Higher Order Derivatives

If $$\frac{\partial f}{\partial x_{j} }$$ is difinied, it is also a real valued function of n variables. The partial derivatives of $$\frac{\partial f}{\partial x_{1} } ,\ldots , \frac{\partial f}{\partial x_{n}}$$ are called second order partial derivatives (二階偏導函數) of f. It is customary to write $$\frac{\partial^{2} f}{\partial x_{j}\partial x_{k}}$$ or $$f_{x_{k} x_{j} }$$ for $$\frac{\partial}{\partial x_{j} } \left( \frac{\partial f}{\partial x_{k} } \right)$$. Derivatives of orders higher than 2 are definied similarly. Thus,

$$\frac{\partial^{m} f}{\partial x_{k_{1}} \partial x_{k_{2}} \... ...1true cm{\hfill} f_{x_{k_{m} }x_{k_{m-1} }\cdots x_{k_{1} } }$$

both denote the m th order partial derivative $$\frac{\partial}{\partial x_{k_{1} } } \cdot \frac{\partial}{\partial x_{k_{2} } } \cdots \left( \frac{\partial f}{\partial x_{k_{m} } }\right)$$.

我們舉兩個簡單的例子如下:

例 9   令 $f(x,y)=-\cos(x^{2},y)$. 則

$$f_{x} =2xy\sin(x^{2}y), \hbox to 1.5truecm{\hfill} f_{y}=sin(x^{2}y).$$

於是

$$\begin{eqnarray*}f_{xx} & = & 2y\sin{(x^{2} y)}+4x^{2} y^{2}\cos{(x^{2}y)}, \\ ... ...+2x^{3} y\cos{(x^{2}y)}, \\ f_{yy} & = & x^{4}\cos{(x^{2}y)}. \end{eqnarray*}$$

例 10   令 f(x, y, z)=x³y²z. 則 f_xy=f_yx=6xyz, 且 f_xyz, f_xzy, f_yzx, f_yxz, f_zxy, f_zyx 均等於 6x²y.

在這兩個例子裡都有 f_xy=f_yx, 而且關於不同變數的混合偏微分均似與變數的次序無關. 這項觀察大體不錯, 因為我們有下定理:

證明. 選擇 (x,y) 使 $x\neq a$, $y\neq b$. 令

Q=f(x,y)-f(a,y)-f(x,b)+f(a,b)

先暫時固定 x, 定義 g(y)=f(x,y)-f(a,y). 則由均值定理得

$$Q=g(y)-g(b)=g'(\eta)(y-b)=[f_{y}(x,\eta )-f_{y }(a,\eta )](y-b)$$

式中 $\eta$ 之值在 y 和 b 之間. 現在把 x 當成變數, 再用一次均值定理, 乃得

$$Q=f_{yx} (\xi ,\eta )(x-a)(y-b)$$

式中 $\xi$ 之值在 x 和 a 之間.

仿此先固定 y, 定義 h(x)=f(x, y)-f(x, b). 引用均值定理兩次乃得

$$Q=f_{xy}(\xi ', \eta ')(x-a)(y-b),$$

式中 $\xi'$ 和 $\eta'$ 之值分別在 x, a 和 y, b 之間. 比較兩個 Q 式, 遂有 $f_{yx} (\xi,\eta )=f_{xy}(\xi ', \eta ').$ 令 $(x,y) \rightarrow (a,b)$, 由 f_xy 和 f_yx 的連續性知

$$f_{yx} (a,b)=\lim_{(x,y) \rightarrow (a,b)} f(\xi,\eta)=\lim_{(x,y) \rightarrow (a, b)} f_{xy} f(\xi' , \eta ')=f_{xy}(a,b).$$

明所欲證.

