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上節課

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我給大家介紹了

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偏導下的連鎖律

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我們說 如果有一個函數Ψ

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這是希臘字母Ψ 它是x、y的函數

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如果我要求它的偏導數

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關於。。。 不對 我要求導數

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不是偏導 求它關於x的導數

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那就是?Ψ

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除以?x 加上?Ψ

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除以?y 乘以dy/dx

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上一個影片中 我沒有證明

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但我給了大家一種直觀

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所以相信我吧

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但可能某天

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我會嚴格地證明它

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不過如果有興趣的話

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也能在網絡上找到

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偏導下的連鎖律的證明

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放一邊吧

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下面來看看偏導的另一個性質

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這之後 我們就能直觀地感受

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恰當方程了

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因爲你會發現

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這些足夠讓我們去解恰當方程了

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但直覺這東西吧

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好吧 我不想說它有點難

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因爲直覺有了就是有了

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所以 如果有一個函數Ψ

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我要求Ψ的偏導數

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首先是關於x的偏導

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寫下Ψ

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我不用每次都寫上x、y

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然後我求關於y的

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偏導數

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正如記號 可以寫成。。。

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多多少少可以看做

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把操作符（求導符號）相乘

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可以寫成這樣

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上面是?2Ψ

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下面是?y 或者?x

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也可以寫成。。。

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這是我最喜歡的符號

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因爲它沒有多余的符號

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你可以說

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求偏導 先是x

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這意味著 對Ψ求關於x的偏導

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然後求關於y的偏導

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這是其中一種情況

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先求關於x 再求關於y的偏導

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是怎樣做的呢？

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先是關於x

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把y固定 求關於x的偏導

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關於x的 把y忽略

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然後把x固定

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求關於y的偏導

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那交換x和y的順序

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會發生什麽呢？

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會發生的是。。。

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用另一種顏色 寫下Ψ

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然後求偏導

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先是關於y

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然後是關於x 這是什麽呢？

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這只是記號罷了

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大家應該適應了吧

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這是?x和?y

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這是算符

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這裡可能會引起誤會

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這兩個記號

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盡管是一樣的

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但順序變了

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這不過是因爲

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看待事物的方法不一樣

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這是說 先求關於x的偏導 再y

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這看上更像算符

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先求關於x的偏導 然後求關於y的

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就像是算符乘積那樣

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無論怎樣 這也可以寫成

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先是y 然後才是x

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不好意思 關於y

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然後才是關於x的偏導

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現在 我要告訴大家

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如果求偏之後函數都是連續的

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我們處理的

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大部分函數的定義域都是平凡的

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也就是 是連續的 沒有洞的

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函數的定義中也沒有詭異的地方

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它們通常都是連續的

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特別地 在第一年的微積分或微分課程中

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我們處理的

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大部分是連續函數

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定義域是好的

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如果這兩個函數是連續的

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求偏之後還都是連續的

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那它們就是相等的

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Ψxy等於Ψyx

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現在 我們要應用它了

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求偏下的連鎖律

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應用它去解

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一種類型的微分方程

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一階的微分方程

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叫做“恰當方程”

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恰當方程是怎樣的呢？

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它們是這樣的

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選擇顏色真不容易啊

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這是我的微分方程

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關於x和y的函數

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不確定是什麽

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它可能是x2<i>cosy 或者其他</i>

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不確定是什麽 可以是任意x、y的函數

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加上另一個x、y的函數

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稱之爲N 乘以dy/dx之後等於0

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這是。。。

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我不確定是否爲恰當方程

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不過你看到這樣的形式

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首先要做的是。。。

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首先考慮它是否可隔離變量

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你們應該做一些代數練習

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看看變量是否可隔離

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因爲那可以直接解出來

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如果不可隔離

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但還是這樣的形式

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你就會問“喔 這是恰當方程麽？”

