Thursday, June 25, 2015

▽读Nabla,奈不拉 △二次函数根的判别式或者指三角形 The volatility indices measure the implied volatility for a basket of put and call options ... A daily number would be found by dividing by the square root of 252

增量资金被堵在门外,依靠这溃不成军的存量,想再拉出几根长阳的牛气,未免心有余而力不足。

entropy crowd ∇ Nabla operator or gradient to turn a "landscape" into a directed force (a force field). T is a kind of temperature, that defines the overall strength (available resources) the intelligent system has (heat can do work, think of a steam engine: the more heat the more power).  Sτ is the "freedom of action" of each state that can be reached by the intelligence within a time horizon τ (tau).

 

△二次函数根的判别式或者指三角形
▽读Nabla,奈不拉,也可以读作“Del” 这是场论中的符号,是矢量微分算符。 高等数学中的梯度,散度,旋度都会用到这个算符。 其二阶导数中旋度的散度又称Laplace算符

 

倒三角和正三角

(2011-11-02 10:59:04)

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杂谈

▽ 是梯度算子(在空间各方向上的全微分),
 △二次函数根的判别式或者指三角形 ▽读Nabla,奈不拉,也可以读作“Del” 这是场论中的符号,是矢量微分算符。 高等数学中的梯度,散度,旋度都会用到这个算符。 其二阶导数中旋度的散度又称Laplace算符 
 
 
梯度
 1 散度 δP/δx + δQ/δy + δR/δz 叫做向量场 A 的散度,记作 div A,即 div A = δP/δx + δQ/δy + δR/δz 2 梯度 在二元函数的情形,设函数z=f(x,y)在平面区域D内具有一阶连续偏导数,则对于每一点P(x,y)∈D,都可以定出一个向量 (δf/x)*i+(δf/y)*j 这向量称为函数z=f(x,y)在点P(x,y)的梯度,记作gradf(x,y) 类似的对三元函数也可以定义一个:(δf/x)*i+(δf/y)*j+(δf/z)*k 记为grad[f(x,y,z)] 
i,j是方向

物理学计算中经常出现一个正三角或倒三角,是什么算符?_ ...

wenda.tianya.cn/question/5790b391daa8e85b
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2009年8月14日 - 正三角形是在高中物理上经常出现的一个符号,它是希腊字母,读作:delta,它表示的是某个物理量的变化。例如: Δv=v2-v1. Δt=t2-t1 而倒三角形是在高等 ...

Volatility and the Square Root of Time


Friday, April 17th, 2015 | Vance Harwood
It’s not obvious (at least to me) that volatility theoretically scales with the square root of time (sqrt[t]).  For example if the market’s daily volatility is 0.5%, then the volatility for two days should be the square root of 2 times the daily volatility (0.5% * 1.414 = 0.707%), or for a 5 day stretch 0.5% * sqrt(5) = 1.118%.
This relationship holds for ATM option prices too.  With the Black and Scholes model if an option due to expire in 30 days has a price of $1, then the 60 day option with the same strike price and implied volatility should be priced at sqrt (60/30) = $1 * 1.4142 = $1.4142  (assuming zero interest rates and no dividends).


Free Will and the Square Root of Entropy | Multisense Realism

multisenserealism.com/2013/12/.../free-will-and-the-square-root-of-entropy...
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    by K Marton - ‎2010 - ‎Cited by 3 - ‎Related articles
    We correct an error in the paper identified in the title. The error affects the factor before the square root of entropy in Theorem 1; the same factor appears in ...

    1. out the last of my shares here. $3 bounce off lows. It was just a day trade. So +$570 in whole trade. Wish I covered balance lower
    1. covered 100 shares $165.65 ( was out in the garage when i got the alert fill ) +$500 in the trade. Still hold 100 shares short
    1. covered a partial to give me some room to breathe on the remaining shares. $168.61 cover +$100 still hold 200 shares.
    1. short $169.20 - the dip buyers on the offering are underwater. This will flush fast on new lows.
  • Turn VIX into information you can use


