2\u5909\u6570\u95a2\u6570 \\(z = f(x, y)\\) \u306b\u304a\u3044\u3066\uff0c\\(x = x(t), \\ y = y(t)\\) \u306e\u5834\u5408\uff0c$ x = x(u, v), \\ y = y(u, v)$ \u306e\u5834\u5408\uff0c\u306a\u3069\u306b\u304a\u3051\u308b\u5fae\u5206\u30fb\u504f\u5fae\u5206\u306b\u3064\u3044\u3066\u3002
\n<\/p>\n
2\u5909\u6570\u95a2\u6570 \\(z = f(x, y)\\) \u306b\u304a\u3044\u3066\uff0c\\(x = x(t), \\ y = y(t)\\) \u306a\u3089\uff0c\u30d1\u30e9\u30e1\u30fc\u30bf\uff08\u5a92\u4ecb\u5909\u6570\uff09\\(t\\) \u3092\u6c7a\u3081\u308c\u3070 \\(x\\) \u3068 \\(y\\) \u306e\u5024\u304c\u4e00\u610f\u306b\u6c7a\u307e\u308a\uff0c\u305d\u308c\u306b\u3088\u3063\u3066 \\(z\\) \u306e\u5024\u3082\u6c7a\u307e\u3063\u3066\u3057\u307e\u3046\u306e\u3067\uff0c\u7d50\u679c\uff0c\\(z\\) \u306f \\(t\\) \u306e1\u5909\u6570\u95a2\u6570 \\(z = z(t) \\) \u3068\u306a\u308b\u3002\u3064\u307e\u308a\uff0c $$z = f(x(t), y(t)) \\quad \\rightarrow\\quad\u00a0 z = z(t)$$<\/p>\n
\\(z\\) \u306e\u5168\u5fae\u5206\u306f\uff0c $$ dz = \\frac{\\partial z}{\\partial x} dx + \\frac{\\partial z}{\\partial y} dy$$ \u4e21\u8fba\u3092 \\(dt\\) \u3067\u300c\u5272\u3063\u3066\u300d $$ \\frac{dz}{dt} = \\frac{\\partial z}{\\partial x} \\frac{dx}{dt} + \\frac{\\partial z}{\\partial y} \\frac{dy}{dt} $$<\/p>\n
\\( z = f(x, y) \\) \u306b\u3064\u3044\u3066\uff0c$$ x = x(u, v), \\quad y = y(u, v)$$ \u306a\u3089\uff0c
\n$$ z = f(x(u, v), y(u, v)) \\quad \\rightarrow\\quad\u00a0 z = z(u, v)$$<\/p>\n
$$ \\frac{\\partial z}{\\partial u} = \\frac{\\partial z}{\\partial x}\\frac{\\partial x}{\\partial u} + \\frac{\\partial z}{\\partial y} \\frac{\\partial y}{\\partial u}$$ $$ \\frac{\\partial z}{\\partial v} = \\frac{\\partial z}{\\partial x}\\frac{\\partial x}{\\partial v} + \\frac{\\partial z}{\\partial y} \\frac{\\partial y}{\\partial v}$$<\/p>\n
\u3053\u306e\u3088\u3046\u306a\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206\u306e\u95a2\u4fc2\u304c\u5229\u7528\u3055\u308c\u308b\u72b6\u6cc1\u3068\u3057\u3066\uff0c\u5ea7\u6a19\u5909\u63db\u304c\u3042\u3052\u3089\u308c\u308b\u3002<\/p>\n
\u3042\u3068\uff0c\u3053\u3093\u306a\u30b1\u30fc\u30b9\u3082\u3002\\(z = f(u)\\) \u3068 \\(z\\) \u306f1\u5909\u6570 \\(u\\) \u306e\u95a2\u6570\u306a\u306e\u3060\u304c\uff0c\\(u\\) \u306f 2\u5909\u6570 \\(x, y\\) \u306e\u95a2\u6570\u3067\u3042\u308a \\(u = u(x, y)\\)\uff0c\u7d50\u5c40 \\(z = z(x, y)\\) \u306f2\u5909\u6570\u95a2\u6570\u3068\u306a\u308b\uff0c\u3068\u3044\u3046\u5834\u5408\u3002
