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

演習問題

1.

$\displaystyle{\frac{dz}{dt}}$ を求めよう．ただし，

は $C^{1}$ 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)

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

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

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