And what's fallen out when we do that, because in each case, one of the first derivatives gives us the entropy.
当我们这样做时就得到了结果,因为在这些例子中,一阶导数是熵。
And all this is, is saying that when you take a mixed second derivative, it doesn't matter in which order you take the two derivatives.
麦克斯韦关系的本质是,当你考虑混合的二阶导数时,求导的顺序不影响最后的结果,现在,我们利用这些关系。
If you knew only the third derivative of the function, you can have something quadratic in t without changing the outcome.
如果方程里有三阶导数,你就可以引入一个二次项,但是结果却不会变
All right. If the derivative is small, it's not changing, maybe want to take a larger step, but let's not worry about that all right?
好,如果导数很小的话,函数就基本没什么变化,可能我们就想把步子迈大一点儿了,但是别为这个担心?
So here I've written for the hydrogen atom that deceptively simple form of the Schrodinger equation, where we don't actually write out the Hamiltonian operator, but you remember that's a series of second derivatives, so we have a differential equation that were actually dealing with.
这里我写出了,氢原子薛定谔方程的,最简单形式,这里我们实际上,没有写出哈密顿算符,但是请记住那你有,一系列的二次导数,所有我们实际上会处理一个微分方程。
So if we differentiate this object, I'm gonna find a first order condition in a second.
想要求它的导数,先让我想想一阶条件
And, coincidently, what is the partial of energy with respect to distance?
巧合的是,能量对距离的导数是什么?
So I'm hoping you guys are comfortable with the notion of taking one or two or any number of derivatives.
我希望你们,能习惯求一阶,二阶,或者任意阶导数的概念
It's constant pressure. OK, so now, last time you looked at the Joule expansion to teach you how to relate derivatives like du/dV.
这是恒压的,好,上节课你们,学习了焦耳定律,以及怎样进行导数间的变换。
So, all I want to do now is look at the derivatives of the free energies with respect to temperature and volume and pressure.
我现在所要做的一切就是,考察自由能对,温度,体积和压强的偏导数。
What's a function, what's a derivative, what's a second derivative, how to take derivatives of elementary functions, how to do elementary integrals.
什么是函数,什么是导数,什么是二阶导数,如何对初等函数求导,如何进行初等积分
I tell you something about the second derivative of a function and ask you what is the function.
我告诉你一个函数的二阶导数,然后问你这个函数是什么
That is, it's easy to write down straight away that dG with respect to temperature at constant pressure S is minus S.
这就是说,可以很简单的写出dG在,恒定压强下对温度的偏导数,是负。
One way to think about this intuitively if the derivative is very large the function is changing quickly, and therefore we want to take small steps.
关于这个方法很直观的一点想法是,如果导数非常大,函数也就变化的非常快,因此我们想一小步一小步的来。
It's three different second derivatives in terms of the three different parameters.
它是用三个不同常数表示的,三个不同的二阶导数。
But, of course, it's going to come from the fact that these second derivatives are also equal.
但是,结果同样是依赖于,二阶混合偏导数相等。
Then the second derivative gives the change in entropy with respect to the variable that we're differentiating, with respect to which is either pressure or volume.
二阶导数给出熵,随着变量变化的情况,这些变量包括压强或者体积。
Now, I want to do one concrete problem where you will see how to use these derivatives.
现在我要解决一个具体的问题,从中你会明白如何运用这些导数
The third derivative, unfortunately, was never given a name, and I don't know why.
遗憾的是三阶导数没有专门的名称,我也不知道这是为什么
Of course, you can take a function and take derivatives any number of times.
当然,你可以随意拿一个函数,对它求任意阶的导数
I take a look at the second derivative, which is the second order condition.
我需要求出二阶导数,即需要寻找二阶条件
At my maximum, I'll put a hat over it to indicate this is the argmax; at my maximum I'm going to set this thing equal to 0.
给每个最大值都标注上一个帽来,在最大值处导数方程等于0
So now we have this derivative, in terms of physical quantities, things that we can measure.
现在我们得到了这个导数,它可以用实验可测的与。
Let's sneak in one more derivative here, which is to take the derivative of the derivative.
我们再求一次导数,也就是对导数求导
And, of course, see that either way we do that we'll have an equality.
利用两种不同的顺序求二阶导数,就可以得到一个等式。
And all we did, further, is take that second derivative.
总而言之,我们所做的就是求二阶导数。
So the slope of the guess is the first derivative.
因此斜率等于此处的一阶导数。
Then you're supposed to know derivatives of simple functions like sines and cosines.
你应该知道一些简单函数的导数,比如正弦函数和余弦函数
To find a maximum I want the second derivative to be negative.
最大值处的二阶导数是负数
Now let's take it in the other order.
我们用另一种顺序求二阶偏导数。
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