    Most people think of VIX as simply an index.  This makes sense — the “I” in VIX stands for that very word.  But VIX is more useful than your average index.  It could easily be grouped with economic indicators, like the unemployment percentage or new home sales.  Why?  Because the VIX level — not just the changes in that level — contains valuable information.  In this post I will show you how, with some high school math, you can extract this information and arm yourself with knowledge that will inform your investment decisions.
    VIX is not your typical indexWhen an index provider starts calculating a new index, they will choose a starting point, a base level.  This can be any number —  100, 1000, or even a million.  Investors correctly don’t pay much attention to that.  It’s the percentage changes in this index, driven by the increases and decreases in value of the securities it comprises, which are of most value to the market.
    In the case of VIX, movements up and down are important as well.  However, the actual VIX level means something too.  It communicates the 30-day implied volatility of the S&P 500.  In a previous post, I explained what this means.
    But how can we arrive at the 30-day implied volatility?  How do we convert the current VIX level into this information?  This isn’t hard, but it does require some simple math.  Here are the two steps.
    1. Turn the VIX level into a percentage.  So, a VIX level of 15 equates 15%.  This is the annualized implied 30-day volatility for the S&P 500.
    2. Deannualize VIX to turn it back into its true monthly measure.  The formula for this is to divide the VIX level, as a percentage, by the square root of 12, the number of months in a year.
    This table shows how VIX levels translate into 30-day implied volatility figures using this approach.

    You may wonder why we would divide by the square root of 12.  My advice is to put this deep question aside and to concentrate on the basics.  For now, it is sufficient to know that you can back out the 30-day implied volatility from the VIX level with this simple formula.
    Understanding what “volatility” is in the context of VIXThe implied 30-day volatility of the S&P 500 tells you the likely range of possible index levels the market “expects” in a month.  If the implied 30-day volatility is 4.3% — which equates a VIX level of 15 — then this means that the market expects the index to have a return that is within a range that spans 4.3% higher and 4.3% lower than the index level.*
    We say “expects” because the market isn’t sure and is instead communicating what will probably happen.  You will remember from your basic statistics courses that when someone brings up probability, a bell curve soon follows.  This blog post is no exception.
    Below is a normal distribution, the most common type of probability distribution that supposes that most data points will fall near the average.  On this type of curve, one standard deviation comprises about 68% of likely outcomes.  In our case, the data points in the bell curve are possible index levels the S&P 500 could arrive at 30 days from now.  The 30-day implied “volatility” that VIX conveys is one standard deviation in the bell curve, which is centered on the index level.

    Putting this knowledge into practiceNow that you know the basics, let’s use them.  Assume that today, the S&P 500 level is 1710 and the VIX level is 18.  What additional information can you extract from these two data points?
    As a first step, turn the VIX level into the 30-day implied volatility:
    Turn VIX into a percentage: VIX = 18 = 18%
    Then deannualize it:  18% / √12 = 5.20%
    Second, apply this percentage to the index level:
    1710 x 5.20% ≈ 89
    Third, subtract and add this amount to the index level to tell you the range the market expects the S&P 500 to trade in 30 days from now, with a 68% confidence level:
    1710 + 89 = 1799
    1710 – 89 =  1621
    The range between these numbers, 1621 to 1799, is where the market expects the S&P 500 to trade in 30 days, with a reasonably high level of confidence.  You may agree or disagree with this, but this is what the market believes, and this is a powerful starting point around which to build an investment strategy.
    * Using the current index level is a helpful shortcut.  In reality, the probability curve is centered around the forward price of the index, which is the current index level adjusted for various payments the investor hypothetically might have received or lost, including interest, dividends, and income from stock lending.  Making these adjustments matters less if you are only looking ahead a short time and if interest rates and dividends are low.
    - See more at: http://www.indexologyblog.com/2013/10/18/turn-vix-into-information-you-can-use/#sthash.Wydld9jm.dpuf



    Volatility Index (VIX) [ChartSchool] - StockCharts.com

    stockcharts.com/school/doku.php?id=chart_school...volatility_index
    The volatility indices measure the implied volatility for a basket of put and call options ... A daily number would be found by dividing by the square root of 252 ... VIX levels with the expected volatility in the S&P 500 on a monthly or daily basis.
     
     
    The resulting VIX provides us with the weighted 30-day standard deviation of annual movement in the S&P 500. A reading of 20% would expect a 20% move, up or down, in the next 12 months. This annualized number can be transformed into a monthly number by dividing it by the square root of 12 (~3.464). A daily number would be found by dividing by the square root of 252 (~15.874), which is the number of trading days in a year. The table below shows VIX levels with the expected volatility in the S&P 500 on a monthly or daily basis. Keep in mind that we are talking about volatility, not the expected return or change.



