# Quod Erat Demonstrandum

## 2009/06/21

### 應用純數

Filed under: Additional / Applied Mathematics,HKALE,Pure Mathematics — johnmayhk @ 12:02 上午

$f(\theta)$ = 對應 A, C 兩腿與地面距離之和
$g(\theta)$ = 對應 B, D 兩腿與地面距離之和

$f(\theta)g(\theta) = 0$　$\forall \theta$ – – – – – – (*)

$f(0) = 0$
$g(0) > 0$

$h(\theta) = f(\theta) - g(\theta)$

$h(0) = f(0) - g(0) < 0$

$f(\frac{\pi}{2}) > 0$
$g(\frac{\pi}{2}) = 0$

$h(\frac{\pi}{2}) = f(\frac{\pi}{2}) - g(\frac{\pi}{2}) > 0$

$h(c) = 0$

$f(c) = g(c) = 0$（第二個等式由 (*) 而得）亦即四腿在 $\theta = c$ 之位置著地。

……這例子讓我憶起兒時和弟妹在同一張小桌子上做家課的景象……

$\frac{x^2}{9} + \frac{y^2}{36} + \frac{z^2}{9} = 1$

$\nabla z = \{\frac{\partial z}{\partial x} , \frac{\partial z}{\partial y}\}$

$L_{xy} : f(x,y) = 0$

$L_{xy}$ 的切向量 $\{dx , dy\}$ 應與 $\nabla z$ 平行，故

$\frac{dx}{(\frac{\partial z}{\partial x})} = \frac{dy}{(\frac{\partial z}{\partial y})}$

$\frac{2x}{9}dx + \frac{2y}{36} + \frac{2z}{9}dz = 0$
$dz = -\frac{x}{z}dx - \frac{y}{4z}dy$
$\frac{\partial z}{\partial x} = -\frac{x}{z}$, $\frac{\partial z}{\partial y} = -\frac{y}{4z}$

$\frac{dx}{-\frac{x}{z}} = \frac{dy}{-\frac{y}{4z}}$
$\frac{dx}{x} = \frac{4dy}{y}$

$x = Cy^4$$C$ 由雨滴初始位置決定），因此所求的曲線為

$L: \left \{ \begin{array}{ll} x = Cy^4\\\frac{x^2}{9} + \frac{y^2}{36} + \frac{z^2}{9} = 1\end{array}\right.$

……趁我還未忘記記下……

……無聊聯想到：一分鐘「驗毒」、中學生「驗毒」獎勵計劃……

……我做數也是如此，很人性的感覺吧……

## 4 則迴響 »

1. 第二個例消化中…有D似之前final見過的題目…這不是’有點’out c吧XD.
Multivariable cal的final炒了…

迴響 由 Justin — 2009/06/21 @ 2:17 上午 | 回覆

2. 炒了也沒事吧？

第一例，好像愈看愈有問題…真的可以保證三腳著地嗎？

迴響 由 johnmayhk — 2009/06/22 @ 1:49 下午 | 回覆

3. multi. cal.係core course…炒了+拉低成績…

但如果由對角的兩腳着地應該也可以做到結果吧?不太肯定…
請問介值定理是否即是Bolzano theorem?
另外,第一例(*)的部分是for all pheta?,還是there exist?

迴響 由 Justin — 2009/06/27 @ 2:29 下午 | 回覆

4. Yes, Justin you are right. It seems that we can ensure 2 legs touching the ground instead of 3, and we still have

$f(\theta)g(\theta) = 0$ for ALL $\theta$

And the proof is OK.

Bolzano theorem and intermediate value theorem are “two faces of a coin", just read

http://www.cut-the-knot.org/Generalization/ivt.shtml

If mathematics is your favour, never give up. Study mathematics in university is quite different from secondary level, the pattern “definition-theorem" may be a bit boring without knowing the whole context of development of certain topics. Fed with speedy content, seems know nothing more after a semester…Be patient!

迴響 由 johnmayhk — 2009/06/29 @ 5:27 下午 | 回覆