The theorem that alternativity can replace associativity is true.
交替性可以代替关联性的定理是正确的。
下面来证明这个定理。
So it is really the same theorem.
还是原来的的定理。
We just use the fundamental theorem.
只需要应用基本定理就行。
Here is an argument for this theorem.
这里我们再看看一个关于此定理的论点。
We saw yesterday Green's theorem.
昨天讲了格林公式。
We can just use the fundamental theorem.
仅需应用基本定理。
我不知道那个定理。
If you want, in a way, it's this theorem.
可以这样想，这就是定理告诉我们的。
OK, so this way is a fundamental theorem.
解决办法是基本定理。
The proof of the theorem is elementary.
完成了该定理的初等证明。
More complicated example Pythagoras's theorem.
在毕达哥拉斯定理中还有更复杂的例子。
Why is this not like conceptually a new theorem?
为什么这个看上去不是一个新的定理呢？
In the plane that is called Green's theorem.
平面上的就是格林公式。
OK, and actually that's how we prove the theorem.
实际上，这也是我们证明的思路。
And for each of them, you will apply Green's theorem.
对于它们中的每一个，使用格林公式。
I would say, "Hey, why not apply the work-energy theorem?"
我会说，为什么不用，功能定理？
This is really one of the features of Green's theorem.
这是格林公式的一种表示。
We have a connection between that, which is Stokes theorem.
还有一种联系的方式就是Stokes定理。
At that part is actually math, namely, the divergence theorem.
那部分是数学的东西，即散度定理。
Well, remember in the case of work we had Green's theorem.
请记得做功的例子，可以用格林公式的。
On the other hand, we give a different proof of a theorem.
另外，我们给出一个定理的一种不同的证明。
Well, that's the statement of the theorem we are trying to prove.
这就是我们试图证明的定理。
The circuit theorem is the important part in circuit theory.
电路定理是电路理论的重要组成部分。
So, we've seen the statement of Green's theorem in two forms.
我们已经了解了，格林公式的两种表达形式。
This is a simple example of the equipartition theorem at work.
这就是均分定理应用的一个简单例子。
What have we seen? Well, we have seen Green's theorem for work.
能看到什么？，可以发现做功的格林公式。
One is direct calculation, and the other one is Green's theorem.
一种是直接计算，另一种则是格林公式。
Now, let's instead try to use Stokes' theorem to do the calculation.
现在使用Stokes定理来算。
A complete analysis must build on the implicit function theorem.
完整的分析必须建立在隐函数定理的基础上。
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