    统计物理学中的“最小作用量原理” - 心情随笔网

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    经典力学中力学系统的运动规律的最一般表述由“最小作用量原理”给出。我觉得,“最小作用量 ... 因此我们说,分布函数是关于力学不变量的函数。我们选择能量就成为 ...
  • 关于最小作用量原理2_百度文库

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    2010年12月12日 - 物理学关于最小作用量原理刘大为(甘肃联合大学理工学院, ... 作用量函数L=一mc^/1一%=一mc癣(4) ~ C' 质点动量1一曲一劬一廊。 p一照 ...
  • 作用量,熵(体系,理想氣體)狀態函數,路徑無關;最小作用量原理 ...

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  • 熵- 维基百科,自由的百科全书

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    熵亦被用於計算一個系統中的失序現象,也就是計算該系統混亂的程度。熵是一个描述系统状态的函数,但是经常用熵的参考值和变化量进行分析比较,它在控制论、 ...
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    关于最小作用量原理-1对于三种基本原理目前被认知的普遍程度的评述任何自然事物的变化 ... 沿用普通分析力学中的研究思路,引入两个新广义动量和新Hamil-ton函数,将保守系统的四 ... 从数学角度看:(2)式给出的作用量是定义于集{L(q,q,t)}上的泛函.
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    2014年10月6日 - Nottale的标度相对论主要分成两大块,一块是纯粹的相对论,他仿照爱因 ... 信息熵是整个系统的平均消息量,即: ... 其中s是作用量,ψ是波函数。 )
  • 王彬

    survivor99.com/entropy/paper/p50.htm
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    信息熵是几率的函数,自然与几率密度----分布函数建立了对应关系. ... 力学的最小作用量原理指出,质点运动的真实轨道是作用量取极小值的轨道.光学的费马原理 ...
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    系统的拉氏量L可以减去“任意函数f对时间的全导数”而不改变其物理演化。 2. 场论中的作用量密度L是t,x,y,z的函数,经典力学中的作用量是t的函数. 3. 四维磁矢势A ...
  • [PDF]咬文嚼字027-熵非商—theMythofEntropy - 中国科学院物理 ...

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    由 曹则贤 著作 - ‎被引用 1 次 - ‎相關文章
    相联系的,实际上温度是熵关于能量的共轭,但温度. 并不总是 ... 个关于热力学体系的态函数,用来表述热力学第二 ..... 本来就有关于作用量(不过那里不叫作用量,而是被.




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    2011年6月10日 - ?Tad ad 比热容测量法:磁比热~温度曲线?熵~温度曲线? ?T ?S M 7.2 磁制冷技术1、磁制冷实现的过程(1)等温磁化过程(2)绝热去磁过程(3)等温 ...
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    ... 相起伏就是晶核的胚芽,称为晶胚。 rmax 过冷度△T. 6.2.2. 晶体凝固的热力学条件. 一、自由能和温度曲线熵的物理意义是表征系统中原子排列混乱程度的参数。
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  • A new Equation for Intelligence F = T ∇ Sτ - a Force that Maximises the Future Freedom of Action

    Intelligence is a Force with the Power to Change the World


    Describing intelligence as a physical force that maximises the future freedom of action, adds a new aspect to intelligence that is often forgotten: the power to change the world. This, I think, was the biggest revelation for me, when I started thinking about the the new equation for intelligence. The second revelation was, that intelligent systems are survival engines, that increase their chances of survival by maximising a single quantity: the freedom of action. Both insights may sound trivial or obvious, but I don't think they are.

    A few days ago a saw the TED talk "A new equation for intelligence" by Alex Wissner-Gross. He presents an equation he published in April 2013 in a physics journal. It may not be the most impressive talk I have ever seen. And I had to watch it twice to fully understand it. But the message excites me so much, that I don't sleep well since a few days. I thought everybody must be excited about this equation. But, it seems that this is not the case. Either I am not understanding it correctly or others don't get it. Or maybe it resonates with me, because I am physicist, with a strong background in computing, who has done research in computational biology. To find this out, let me explain my understanding of the equation. Please tell what your think and what's wrong with my excitement (I need sleep)....

    So, why did the equation blow me away? Because this very simple physical equation can guide us in our decisions and it makes intelligent behaviour measurable and observable. It adds a new real physical force to the world, the force of intelligence. From the equation we can deduce algorithms to act intelligently, as individuals, as societies and as mankind. And we can build intelligent machines using the equation. Yes, I know, you may ask: "How can the simple equation F = T  Sτ do all of that?"

