} y^{2} z^{2} + 3 x y z + 1) e^{x y z}

D_{1} D_{2} D_{3} \sin (x y z)

= D_{1} D_{2} x y \cos (x y z)

= D_{1} (x \cos (x y z) - x^{2} y z \sin (x y z))

= \cos (x y z) - x y z \sin (x y z) - 2 x y z \sin (x y z) - x^{2} y^{2} z^{2} \cos (x y z)

D_{3} D_{2} D_{1} \sin (x y z)

= D_{3} D_{2} y z \cos (x y z)

= D_{3} (z \cos (x y z) - x y z^{2} \sin (x y z))

= \cos (x y z) - x y z \sin (x y z) - 2 x y z \sin (x y z) - x^{2} y^{2} z^{2} \cos (x y z)

D_{1} D_{2} D_{3} \cos (x + y + z)

= D_{1} D_{2} (- \sin (x + y + z))

= D_{1} (- \cos (x + y + z))

= \sin (x + y + z)

D_{3} D_{2} D_{1} \cos (x + y + z)

= \sin (x + y + z)

D_{1} D_{2} D_{3} \sin (x + y + z) = - \cos (x + y + z)

D_{3} D_{2} D_{1} \sin (x + y + z) = - \cos (x + y + z)

D_{1} D_{2} D_{3} {(x^{2} + y^{2} + z^{2})}^{- 1}

= D_{1} D_{2} (- 2 z {(x^{2} + y^{2} + z^{2})}^{- 2})

= D_{1} 8 y z {(x^{2} + y^{2} + z^{2})}^{- 3}

= - 48 x y z {(x^{2} + y^{2} + z^{2})}^{- 4}

D_{1} D_{2} D_{3} (x^{3} y^{2} z + 2 (x + y + z))

= D_{1} D_{2} (x^{3} y^{2} + 2)

= D_{1} 2 x^{3} y

= 6 x^{2} y

D_{3} D_{2} D_{1} (x^{3} y^{2} z + 2 (x + y + z))

= D_{3} D_{2} (3 x^{2} y^{2} z + z)

= D_{3} 6 x^{2} y z

= 6 x^{2} y

コード(Wolfram Language, Jupyter)

fs = {x y z, x^2 y z, Exp[x y z], Sin[x y z], Cos[x + y + z],
      Sin[x+y+z], (x^2+y^2+z^2)^(-1), x^3 y^2 z +2(x+y+z)}

Column[
    Table[
        {f, D[f, x, y, z], D[f, z, y, x],
          D[f, x, y, z] == D[f, x, z, y] == D[f, y, x, z] == D[f, y, z, x] == D[f, z, x, y] == D[f, z, y, x]},
         {f, fs}
    ]
]

Column[
    Table[
        {f, Factor[D[f, x, y, z]], D[f, z, y, x],
          D[f, x, y, z] == D[f, x, z, y] == D[f, y, x, z] == D[f, y, z, x] == D[f, z, x, y] == D[f, z, y, x]},
         {f, fs}
    ]
]

TraditionalForm[%]

Manipulate[
    Plot3D[x y z, {x, -5, 5}, {y, -5, 5}, PlotRange -> {-5, 5}],
    {z, -5, 5}
]