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In other words, now I know how to tell how the Helmholtz free energy changes as a function of temperature.
换句话说，我现在知道如何写出亥姆赫兹,自由能作为温度的函数。
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So, saying wave functions within molecules might sound a little confusing, but remember we spent a lot of time talking about wave functions within atoms, and we know how to describe that, we know that a wave function just means an atomic orbital.
说分子内的波函数可能,听着有点容易搞混，但记住我们花了很多时间,讨论了原子中的波函数，而且我们知道如何去描述它，我们知道波函数意味着原子轨道。
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It's a state function. If we know the enthalpy of the products minus the enthalpy of the reactants, right. It's a state function.
如果我们知道生成物的焓减,去反应物的焓的值,它是个态函数，原则上我们可以做到。
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If you could image how the heart is moving, you would know a lot about its function.
如果你对心脏进行动态造影,你就会了解它的很多功能了
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same point Cartesian to have 1 representation for a point that supports different ways of getting information out, but has gathered within it, a method or a function for dealing with things like how do I know if it's the same point or not.
还有,这是一个比较极坐标的点,是否相同的不同的方法，但是这开始成了一个麻烦事儿了，我真正想要做的事情是,创建一个能支持不同的方式,输出已经收集的信息的点的表示。
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If I give you a function, you know how to take the derivative.
如果我给你一个函数,你就知道如何求导
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As we'll discuss in the class, and most of you probably know this, your body weight is a function of how many calories you take in and how many calories you burn off through metabolic processes but also physical activity.
我们将会讲到,而且你们中的大部分人可能也知道,你的体重随着卡路里的摄取量,和代谢中的消耗量而变化,还跟肢体活动有关
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X So here I'm declaring another variable called X, and this is totally legitimate because I already know that if I'm implementing a function like swap or increment, I can absolutely take input.
这里我声明另一个变量，这个完全是合法的因为我已经知道,如果我执行一个像swap，或，increment的函数，我完全可以携带输入。
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Once we can do that we can go on and say okay, what actually is a wave function, but first we need to know how to describe which ones were talking about.
我们就可以继续并说波函数到底是什么,但首先，我们需要知道，如何描述,我们说的,是哪一个波函数。
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So, what's the idea? If I know what my hash function does, it maps, in this case characters into a range zero to 256, which is zero to 255, I create a list that long, and I simply mark things.
我知道我的哈希函数做什么,在这个例子中，它将字母，映射成0到255的256个数字，我创建了一个这么长的列表，并且我只是简单的标志这些东西。
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So, what we can do to actually get a probability instead of a probability density that we're talking about is to take the wave function squared, which we know is probability density, and multiply it by the volume of that very, very thin spherical shell that we're talking about at distance r.
我们能得到一个概率,而不是概率密度的方法,就是取波函数的平方，也就是概率密度，然后把它乘以一个在ｒ处的,非常非常小的,壳层体积。
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Everyone knows from calculus that if you're trying to find a function about which you know only the derivative, you can always add a constant to one person's answer without changing anything.
学过微积分的人都知道,如果你想根据已知的导数,求出其原函数,你总是可以给某人的答案,随便加一个常数,且不影响结果
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What is this? This is a wonderful piece of abstraction. It is saying, you don't need to know squat about what's inside the body of this function. You don't have to worry about the parameter names, because they're going to be preserved, you don't need to worry about how I'm doing it, this tells you how you can use this, in order to use it correctly. Of course, I can then close it off, and off we go.
但是这能告诉你一些信息,这是什么？,这是一段完美的抽象,它的意思是你不需要知道,这个函数的内部构造，你不必担心参数是否重名，因为它们已经被预存了,你不需要担心我如何使用它,它告诉了你你该如何操作，从而正确的使用它，当然我也可以把它关掉。
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I would say, okay, this guy wants me to find a function which reduces to the number a when I take two derivatives, and I know somewhere here, this result, which says that when I take a derivative, I lose a power of t.
出题者想要我找出一个函数,它在经过两次求导后得到数字a,我知道这里的某个地方,这个结论告诉我,我每求一次导,t就降一次幂
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Now, I hope you guys know that much calculus, that when you take a derivative of a function of a function, namely v square over 2 is a function of v, and v itself is a function of t, then the rule for taking the derivative is first take the v derivative of this object, then take the d by dt of t, which is this one.
我希望你们了解更多的微积分知识,当你对复合函数求导时,也就是说v^/2是关于v的函数,而v本身是关于t的函数,求导的法则应该是,第一步是这一部分对v求导,然后v再对t求导,得到这一部分
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You know that the function of a protein is intimately related to its structure.
蛋白质的功能与结构密切相关
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So, also about Max Born, just to give you a little bit of a trivial pursuit type knowledge, he not only gave us this relationship between wave function squared, This is her grandfather, I don't know if you can see from the eyes, I feel like there's a little bit of a resemblance there.
这里有些，关于它的，花边新闻,他不仅带给我们，这个波函数平方的关系,还给我们带来了，他是她的外祖父，我不知道,你们能不能看出来，我觉得,他们眼睛长得很像。
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It makes sense to draw the wave function as a circle, because we do know that 1 s orbitals are spherically symmetric.
把波函数画成一个圆是有道理的，因为我们知道1s轨道是球对称的。
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So if, in fact, we want to describe a wave function, we know that we need to describe it in terms of all three quantum numbers, and also as a function of our three positional factors, which are r, the radius, phi plus the two angles, theta and phi.
实际上，我们想描述波函数，我们知道我们需要,用这三个量子数来描述它，同样，波函数还是,三个位置变量的函数，它们是r半径，还有两个角度theta和。
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