    Intelligence is a Force that Maximises the Future Freedom of Action

    Before we look at the equation in more detail, let me describe its essence in every day terms. Like many physical laws or equations the idea behind it is simple:
    • Intelligence is a force that maximises the future freedom of action.
    • It is a force to keeps options open.
    • Intelligence doesn't like to be trapped.
    But what is necessary to keep options open and not to be trapped? Intelligence has to to predict the future and change the world in a direction that leads to the "best possible future".  In order to predict the future, an intelligent system has to observe the world and create a model of the world. Since the future is not deterministic the prediction has to be based on some heuristics. Prediction is a kind of statistical process. In order to change the world, the intelligence has to interact with the world. Just thinking about the world, without acting, is not intelligence, because it produces no measurable force (well, sometimes it is intelligent not to act, because the physical forces drive you already in the right direction, but that is a way of optimising resources). The better in can predicted the future and the better a it can change the world in the desired direction, the more intelligent the system is.

    The new Equation for Intelligence F = T ∇ Sτ

    Note: skip this section, if you are not interested in understanding the mathematics of the equation!


    This is the equation:

       F = T

    Where F is the force, a directed force (therefore it is bold), T is a system temperature, Sτ is the entropy field of all states reachable in the time horizon τ (tau). Finally, ∇ is the nabla operator. This is the gradient operator that "points" into the direction of the state with the most freedom of action. If you are not a physicist this might sound like nonsense. Before I try to explain the equation in more detail, let's look at a another physical equation of force.

    The intelligence equation very similar to the equation for potential energy F = ∇ Wpot. Wpot is the potential energy at each point is space. The force F pulls into the direction of lower energy. This is why gravitation pulls us in direction of the center of the earth. Or think of a landscape. At each point the force points downhill. The direction is the direction a ball would roll starting at that point. The strength of the force is determined by the steepness of the slope. The steeper the slope, the stronger the force. Like the ball is pulled downhill by the gravitational force to reach the state with the lowest energy, an intelligent system is pulled by the force of intelligence into a future with lowest number of limitations. In physics we use the  Nabla operator or gradient to turn a "landscape" into a directed force (a force field).

    Back to our equation F = T  Sτ. What it says is that intelligence is a directed force F that pulls into the direction of states with more freedom of action. T is a kind of temperature, that defines the overall strength (available resources) the intelligent system has (heat can do work, think of a steam engine: the more heat the more power).   is the "freedom of action" of each state that can be reached by the intelligence within a time horizon τ (tau).  The time horizon is how far into future the intelligence can predict. Alex Wissner-Gross uses the notion of entropy S to express the freedom of action in the future. The force of intelligence is pointing into that direction. As we have seen, in physics the direction of the force at each state is calculated by a gradient operation ∇ (think of the direction the ball is pulled). The Nabla operator ∇ is used to assign a directional vector (the direction of the force of intelligence) to each state (in our case: all possible future states). The more freedom of action a state provides the stronger the force is pulling in that direction. So, Sτ is the pointing into the direction with the most freedom of action. The multiplication with T means the more power we have to act, the stronger the force can be.

    Note: the optimal future state is the optimal state form the viewpoint of the intelligent system. It might not the optimal state for other systems or for the entire system.

    If you want to understand the equation in more detail read the original paper 'Causal Entropic Forces - by A. D. Wissner-Gross and C. E. Freer'.


    Understanding the Laplace operator conceptually

    The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
    Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend?
    share|improve this question
        
        
    @JohnM You are very right! I feel terrible for asking a duplicate now. Can it be merged? Should it be made into a community wiki? –  user89 Jun 3 '14 at 5:05
        
    I just wanted to point out those other answers in case they are of interest to you. –  John M Jun 3 '14 at 15:44

    5 Answers 5


    up vote38down voteaccepted
    +50
    The Laplacian Δf(p)  is the lowest-order measurement of how f  deviates from f(p)  "on average" - you can interpret this either probabilistically (expected change in f  as you take a random walk) or geometrically (the change in the average of f  over balls centred at p  ). To make this second interpretation precise, write the Taylor series

    f(p+x)=f(p)+f i (p)x i +12 f ij (p)x i x j + 

    and integrate:

     B r (p) f=f(p)V(B r )+f i (p) B r (0) x i dx+12 f ij (p) B r (0) x i x j dx+. 

