Essay -Partial differentiation

偏微分是把一個多變量的函數取其中一個變量微分，而其他變量保持恆定。

例子：
z = x2 + xy + y2
∂z/∂x = 2x + y
∂z/∂y = x + 2y


偏微分的來由？為什麼要發展偏微分？

真正的原因恐怕連數學家都無法知道，不過我估計發展偏微分的原因最可能與坐標幾何有關。

首先我們知道對於一個函數y = f(x)來說，用坐標來表示，它是一條曲線。而dy/dx是指在y = f(x)上任何一點的切線的斜率。

但是對於一個函數z = f(x, y)來說，用坐標來表示，它是一個曲面。但由於曲面上的每一點都有無窮多條切線，因此描述這種函數的微分相當困難。而通常我們最感興趣的是平行於其中一軸的切線，那麼我們有甚麼辦法去求出該切線的斜率？

事實上，我們會這樣做的：

首先把該曲面在要求的一點並平衡於該軸的方向切開，其切口自然是一條曲線，然後求該切口上在要求的一點的切線的斜率。

「把該曲面在要求的一點並平衡於該軸的方向切開，然後求該切口上在要求的一點的切線的斜率」這套動作即是把該軸的變量設為常數，然後取其切口的方程進行微分，這正是「偏微分」的精神。當然這個精神可以推廣至更多變數數量的函數。

因此我相信「偏微分」這個概念就是這樣發展出來。


偏微分的應用：

Non-constraints-containing Optimization

If we want to optimize y where y = f(x1, x2, x3, ...... , xn) , the way is to solve this simultaneous equation:
　╭
　│∂y/∂x1 = 0
　│∂y/∂x2 = 0
　│∂y/∂x3 = 0
─┤：
　│：
　│：
　│∂y/∂xn = 0
　╰

這樣的做法是可以理解的，因為對於一個多變量的函數來說，如果位於該函數的一點是極點的話，那麼在該點無論平衡於任何一軸的方向切開，該點也必然是任何一個方向的切口的極點。

例子：
http://hk.knowledge.yahoo.com/question/question?qid=7009061500801中dndmmokaa的回答


Constraints-containing Optimization

If we want to optimize f(x1, x2, x3, ...... , xn) subject to the below constraints:
g1(x1, x2, x3, ...... , xn) = 0
g2(x1, x2, x3, ...... , xn) = 0
g3(x1, x2, x3, ...... , xn) = 0
：
：
：
gk(x1, x2, x3, ...... , xn) = 0

The way is to optimize y = f(x1, x2, x3, ...... , xn) + Σ(i = 1 to k) λi gi(x1, x2, x3, ...... , xn) , and solve this simultaneous equation:
　╭
　│∂y/∂x1 = 0
　│∂y/∂x2 = 0
　│∂y/∂x3 = 0
　│：
　│：
　│：
＿│∂y/∂xn = 0
　│∂y/∂λ1 = 0
　│∂y/∂λ2 = 0
　│∂y/∂λ3 = 0
　│：
　│：
　│：
　│∂y/∂λk = 0
　╰

Note: This method is also known as Method of Lagrange multipliers.

例子：
http://hk.knowledge.yahoo.com/question/question?qid=7009011600954中myisland8132的回答
http://hk.knowledge.yahoo.com/question/question?qid=7009061500801中nelsonywm2000的回答

2009-07-08 06:17:40 補充：
Linear programming and Nonlinear programming

2009-07-08 06:20:30 補充：
Linear programming and Nonlinear programming

2009-07-08 06:31:52 補充：
Linear programming and Nonlinear programming

Although there are many other methods to do linear programming and nonlinear programming, they can also be done by the method which are as same as the method to do constraints-containing optimization,

2009-07-08 06:34:59 補充：
because the situation of linear programming and nonlinear programming are similar to that of constraints-containing optimization. The only difference is that some or all of the constraint(s) in linear programming or nonlinear programming is(are) in inequality relation(s).

2009-07-08 06:35:49 補充：
However, this is not a problem since the handling method of the constraint(s) in inequality relation can be tread as the same as the constraint(s) in equality relation(s).

2009-07-08 06:36:09 補充：
However, the method to do constraints-containing optimization is not suitable to do integer programming (i.e. some or all of the variables are restricted to be integer(s).) or some or all of the constraint(s) cannot be expressed as function(s).

2009-07-08 06:37:14 補充：
Calculus of Variations

我不在這裡介紹，詳情請見http://en.wikipedia.org/wiki/Calculus_of_variations。

例子：
http://hk.knowledge.yahoo.com/question/question?qid=7009061500801中myisland8132的回答