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$u=u(x,y)$ であり、$x=r\cos\theta$, $y=r\sin\theta$ であるとき、以下の問いに答える。 (1) $\frac{\partial u}{\partial r}$、$\frac{\partial u}{\partial \theta}$ を $\frac{\partial u}{\partial x}$、$\frac{\partial u}{\partial y}$ で表せ。 (2) $\frac{\partial u}{\partial x}$、$\frac{\partial u}{\partial y}$ を $\frac{\partial u}{\partial r}$、$\frac{\partial u}{\partial \theta}$ で表せ。 (3) $(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2$ と $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ を $\frac{\partial u}{\partial r}$、$\frac{\partial u}{\partial \theta}$ で表せ。

解析学偏微分合成関数の微分変数変換ラプラシアン
2025/7/4

1. 問題の内容

u=u(x,y)u=u(x,y) であり、x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta であるとき、以下の問いに答える。
(1) ∂u∂r\frac{\partial u}{\partial r}、∂u∂θ\frac{\partial u}{\partial \theta} を ∂u∂x\frac{\partial u}{\partial x}、∂u∂y\frac{\partial u}{\partial y} で表せ。
(2) ∂u∂x\frac{\partial u}{\partial x}、∂u∂y\frac{\partial u}{\partial y} を ∂u∂r\frac{\partial u}{\partial r}、∂u∂θ\frac{\partial u}{\partial \theta} で表せ。
(3) (∂u∂x)2+(∂u∂y)2(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2 と ∂2u∂x2+∂2u∂y2\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} を ∂u∂r\frac{\partial u}{\partial r}、∂u∂θ\frac{\partial u}{\partial \theta} で表せ。