    The integrals x i dx  vanish because x i   is an odd function under reflection in the x i   direction, and similarly the integrals x i x j dx  vanish whenever ij  ; so this simplifies to

    1V(B r )  B r (p) f=f(p)+CΔf(p)r 2 + 

    where C  is a constant depending only on the dimension.
    The Laplace-Beltrami operator is essentially the same thing in the more general Riemannian setting - all the nasty curvy terms will be higher order, so the same formula should hold.
    share|improve this answer
        
    Dear Anthony, I have accepted your answer, but wonder if you could write a little about why the Laplace operator is sometimes called the "diffusion operator"? Also, do I understand it correctly that in only one "spatial dimension" x  , it simply reduces to the second derivative? I.e. u xx =Δu  ? –  user89 May 28 '14 at 8:35
    3  
    @user89: Correct on the second question. Diffusion is a process that smooths out some density function by changing the value at each point to be closer to the values at surrounding points, and is modelled by the equation f/t=Δf  . –  Anthony Carapetis May 28 '14 at 10:08
        
    +1 for "all the nasty curvy terms will be higher order" –  Neal Jun 12 '14 at 13:41

    I think the most important property of the Laplace operator Δ  is that it is invariant under rotations. In fact, if a differential operator on Euclidean space is rotation and translation invariant, then it must be a polynomial in Δ  . That is why it is of such prominence in physical problems.
    Some good books on the subject:
    1. Rosenberg's The Laplacian on a Riemannian Manifold.
    2. Gurarie's Symmetries and Laplacians.
    share|improve this answer
        
    Is the Laplacian invariant under rotations because it takes into account all points in the neighbourhood of the point in question? –  user89 May 28 '14 at 8:32
    3  
    Yes, because it takes into account all the points in a symmetric way. –  John M May 28 '14 at 12:59
        
    John, if I consider the function f(x)=x 2   in 2D, rotating my axes can give me different values for the second derivative at a particular point (the second derivative being what the Laplace operator reduces to in one spatial dimension). This does not seem to be invariant -- why is that? –  user89 May 29 '14 at 0:23
        
    Nice thing to try to check! It is invariant: If f(x,y)=x 2   in R 2   , then Δf=( f /x 2 )+( f /y 2 )=2+0=2  . On the other hand, if you rotate your axes 45 degrees clockwise, you get f(x,y)=x 2 /2+y 2 /2  . This also has Δf=1+1=2  . Rotate you axes another 45 degrees to get f(x,y)=y 2   . Again Δf=0+2=2  . –  John M May 29 '14 at 1:34
        
    Ah. Hmm. I was simply flipping the graph of f(x)=x 2   , so that in some new axis system, it would be f  (x  )=(x 2 )  (i.e. rotating it 180 degrees). Then, the second derivative of that would be 2  . I know I am making an incredibly silly error here, but I can't seem to be able to catch it. –  user89 May 29 '14 at 5:13