從以上的討論我們大體上可以相信 f_yx=f_xy 一般都成立了. 但事實上也有二者不相等的例子:

例 11   令

$$f(x,y)=\left\{ \begin{array}{ll} \displaystyle\frac{xy(x^{2}-... ...{\smallskip } 0 & \mbox{if } (x, y)=(0, 0) \end{array}\right.$$

易證 f 在整個平面上連續. 若 $(x, y)\neq(0, 0)$ 則

$$f_{x}(x, y)=\frac{4x^{2}y^{3}+x^{4}y-y^{5}}{(x^{2}+y^{2})^{2}},\hbox to 1truecm{\hfill} f_{x}(0, 0)=0$$

從而知 f_x 在整個平面上連續. 因 f(y, x)=-f(x, y), 故有

$$f_{y}(x, y)=-f_{y}(y, x)=-\frac{4x^{2}y^{3}+x^{4}y-x^{5}}{(x^{2}+y^{2})^{2}}, \hbox to 1truecm{\hfill} f_{y}(0, 0)=0$$

f_x 也在整個平面上連續. 由以上的結果很容易算出

$$f_{xy}(0, 0)=-1,\quad f_{yx}(0, 0)=1 .$$

習$\quad$題

1.

Show that the minimal distance from a point $y\in \mathbb{R} ^n$to the hyperplane through a point p with an unit normal vector u is given by $\vert (y-p)\cdot u \vert$.

2.

A curve which satisfies an equation of the form

Ax²+Bxy+Cy²+Dx+Ey+F=0.

is called a conic. Assume that this conic passes through the point (a,b). Show that the tangent line to this conic at (a,b) is given by

$$Aax+B\frac {bx+ay}{2}+Cby+D\frac {x+a}{2}+E\frac {y+b}{2}+F=0.$$

3.

Let $u\in \mathbb{R} ^n, \prod=\{v\in \mathbb{R} ^n:u\cdot v=0\}$. If $w\cdot v=0$ for all $v\in \prod$, show that $w=\lambda u$ for some $\lambda \in \mathbb{R}$.

4.

Show that every tangent plane to cone

z²=x²+y².

intersects the cone in a straight line.

5.

試計算下列三函數的各二階偏導函數:(a) $$\sin(x-y)+\sin(x+y)$$,     (b) $$\arcsin \frac {x}{\sqrt {x^2+y^2}}$$,     (c) $$\cosh\ln(xy)$$.

6.

Let u=x²siny+y²sinx. Find $$\frac {\partial ^6u}{\partial x^3\partial y^3}$$.

7.

設 $(x, y)\neq(0, 0)$. 令 $u=ln \sqrt{x^2+y^2}$. 試證

$$\frac {\partial ^2u}{\partial x^2}+\frac {\partial ^2u}{\partial y^2}=0.$$

8.

設 $(x,y,z)\neq (0,0,0)$. 令 $u=\frac {1}{\sqrt{x^2+y^2+z^2} }$. 試證

$$\frac {\partial ^2u}{\partial x^2}+\frac {\partial ^2u}{\partial y^2}+\frac {\partial ^2u}{\partial z^2}=0.$$

9.

定義函數 $u(x,t)=e^{-an^2t}\sin \,nx$, 式中 a 和 n 為常數. 試證明 u 滿足微分方程式 u_t=au_xx.

10.

(a) Let F(t)=f(x,y) where $f(x,y)=e^x\cos y$, x=t², y=1-t³. Compute F'(t) and F''(t) first by substitution and then by the chain rule.
(b) Do the same thing with f(x,y,z)=e^x2+y2+z2, $x=\cos t$, $y=\sin t$, z=t.

11.

若 $u=u(x,y), x=r\cos \theta, y=r\sin \theta$, 試證明

$$u_{xx}+u_{yy}=u_{rr}+\frac {1}{r}u_r+\frac {1}{r^2}u_{\theta \theta}.$$

12.

若 u=u(x, y, z), $x=\rho \cos\theta \cos\varphi$, $y=\rho \sin\theta \cos\varphi$, $z=\rho \sin\varphi$, 試將 u_xx+ u_yy+u_zz 用關於 $\rho$, $\theta$和 $\varphi$ 的偏導函數表出.