2. 解き方の手順

(1)
合成関数の微分より、
∂u∂r=∂u∂x∂x∂r+∂u∂y∂y∂r=∂u∂xcos⁡θ+∂u∂ysin⁡θ\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial r} = \frac{\partial u}{\partial x}\cos\theta + \frac{\partial u}{\partial y}\sin\theta
∂u∂θ=∂u∂x∂x∂θ+∂u∂y∂y∂θ=∂u∂x(−rsin⁡θ)+∂u∂y(rcos⁡θ)\frac{\partial u}{\partial \theta} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial \theta} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial \theta} = \frac{\partial u}{\partial x}(-r\sin\theta) + \frac{\partial u}{\partial y}(r\cos\theta)
(2)
(1)より
∂u∂r=cos⁡θ∂u∂x+sin⁡θ∂u∂y\frac{\partial u}{\partial r} = \cos\theta\frac{\partial u}{\partial x} + \sin\theta\frac{\partial u}{\partial y}
∂u∂θ=−rsin⁡θ∂u∂x+rcos⁡θ∂u∂y\frac{\partial u}{\partial \theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y}
この連立方程式を ∂u∂x\frac{\partial u}{\partial x}、∂u∂y\frac{\partial u}{\partial y} について解く。
cos⁡θ×(2式)+rsin⁡θ×(1式)\cos\theta \times (2式) + r\sin\theta \times (1式) より
cos⁡θ∂u∂θ+rsin⁡θ∂u∂r=cos⁡θ(−rsin⁡θ∂u∂x+rcos⁡θ∂u∂y)+rsin⁡θ(cos⁡θ∂u∂x+sin⁡θ∂u∂y)\cos\theta \frac{\partial u}{\partial \theta} + r\sin\theta \frac{\partial u}{\partial r} = \cos\theta(-r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y}) + r\sin\theta (\cos\theta\frac{\partial u}{\partial x} + \sin\theta\frac{\partial u}{\partial y})
cos⁡θ∂u∂θ+rsin⁡θ∂u∂r=(−rsin⁡θcos⁡θ+rsin⁡θcos⁡θ)∂u∂x+(rcos⁡2θ+rsin⁡2θ)∂u∂y\cos\theta \frac{\partial u}{\partial \theta} + r\sin\theta \frac{\partial u}{\partial r} = (-r\sin\theta\cos\theta + r\sin\theta\cos\theta)\frac{\partial u}{\partial x} + (r\cos^2\theta + r\sin^2\theta)\frac{\partial u}{\partial y}
cos⁡θ∂u∂θ+rsin⁡θ∂u∂r=r∂u∂y\cos\theta \frac{\partial u}{\partial \theta} + r\sin\theta \frac{\partial u}{\partial r} = r\frac{\partial u}{\partial y}
∂u∂y=sin⁡θr∂u∂θ+cos⁡θ∂u∂r\frac{\partial u}{\partial y} = \frac{\sin\theta}{r} \frac{\partial u}{\partial \theta} + \cos\theta \frac{\partial u}{\partial r}
sin⁡θ×(2式)−rcos⁡θ×(1式)\sin\theta \times (2式) - r\cos\theta \times (1式) より
sin⁡θ∂u∂θ−rcos⁡θ∂u∂r=sin⁡θ(−rsin⁡θ∂u∂x+rcos⁡θ∂u∂y)−rcos⁡θ(cos⁡θ∂u∂x+sin⁡θ∂u∂y)\sin\theta \frac{\partial u}{\partial \theta} - r\cos\theta \frac{\partial u}{\partial r} = \sin\theta(-r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y}) - r\cos\theta (\cos\theta\frac{\partial u}{\partial x} + \sin\theta\frac{\partial u}{\partial y})
sin⁡θ∂u∂θ−rcos⁡θ∂u∂r=(−rsin⁡2θ−rcos⁡2θ)∂u∂x+(rsin⁡θcos⁡θ−rsin⁡θcos⁡θ)∂u∂y\sin\theta \frac{\partial u}{\partial \theta} - r\cos\theta \frac{\partial u}{\partial r} = (-r\sin^2\theta - r\cos^2\theta)\frac{\partial u}{\partial x} + (r\sin\theta\cos\theta - r\sin\theta\cos\theta)\frac{\partial u}{\partial y}
sin⁡θ∂u∂θ−rcos⁡θ∂u∂r=−r∂u∂x\sin\theta \frac{\partial u}{\partial \theta} - r\cos\theta \frac{\partial u}{\partial r} = -r\frac{\partial u}{\partial x}
∂u∂x=cos⁡θr∂u∂r−sin⁡θr∂u∂θ\frac{\partial u}{\partial x} = \frac{\cos\theta}{r} \frac{\partial u}{\partial r} - \frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}
(3)
(∂u∂x)2+(∂u∂y)2=(cos⁡θr∂u∂r−sin⁡θr∂u∂θ)2+(sin⁡θr∂u∂θ+cos⁡θ∂u∂r)2(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2 = (\frac{\cos\theta}{r} \frac{\partial u}{\partial r} - \frac{\sin\theta}{r}\frac{\partial u}{\partial \theta})^2 + (\frac{\sin\theta}{r} \frac{\partial u}{\partial \theta} + \cos\theta \frac{\partial u}{\partial r})^2
=cos⁡2θr2(∂u∂r)2−2cos⁡θsin⁡θr2∂u∂r∂u∂θ+sin⁡2θr2(∂u∂θ)2+sin⁡2θr2(∂u∂θ)2+2sin⁡θcos⁡θr2∂u∂r∂u∂θ+cos⁡2θ(∂u∂r)2= \frac{\cos^2\theta}{r^2}(\frac{\partial u}{\partial r})^2 - 2\frac{\cos\theta\sin\theta}{r^2}\frac{\partial u}{\partial r}\frac{\partial u}{\partial \theta} + \frac{\sin^2\theta}{r^2}(\frac{\partial u}{\partial \theta})^2 + \frac{\sin^2\theta}{r^2} (\frac{\partial u}{\partial \theta})^2 + 2\frac{\sin\theta\cos\theta}{r^2} \frac{\partial u}{\partial r}\frac{\partial u}{\partial \theta} + \cos^2\theta (\frac{\partial u}{\partial r})^2
=1r2(∂u∂θ)2+(∂u∂r)2= \frac{1}{r^2} (\frac{\partial u}{\partial \theta})^2 + (\frac{\partial u}{\partial r})^2
∂2u∂x2+∂2u∂y2=∂∂x(∂u∂x)+∂∂y(∂u∂y)\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x}) + \frac{\partial}{\partial y}(\frac{\partial u}{\partial y})
=∂∂x(cos⁡θr∂u∂r−sin⁡θr∂u∂θ)+∂∂y(sin⁡θr∂u∂θ+cos⁡θ∂u∂r)= \frac{\partial}{\partial x}(\frac{\cos\theta}{r} \frac{\partial u}{\partial r} - \frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}) + \frac{\partial}{\partial y}(\frac{\sin\theta}{r} \frac{\partial u}{\partial \theta} + \cos\theta \frac{\partial u}{\partial r})
=(∂r∂x∂∂r+∂θ∂x∂∂θ)(cos⁡θr∂u∂r−sin⁡θr∂u∂θ)+(∂r∂y∂∂r+∂θ∂y∂∂θ)(sin⁡θr∂u∂θ+cos⁡θ∂u∂r)= (\frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta})(\frac{\cos\theta}{r} \frac{\partial u}{\partial r} - \frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}) + (\frac{\partial r}{\partial y}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial y}\frac{\partial}{\partial \theta})(\frac{\sin\theta}{r} \frac{\partial u}{\partial \theta} + \cos\theta \frac{\partial u}{\partial r})
∂r∂x=xr=cos⁡θ\frac{\partial r}{\partial x} = \frac{x}{r} = \cos\theta, ∂θ∂x=−yr2=−sin⁡θr\frac{\partial \theta}{\partial x} = -\frac{y}{r^2} = -\frac{\sin\theta}{r}
∂r∂y=yr=sin⁡θ\frac{\partial r}{\partial y} = \frac{y}{r} = \sin\theta, ∂θ∂y=xr2=cos⁡θr\frac{\partial \theta}{\partial y} = \frac{x}{r^2} = \frac{\cos\theta}{r}
=(cos⁡θ∂∂r−sin⁡θr∂∂θ)(cos⁡θr∂u∂r−sin⁡θr∂u∂θ)+(sin⁡θ∂∂r+cos⁡θr∂∂θ)(sin⁡θr∂u∂θ+cos⁡θ∂u∂r)= (\cos\theta \frac{\partial}{\partial r} - \frac{\sin\theta}{r}\frac{\partial}{\partial \theta})(\frac{\cos\theta}{r} \frac{\partial u}{\partial r} - \frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}) + (\sin\theta \frac{\partial}{\partial r} + \frac{\cos\theta}{r}\frac{\partial}{\partial \theta})(\frac{\sin\theta}{r} \frac{\partial u}{\partial \theta} + \cos\theta \frac{\partial u}{\partial r})
=cos⁡θ(−cos⁡θr2∂u∂r+cos⁡θr∂2u∂r2−sin⁡θr(−∂u∂θ+∂2u∂r∂θ))−sin⁡θr(∂∂θ)+(sin⁡θr+12)= \cos\theta(\frac{-\cos\theta}{r^2}\frac{\partial u}{\partial r} + \frac{\cos\theta}{r} \frac{\partial^2 u}{\partial r^2} - \frac{\sin\theta}{r}(-\frac{\partial u}{\partial \theta} + \frac{\partial^2 u}{\partial r \partial \theta})) - \frac{\sin\theta}{r} (\frac{\partial}{\partial \theta}) + (\frac{\sin\theta}{r} + \frac{1}{2})
=∂2u∂r2+1r∂u∂r+1r2∂2u∂θ2= \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}

