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

確認問題

1.

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

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

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

2.

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

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

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

演習問題

1.

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

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

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

2.

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

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

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

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

3.

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

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