从视觉上看，我们要做的是，想办法，在这个反函数，在反函数上取一个点，设法找出切线。

And visually what we're accomplishing is somehow to take the hyperbola, and take a point on the hyperbola, and figure out some tangent line.

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对这个函数取梯度，你应该又得到这个向量场。

If you take the gradient of this, you should get again this vector field over there.

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相当于在一元微积分中，取一个函数的不定积分，仅仅需要在结果后加一个常数。

It is the same thing as, in one variable calculus, when you take the anti-derivative of a function it is only defined up to adding the constant.

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No matter where you specify your electron is in terms of those two angles, it doesn't matter the angular part of your wave function is going to be the same.

不论你将，这两个角度,取成什么值，波函数的角向部分,都是，相同的。

麻省理工公开课 - 化学原理课程节选

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。

麻省理工公开课 - 化学原理课程节选

R is a function of n and takes on discrete values.

在这儿r是n的函数，取的是不连续的值。

麻省理工公开课 - 固态化学导论课程节选