合成関数の偏微分法(differentiation of composite functions

確認問題

1.
$\displaystyle{\frac{dz}{dt}}$ を求めよう.ただし, $f$$C^{1}$ 級とする.

(a) $\displaystyle{z = x^{2} + 2y, x = 2t, y = t^{3}}$ (b) $\displaystyle{z = x^{2} + y^{2}, x = \cos{t}, y = \sin{t}}$

(c) $\displaystyle{z = x^{2} + xy + 2y^{2}, x = \cos{t}, y = \sin{t}}$ (d) $\displaystyle{z = x^{3} y^{2}, x = t^{2}, y = t^{3}}$

2.
次の関数について, $\displaystyle{\frac{\partial z}{\partial u},  \frac{\partial z}{\partial v}}$を求めよう.

(a) $\displaystyle{z = x^{2} + y^{2}, x = u - 2v,  y = 2u + v}$

(b) $\displaystyle{z = x^{2} + xy + 2y^{2}, x = u+v, y = uv}$ (c) $\displaystyle{f(x,y) = x^{2}y^{2}, x = uv, y = v^{2}}$

演習問題

1.
$\displaystyle{\frac{dz}{dt}}$ を求めよう.ただし, $f$$C^{1}$ 級とする.

(a) $\displaystyle{z = \log{(x^2 + y^2)}, x = t + \frac{1}{t}, y = t(t-1)}$ (b) $\displaystyle{z = f(t^2,e^t)}$

(c) $\displaystyle{z = f(2t, 4t^2)}$ (d) $\displaystyle{z = x^2 - 2y^2, x = \cos{t}, y = \sin{t}}$

2.
次の関数について, $\displaystyle{\frac{\partial z}{\partial r},  \frac{\partial z}{\partial s}}$を求めよう.

(a) $\displaystyle{z = \tan^{-1}{\frac{y}{x}}, x = r^3 - 3rs^2,  y = 3r^2 s - s^3}$

(b) $\displaystyle{z = \log{\frac{y}{x}}, x = (r-1)^2 + s^2, y = (r+1)^2 + s^2}$

(c) $\displaystyle{z = \sqrt{x^{2} + y^{2}}, x = r\cos{s}, y = r\sin{s}, (r > 0)}$

3.
$\displaystyle{z = f(x,y), x = r\cos{\theta}, y = r\sin{\theta}}$ のとき,次の式が成り立つことを示そう.

$\displaystyle z_{r} = z_{x}\cos{\theta} + z_{y}\sin{\theta},  z_{\theta} = r(-z_{x}\sin{\theta} + z_{y}\cos{\theta}). $