\n$$ \\frac{\\partial z}{\\partial x} = \\frac{dz}{du} \\frac{\\partial u}{\\partial x} $$\u00a0 $$ \\frac{\\partial z}{\\partial y} = \\frac{dz}{du} \\frac{\\partial u}{\\partial y} $$<\/p>\n
2\u6b21\u5143\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 \\(x, y\\) \u304b\u30892\u6b21\u5143\u6975\u5ea7\u6a19 \\(r, \\phi\\) \u3078\u306e\u5ea7\u6a19\u5909\u63db<\/p>\n
\\begin{eqnarray}
\nr &=& \\sqrt{x^2 + y^2} \\\\
\n\\phi &=& \\tan^{-1} \\frac{y}{x}
\n\\end{eqnarray}<\/p>\n
\u304a\u3088\u3073\u305d\u306e\u9006\u5909\u63db
\n\\begin{eqnarray}
\nx &=&\u00a0 r \\cos\\phi\\\\
\ny &=&\u00a0 r \\sin\\phi
\n\\end{eqnarray}<\/p>\n
\u304b\u3089\uff0c\u4ee5\u4e0b\u306e\u504f\u5c0e\u95a2\u6570\u3092\u8a08\u7b97\u3057\uff0c\u7d50\u679c\u3092\u6975\u5ea7\u6a19 \\(r, \\phi\\) \u3067\u8868\u3059\u3002<\/p>\n
$u \\equiv x^2 + y^2$ \u3068\u304a\u304f\u3068 $r = u^{\\frac{1}{2}}$ \u3067\u3042\u308b\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n\\frac{\\partial r}{\\partial x} &=& \\frac{d}{du} u^{\\frac{1}{2}}\\cdot \\frac{\\partial u}{\\partial x} \\\\
\n&=& \\frac{1}{2} u^{-\\frac{1}{2}} \\cdot 2 x\\\\
\n&=& \\frac{x}{r} \\\\
\n&=& \\cos\\phi
\n\\end{eqnarray}<\/p>\n
$u \\equiv x^2 + y^2$ \u3068\u304a\u304f\u3068 $r = u^{\\frac{1}{2}}$ \u3067\u3042\u308b\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n\\frac{\\partial r}{\\partial y} &=& \\frac{d}{du} u^{\\frac{1}{2}}\\cdot \\frac{\\partial u}{\\partial y} \\\\
\n&=& \\frac{1}{2} u^{-\\frac{1}{2}} \\cdot 2 y\\\\
\n&=& \\frac{y}{r} \\\\
\n&=& \\sin\\phi
\n\\end{eqnarray}<\/p>\n
$u \\equiv \\dfrac{y}{x}$ \u3068\u304a\u304f\u3068<\/p>\n
\\begin{eqnarray}
\n\\frac{\\partial \\phi}{\\partial x} &=& \\frac{d}{du} \\tan^{-1} u \\cdot \\frac{\\partial u}{\\partial x} \\\\
\n&=& \\frac{1}{1+u^2}\\cdot \\left(-\\frac{y}{x^2}\\right) \\\\
\n&=& -\\frac{y}{x^2 + y^2} \\\\
\n&=& -\\frac{\\sin\\phi}{r}
\n\\end{eqnarray}<\/p>\n
$u \\equiv \\dfrac{y}{x}$ \u3068\u304a\u304f\u3068<\/p>\n
\\begin{eqnarray}
\n\\frac{\\partial \\phi}{\\partial y} &=& \\frac{d}{du} \\tan^{-1} u \\cdot \\frac{\\partial u}{\\partial y} \\\\
\n&=& \\frac{1}{1+u^2}\\cdot \\left(\\frac{1}{x}\\right) \\\\
\n&=& \\frac{x}{x^2 + y^2} \\\\
\n&=& \\frac{\\cos\\phi}{r}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"
2\u5909\u6570\u95a2\u6570 \\(z = f(x, y)\\) \u306b\u304a\u3044\u3066\uff0c\\(x = x(t), \\ y = y(t)\\) \u306e\u5834\u5408\uff0c$ x = x(u, v), \\ y = y(u, v)$ \u306e\u5834\u5408\uff0c\u306a\u3069\u306b\u304a\u3051\u308b\u5fae\u5206\u30fb\u504f\u5fae\u5206\u306b\u3064\u3044\u3066\u3002<\/p>