問題1.13の(1)から(5)について、$\frac{dz}{dt}$ を求め、問題2について、$\frac{\partial z}{\partial u}$ と $\frac{\partial z}{\partial v}$ を求め、問題3について、$\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$ を示す。

1. 問題の内容

問題1.13の(1)から(5)について、$\frac{dz}{dt}$ を求め、問題2について、$\frac{\partial z}{\partial u}$ と $\frac{\partial z}{\partial v}$ を求め、問題3について、$\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$ を示す。

2. 解き方の手順

問題1: $\frac{dz}{dt}$ を求める。

(1) $z = f(2t, 3t)$ の場合:

連鎖律より、

$$\frac{dz}{dt} = \frac{\partial f}{\partial (2t)} \frac{d(2t)}{dt} + \frac{\partial f}{\partial (3t)} \frac{d(3t)}{dt} = 2\frac{\partial f}{\partial (2t)} + 3\frac{\partial f}{\partial (3t)}$$

ここで、$\frac{\partial f}{\partial (2t)}$ は、$f$ の第一引数に関する偏微分を表す。同様に、$\frac{\partial f}{\partial (3t)}$ は、$f$ の第二引数に関する偏微分を表す。

(2) $z = f(\cos 2t, \sin 3t)$ の場合:

連鎖律より、

$$\frac{dz}{dt} = \frac{\partial f}{\partial (\cos 2t)} \frac{d(\cos 2t)}{dt} + \frac{\partial f}{\partial (\sin 3t)} \frac{d(\sin 3t)}{dt} = -2\sin 2t \frac{\partial f}{\partial (\cos 2t)} + 3\cos 3t \frac{\partial f}{\partial (\sin 3t)}$$

(3) $z = f(e^t, e^{2t})$ の場合:

連鎖律より、

$$\frac{dz}{dt} = \frac{\partial f}{\partial (e^t)} \frac{d(e^t)}{dt} + \frac{\partial f}{\partial (e^{2t})} \frac{d(e^{2t})}{dt} = e^t \frac{\partial f}{\partial (e^t)} + 2e^{2t} \frac{\partial f}{\partial (e^{2t})}$$

(4) $z = f(\frac{1}{t}, \frac{1}{t^2})$ の場合:

連鎖律より、

$$\frac{dz}{dt} = \frac{\partial f}{\partial (\frac{1}{t})} \frac{d(\frac{1}{t})}{dt} + \frac{\partial f}{\partial (\frac{1}{t^2})} \frac{d(\frac{1}{t^2})}{dt} = -\frac{1}{t^2} \frac{\partial f}{\partial (\frac{1}{t})} - \frac{2}{t^3} \frac{\partial f}{\partial (\frac{1}{t^2})}$$

(5) $z = f(\varphi(t) \cos(\psi(t)), \varphi(t) \sin(\psi(t)))$ の場合:

連鎖律より、

$$\frac{dz}{dt} = \frac{\partial f}{\partial (\varphi(t) \cos(\psi(t)))} \frac{d(\varphi(t) \cos(\psi(t)))}{dt} + \frac{\partial f}{\partial (\varphi(t) \sin(\psi(t)))} \frac{d(\varphi(t) \sin(\psi(t)))}{dt}$$

$$\frac{d(\varphi(t) \cos(\psi(t)))}{dt} = \varphi'(t)\cos(\psi(t)) - \varphi(t)\sin(\psi(t))\psi'(t)$$

$$\frac{d(\varphi(t) \sin(\psi(t)))}{dt} = \varphi'(t)\sin(\psi(t)) + \varphi(t)\cos(\psi(t))\psi'(t)$$

よって、

$$\frac{dz}{dt} = (\varphi'(t)\cos(\psi(t)) - \varphi(t)\sin(\psi(t))\psi'(t)) \frac{\partial f}{\partial (\varphi(t) \cos(\psi(t)))} + (\varphi'(t)\sin(\psi(t)) + \varphi(t)\cos(\psi(t))\psi'(t)) \frac{\partial f}{\partial (\varphi(t) \sin(\psi(t)))}$$

問題2: $z = f(x, y), x = uv, y = u^2 + v^2$ のとき $\frac{\partial z}{\partial u}$ と $\frac{\partial z}{\partial v}$ を求める。

連鎖律より、

$$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} = v \frac{\partial z}{\partial x} + 2u \frac{\partial z}{\partial y}$$

$$\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v} = u \frac{\partial z}{\partial x} + 2v \frac{\partial z}{\partial y}$$

問題3: $z = f(x, y), x = e^u + e^v, y = e^{-u} + e^{-v}$ のとき $\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$ を示す。

連鎖律より、

$$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} = e^u \frac{\partial z}{\partial x} - e^{-u} \frac{\partial z}{\partial y}$$

$$\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v} = e^v \frac{\partial z}{\partial x} - e^{-v} \frac{\partial z}{\partial y}$$

$$\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = (e^u + e^v) \frac{\partial z}{\partial x} - (e^{-u} + e^{-v}) \frac{\partial z}{\partial y} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$$

3. 最終的な答え

問題1:

(1) $\frac{dz}{dt} = 2\frac{\partial f}{\partial (2t)} + 3\frac{\partial f}{\partial (3t)}$

(2) $\frac{dz}{dt} = -2\sin 2t \frac{\partial f}{\partial (\cos 2t)} + 3\cos 3t \frac{\partial f}{\partial (\sin 3t)}$

(3) $\frac{dz}{dt} = e^t \frac{\partial f}{\partial (e^t)} + 2e^{2t} \frac{\partial f}{\partial (e^{2t})}$

(4) $\frac{dz}{dt} = -\frac{1}{t^2} \frac{\partial f}{\partial (\frac{1}{t})} - \frac{2}{t^3} \frac{\partial f}{\partial (\frac{1}{t^2})}$

(5) $\frac{dz}{dt} = (\varphi'(t)\cos(\psi(t)) - \varphi(t)\sin(\psi(t))\psi'(t)) \frac{\partial f}{\partial (\varphi(t) \cos(\psi(t)))} + (\varphi'(t)\sin(\psi(t)) + \varphi(t)\cos(\psi(t))\psi'(t)) \frac{\partial f}{\partial (\varphi(t) \sin(\psi(t)))}$

問題2:

$\frac{\partial z}{\partial u} = v \frac{\partial z}{\partial x} + 2u \frac{\partial z}{\partial y}$

$\frac{\partial z}{\partial v} = u \frac{\partial z}{\partial x} + 2v \frac{\partial z}{\partial y}$

問題3:

$\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$