那么散度定理究竟讲的是什么？
散度的解释，你们可以思考一下，你们知道的，我首先说什么呢？
Interpretation of divergence. You can think of it, you know, what do I want to say first?
所以那是通过散度定理得到的，其中使用了S是封闭曲面这一事实。
So, that's by the divergence theorem using the fact that s is a closed surface.
这个就是散度公式了，告诉大家一些相似的东西，仅仅是对被闭合曲面包围的空间区域来讲的。
This one here, the divergence theorem, tells you something similar but now for a region of space bounded by a closed surface.
这就是散度所度量了的。
散度定理为我们提供了一种，计算向量场通过闭曲面的通量的方法。
So, the divergence theorem gives us a way to compute the flux of a vector field for a closed surface.
答案是，散度是用来度量物体的发散程度。
Well, the answer is divergence measures how much things are diverging.
要注意的是，散度并不是只在圆心处是。
那部分是数学的东西，即散度定理。
At that part is actually math, namely, the divergence theorem.
我将要证明一些稍简单的结论，而不是证明散度定理，也就是写在这儿的等式，接下来证明点简单的东西。
So, instead of proving the divergence theorem, namely, the equality up there, I'm going to actually prove something easier.
在某种方式下它们是有联系的,这就是散度定理。
OK, so we have somehow a connection between these which is the divergence theorem.
散度是什么意思？
散度本身度量了，给定区域的,单位体积内的冒出的，或者流失的量。
So, the divergence itself measures basically the amount of sources or sinks per unit volume in a given place.
如果说函数的梯度是向量，那么向量场的散度就是函数。
So, if the gradient of a function is a vector, the divergence of a vector field is a function.
梯度和散度是完全不同的东西。
OK, so divergence and gradients are completely different things.
第二点信息是，我们从散度定理得到的，这就是我花费时间试图解释的。
And the second piece of information that we got was from the divergence theorem, and that was the one I spent time trying to explain.
另外一个理解方法是，散度是源头的“冒出率”，它是流体往系统输入的量，也就是单位时间内，单位面积新冒出的流体。
The other way to think about it is divergence is the source rate, it is the amount of fluid that is being inserted into the system, that is being pumped into the system per unit time per unit area.
向量场，的散度，这两个定理说了什么呢？
Remember, the divergence of a vector field What do these two theorems say?
你们需要记住，什么是线积分，什么是场的散度？
So, you should remember, what is this line integral, and what's the divergence of a field?
如果取一个有旋的向量场，流体流动方向是环绕某个坐标轴的，那么就会发现它的散度是零。
And, if you take a vector field where maybe everything is rotating, a flow that's just rotating about some axis, then you'll find that its divergence is zero.
如果取的向量场是处处恒定的，所有点都是平移关系，所以没有散度，因为导数为零。
If you take a vector field that is a constant vector field where everything just translates then there is no divergence involved because the derivatives will be zero.
那就是散度。
也就是通量的格林公式——散度公式。
在散度定理中的约定是,将曲面的定向取为外法线的方向。
The convention in the divergence theorem is that we orient the surface with a normal vector pointing always outwards.
这就是散度定理。
它说的是，电场的散度等于，这是一个物理常数，它等于什么，是基于所用的单位。
And it says that the divergence of the electric field is equal to, so this is a just a physical constant, and what it is equal to depends on what units you are using.
散度场的物理解释是什么？
So, what's the physical interpretation of a divergence field?
那就是散度定理。
原因是…，我们看过一个向量场，散度意味着多少东西膨胀，多少东西被创造。
And the reason for that is, again, we have seen for a velocity field that divergence measures how much things are expanding or how much stuff is being created.
取一个向量场的散度。
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