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OK, this is only true for an ideal gas, and we went through that mathematically where the, with a chain rule.
这一关系只对理想气体成立，上节课我们,用链式法则推导出了这一关系。
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he knows how to mathematically structure the problem,
他知道如何用数学思维分析问题的结构，
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That's going to presumably be given--mathematically if we gave it a number, we'd slap a zero on that.
如果用数学解释,给它一个数字，给它一个零。
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So you can see that we're starting to have a very complicated equation, and it turns out that it's mathematically impossible to even solve the exact Schrodinger equation as we move up to higher numbers of electrons.
所有你们可以看到我们得到了,一个非常复杂的方程，结果是它在数学上是,不可能解出确定的,薛定谔方程,当我们考虑更高的电子数目的时候。
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And I can't, myself, go higher than that but you could certainly do it mathematically.
我自己不能再算比那更大的数字了,不过当然可以算出来的。
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All the words mean things mathematically.
都可以从数学上描述。
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What that means is that one way to describe that mathematically is shown here.
用数学方式对此的表达,显示如下
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It is true mathematically, you can add two numbers, but you've got to ask yourself,"What am I doing as a physicist when I add these two numbers?"
在数学上它是正确的,你可以引入这两个常数,但是你需要问问自己,当我引入这两个常数时,这么做有什么物理意义
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We then have a mathematically rigorous discipline.
于是我们有了一门精确严谨的学科作为工具
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It could be mathematically worked out.
这是很简单的一种正比关系。
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Now, some of these things can prove things that are mathematically way over my head.
如今,这些程序都可以证明一些,令我头大的数学题了
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Because this isn't Cv mathematically speaking.
我们不知道到这是什么。
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He was trying to describe electrons, but the theory said there are two roots in the quadratic equation and the second root is mathematically as interesting as the first one.
他当时只是想去描述电子,但是数学理论告诉我们,二次方程有两个根,而第二个根在数学上和第一个根一样有趣
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Mathematically, we can make it 0.
在数学上讲,我们把它变成零
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Mathematically, it's done by taking the expression for R and differentiating everything in sight that can be differentiated as vector t.
从数学上看是这样的,写出 R 的表达式,并对所有可以进行矢量微分的部分,对时间 t 求微商
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Therefore, mathematically at a given time t, we can trade t for v and put it into this formula.
因此,从数学角度来说,在给定的时刻 t,我们可以用 v 来表示 t,并代入这个式子
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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
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So there's going to be a line that's going to connect the initial point to the final point, and that line mathematically is not going to be the same as this one here.
连接绝热过程,初末态的曲线的方程，和等温线的方程,也不会相同。
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- And in fact, mathematically if it's-- don't really have room for this.
事实上，从数学上来说-,这儿没有空间了。
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you have a mathematically complete problem.
这个问题就转变为数学问题了
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So, that's sort of the mathematically how we get to a, but we can also just look at the graph here, because every time we go one wavelength we can see that we're back in a maximum.
这是就我们得到a的数学,办法，但我们可以直接,从图上看，因为每次我们,经过一个波长，都回到一个最大值。
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But mathematically, we've mentioned this before, log N or really to be precise, log base 2 of N, is the way you express this mathematically.
但从数学上说，之前我们已经提到过了，准确地说是log，N，以2为底N的对数，这就是它在数学上的表示。
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Mathematically, you don't have to.
从数学上来讲没这个必要
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