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什麽是恰當方程？

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好吧 首先要看

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這裡的形式

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看上去和這裡很相似

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如果M是?Ψ/?x呢？

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Ψx是否就是M呢？

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這是Ψx嗎？

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又如果這是Ψy呢？

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也就是Ψy=N

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如果。。。

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我只是想說 我們並不確定

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如果你偶然在某處看到這個式子

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你不會知道這是否

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是某函數關於x的偏導數

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或者這也是一個偏導數

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某函數關於y的偏導數

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但我們說 如果是呢？

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如果確實是

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我們就可以重新寫成 Ψ

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關於x的偏導 加上Ψ

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關於y的偏導 乘以dy/dx 等於0

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這裡左邊的式子

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和這裡是一樣的 對吧？

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這是Ψ關於x的導數

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用到了偏導下的連鎖律

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所以可以重寫了

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重寫成 這是Ψ關於x的

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導數

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Ψ是關於x、y的函數 等於0

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看這個微分方程

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寫出這樣的形式

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你會說 哎 還是不能隔離變量吧

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但這是一個恰當方程

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顯然

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如果它出現在

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最近的考試中

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那它很可能是一個恰當方程

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但看到這個形式 你會說

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它可能是一個恰當方程

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如果它是一個恰當方程。。。

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告訴大家

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怎樣最快地作出判斷

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然後就可以寫成

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某函數Ψ的導數了

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這是Ψ關於x的偏導

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這是Ψ關於y的偏導

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如果可以寫成這樣

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就可以對兩邊求導。。。

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不對 應該是兩邊取不定積分

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就能得到Ψ(x,y)=C

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是方程的一個解

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有兩件事

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是我們應該關心的

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之後你可能會說 好的 Sal

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考慮過了Ψ 、偏導數 所有的這些

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首先 怎樣知道這是否是一個恰當方程？

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然後 如果是恰當方程

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也就是存在那樣的一個Ψ

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然後怎樣解出Ψ呢？

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所以 判斷是否恰當方程的辦法

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就是利用這個信息

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我們知道 Ψ和它的偏導們

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在定義域上都是連續的

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然後關於x和y

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求偏導數

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在兩種求偏順序下 它們還是一樣的

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所以我們說 這是偏導

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關於x的 對吧？

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這是關於y的偏導

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如果這是恰當方程

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如果它是恰當的

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對它關於y的

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偏導數 對吧？

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對M求關於y的偏導。。。

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也就是Ψx

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等於M

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如果我們對它求關於y的

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偏導數

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可以重寫成這樣

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它是等於

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Nx 對吧？

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Ψ關於y的偏導 是N

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如果我們對兩邊

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求關於x的偏導

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我們知道它們應該是相等的

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如果Ψ和它的偏導都是連續的話

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所以這是相等的

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因此 這其實是判斷

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恰當與否的辦法

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我來重新寫一下

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總結一番

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如果你看到這樣的形式M(x,y)

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加上N（x,y)dy/dx 等於0

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然後就應該 對M求關於y的

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偏導數

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然後對N求關於x的偏導

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它們會是相等的

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這。。。是若且唯若的

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如果滿足的話 它就是恰當方程

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正合微分方程

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這是恰當的

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如果它是恰當方程

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也就告訴了我們 存在一個Ψ

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它的導數等於0

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或者Ψ(x,y)=C

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這是方程的解

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Ψ關於x的偏導

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等於M

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Ψ關於y的偏導

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等於N

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在下一個影片中

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我會告訴大家 怎麽利用這個信息解方程

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這裡我還是要指出某些東西

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這是Ψ關於x的

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偏導數

239
00:10:14,890 --> 00:10:17,620
當我們要做判斷時

240
00:10:17,620 --> 00:10:19,590
要關於y求偏

241
00:10:19,590 --> 00:10:21,080
因爲我們想得到混合導數

242
00:10:21,080 --> 00:10:21,470
同樣地

243
00:10:21,470 --> 00:10:23,410
這是Ψ關於y的

244
00:10:23,410 --> 00:10:27,030
偏導數 但我們要判斷的話

245
00:10:27,030 --> 00:10:29,500
就要取其關於x的偏導

246
00:10:29,500 --> 00:10:30,730
又得到了混合導數

247
00:10:30,730 --> 00:10:32,570
這是關於y的

248
00:10:32,570 --> 00:10:33,920
這是關於x的 得到這個

249
00:10:33,920 --> 00:10:36,300
無論如何 有點複雜

250
00:10:36,300 --> 00:10:38,360
但希望大家能明白我所做的一切

251
00:10:38,360 --> 00:10:41,390
我想 大家應該有了

252
00:10:41,390 --> 00:10:43,470
一種關於恰當方程的直覺

253
00:10:43,470 --> 00:10:45,950
下節課 我教大家

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00:10:45,950 --> 00:10:49,400
解一些恰當方程 下次見啦~