    To gain some (very rough) intuition for the Laplacian, I think it's helpful to think of the Laplacian on R  , which is just the second derivative d 2 dx 2    . (This answer may be more elementary than the OP was looking for, but I wish I had kept some of these things in mind when I first learned about the Laplacian.)
    Just as Anthony's answer discusses, the second derivative at pR  measures how much f(p)  deviates from average values of f  on either side of it. If the second derivative is positive, then f(p)  is smaller than the average of f(p+h)  and f(ph)  for small h  . (As I would tell my calculus students, the trapezoid rule for Riemann sums is an overestimate when the second derivative is positive.)
    Generally, a function is harmonic if and only if it satisfies the mean value property. In R  , harmonic functions are simply linear polynomials, which of course are precisely the functions that satisfy the mean value property.
    The maximum principle states roughly that if Δu0  , then local maxima of u  do not occur. This is a generalization of the familiar "second derivative test" from calculus, which says that if the second derivative of u  is positive, then local maxima of u  do not occur (the graph of u  is concave up).
    Finally, let me go up one dimension and mention some of my intuition for harmonic functions u(x,y)  of two variables, in which case Δu= 2 ux 2  + 2 uy 2    . If u  is harmonic, then  2 ux 2  = 2 uy 2    . This says that the graph of u  must always look like a saddle: if, say, the graph is concave up in the x  -direction ( 2 ux 2  >0  ), then it must be concave down in the y  -direction ( 2 uy 2  <0  ). When I picture a saddle-shaped graph in my head, I think I can also see why the maximum principle has to hold for harmonic functions, since a saddle has no local extrema.
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    Another view along the lines of the answers above:
    Suppose you have some region in the plane Ω  , and you are given the value of some scalar function f  along the boundary Ω  . You now want to fill in f  on the interior of Ω  "as smoothly as possible." (A common physical interpretation is that f  is the heat of the region: you are fixing the temperature of the boundary of Ω  and want to know what the temperature on the interior will be at steady state.)
    What does "as smooth as possible" mean? Well, one measure of the smoothness of f  is to look at its gradient f  and measure
    E(f)= Ω f 2 dA. 
    Notice that this integral, called the Dirichlet energy of f  , achieves its lowest possible value of 0  when f  is constant. The less smooth (to first order) that f  is, the higher the Dirichlet energy will be. Making f  as smooth as possible means finding the f  that satisfies the boundary conditions and minimizes E  .
    How do we minimize E  ? We "take the derivative and set it to zero":
     f E=0. 
    It may look a little weird to differentiate a scalar (the Dirichlet energy) with respect to a function, but the idea is the same as when you work with the ordinary gradient. Recall that for an ordinary scalar function g(x,y,z):R 3 R  , the gradient g  at a point is the unique vector that, when you dot it with any direction v  , tells you the directional derivative of g  in that direction:
    g(x,y,z)v=ddt g[(x,y,z)+tv]∣ ∣ ∣  t0 . 
    The gradient of E  works the same way: it gives you the unique function over Ω  that, when you take the inner product of E(f)  with any variation δf  of f  , gives you the directional derivative of E  in that "direction":
     Ω E(f)δfdA=ddt E(f+tδf)∣ ∣ ∣  t0 . 
    You can do the multivariable calculus and after some integration by parts, you will see that E(f)=Δf.  Several takeaways from this:
    • The function f  that interpolates the boundary conditions as smoothly as possible (in the sense of minimizing the Dirichlet energy) is the solution to the Laplace equation Δf=0  .
    • Given some function f  that interpolates the boundary conditions but does not minimize the Dirichlet energy, the gradient of E  , Δf  , is the "direction of steepest ascent" of E  -- the direction to change f  if you want to most quickly increase E  . The negative of this, Δf  , is the direction that most quickly decreases E  : if you are trying to smooth f  , this is then the direction that you want to flow f  in. This insight leads to the heat equation
      dfdt =Δf 
      which, given initial temperatures on Ω  , flows in the direction that best decreases the Dirichlet energy until the heat has diffused as smoothly as possible over the surface.
    • Nowhere in the above discussion was it essential that Ω  was a piece of a plane: as long as you can define functions on Ω  and take gradients of f  to get the Dirichlet energy, the above works equally well, and is one way of motivating the Laplace-Beltrami operator on arbitrary manifolds in R 3   . The physical picture here is that you have some conductive plate in empty space, and heat up the boundary of the plate, and look at how the heat equalizes over the plate.
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    It's interesting that some approaches to image denoising / deblurring / restoration minimize an objective function that contains a discrete version of the Dirichlet energy of the restored image, in order to encourage the restored image to be smooth. More sophisticated approaches use a penalty term E(f)= Ω f 1 dA  , which allows sharp edges in an image to be preserved. (It seems interesting to work out a differential equation based on this energy, as you did for the Dirichlet energy.) –  littleO Jun 3 '14 at 6:42

    Here is some intuition:
    I think the most basic thing to know about the Laplacian Δ  is that Δ=div  , and div  is the adjoint of   . Hence, Δ  has the familiar form A T A  which recurs throughout linear algebra. We see that Δ  is a self-adjoint positive semidefinite operator, and so we would expect (or hope) that the familiar properties of positive semidefinite operators in linear algebra hold true for Δ  . Namely, we expect that Δ  has real nonnegative eigenvalues, and that there should exist (in some sense) an orthonormal basis of eigenfunctions for Δ  . This provides some intuition or motivation for the topic of "eigenfunctions of the Laplacian". (By the way, I think the Laplacian should have been defined to be div  .)
    Notice that the integration by parts formula can be interpreted as telling us that ddx   is the adjoint of ddx   (in a setting where boundary terms vanish). Fourier series can be discovered by computing the eigenfunctions of the anti-self-adjoint operator ddx   in an appropriate setting. Moreover, a multivariable integration by parts formula can be interpreted as telling us that div  is the adjoint of   . Green's second identity can be interpreted as expressing the self-adjointness of the Laplacian

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