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

耶鲁公开课 - 基础物理课程节选

But, of course, we want to go all the way to distance, so we take the second derivative and we have this equation for force here.

当然我们最后想要的是距离,所以我们取二阶微,分就有这个式子。

麻省理工公开课 - 化学原理课程节选

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。

麻省理工公开课 - 热力学与动力学课程节选

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

耶鲁公开课 - 博弈论课程节选

Once you can take one derivative, you can take any number of derivatives and the derivative of the velocity is called the acceleration, and we write it as the second derivative of position.

只要你能求一阶导数,你就能求任意阶的导数,速度的导数被称为"加速度",我们把它写成位移的二阶导数

耶鲁公开课 - 基础物理课程节选

To find a maximum I want the second derivative to be negative.

最大值处的二阶导数是负数

耶鲁公开课 - 博弈论课程节选

So, using those, now, what happens if we take the second derivative of A, the mixed derivative, partial with respect to T and the partial with respect to V.

如果我取A的二阶导数,混合导数，对T偏微分,再对V偏微分。

麻省理工公开课 - 热力学与动力学课程节选

If I take the second derivative I get 2, but I want to get a and not 2.

如果我再求二阶导数就能得到2,但我想要a不想得到2

耶鲁公开课 - 基础物理课程节选

If you know the second derivative of y to be a, then the answer looks like this.

如果你知道y的二阶导数是a,那么这就是答案

耶鲁公开课 - 基础物理课程节选

It doesn't matter, as long as the second derivative is a; that's the answer.

这并不重要,只要二阶导数是a就行了,这就是答案

耶鲁公开课 - 基础物理课程节选

If you knew the second derivative of x, the answer looks like that.

如果你知道x的二阶导数,那么答案就是上面那个

耶鲁公开课 - 基础物理课程节选

And the point is that the second derivative of A, with respect to V and T in this order is the same as the second derivative of a with respect to T and V in this order.

问题的关键在于A的二阶导数，对V和T以这样一个顺序求导,和对T和V以这样一个顺序,求导是一样的。

麻省理工公开课 - 热力学与动力学课程节选

Mathematically complete problem is that you can find the function x of t by saying that the second derivative of the function is equal to -k over m times the function.

这个数学问题就是,你可以求出函数 x,只要令函数的二阶导数,等于 -k 除以 m 再乘以函数

耶鲁公开课 - 基础物理课程节选

We did the comparison with the elephant or something; a is the second derivative of x and for this problem, when F is due to a spring, we know the force is that by studying the spring.

我们也已经把它与大象或其它东西作过比较,a 是 x 的二阶导数,在这个问题中,F 是由弹簧产生的,我们在讨论弹簧问题时已经知道了力的大小

耶鲁公开课 - 基础物理课程节选

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。

麻省理工公开课 - 热力学与动力学课程节选

This function will have the right second derivative.

这个函数会得出正确的二阶导数

耶鲁公开课 - 基础物理课程节选

In other words, your job is to guess a function whose second derivative is a, and this is called integration, which is the opposite of differentiation, and integration is just guessing.

换言之,你的任务是要猜出一个二阶导数为a的函数,这就是积分,和微分恰恰相反,积分就是猜

耶鲁公开课 - 基础物理课程节选