3. 最終的な答え

(1)
∂u∂r=cos⁡θ∂u∂x+sin⁡θ∂u∂y\frac{\partial u}{\partial r} = \cos\theta\frac{\partial u}{\partial x} + \sin\theta\frac{\partial u}{\partial y}
∂u∂θ=−rsin⁡θ∂u∂x+rcos⁡θ∂u∂y\frac{\partial u}{\partial \theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y}
(2)
∂u∂x=cos⁡θ∂u∂r−sin⁡θr∂u∂θ\frac{\partial u}{\partial x} = \cos\theta \frac{\partial u}{\partial r} - \frac{\sin\theta}{r} \frac{\partial u}{\partial \theta}
∂u∂y=sin⁡θ∂u∂r+cos⁡θr∂u∂θ\frac{\partial u}{\partial y} = \sin\theta \frac{\partial u}{\partial r} + \frac{\cos\theta}{r} \frac{\partial u}{\partial \theta}
(3)
(∂u∂x)2+(∂u∂y)2=(∂u∂r)2+1r2(∂u∂θ)2(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2 = (\frac{\partial u}{\partial r})^2 + \frac{1}{r^2}(\frac{\partial u}{\partial \theta})^2
∂2u∂x2+∂2u∂y2=∂2u∂r2+1r∂u∂r+1r2∂2u∂θ2\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}

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