The eigenvalue equation and approximate formula for cut-off frequencies of guided modes are obtained by vector-field-method.
采用矢量场法，得到了在弱导条件下传导模式的特征方程和截止频率的近似计算公式。
We dot our favorite vector field with it.
用我们喜欢的向量场来点乘它。
At the origin, the vector field is not defined.
在原点，向量场是没有意义的。
That is called the curl of a vector field.
这个量叫向量场的旋度。
One is the vector field whose flux you are taking.
一个是要取通量的向量场。
Vector field visualization is an important part of visualization in scientific computing.
矢量场可视化是科学计算可视化的重要组成部分。
Vector field is important conception in science and engineering, which describes directional information of physical procedure and developing trends of states.
向量场是科学和工程中的重要概念，描述物理过程中的方向信息和状态的变化趋势。
It's a vector field that just rotates around the origin counterclockwise.
这是一个绕原点逆时针旋转的向量场。
My vector field is really sticking out everywhere away from the origin.
即给定的向量场是以原点为心向外延伸的。
I have a curve in the plane and I have a vector field.
这有一条平面曲线和一个向量场。
Well, I want to figure out how much my vector field is going across that surface.
下面我们要搞清楚，这个向量场是如何穿过曲面的。
Let's say that I have a plane curve and a vector field in the plane.
有一条平面曲线和这个平面上的向量场。
We have a sphere of radius a. I have my vector field.
先作出半径为a的球面，再画出给定的向量场。
You have seen that in the plane it is already pretty hard to draw a vector field.
正如大家所知，在平面中画出向量场已经很困难了。
Once you have such a formula, you do the dot product with this vector field, which is not the same as that one.
一旦你得到一个这样的计算式，你对向量场做点积，这和前面这个不一样。
So, the divergence theorem gives us a way to compute the flux of a vector field for a closed surface.
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。
The problem comes from a vector field satisfying this criterion in a region but it has a hole in it.
如果一个向量场，在一个有洞的区域上，满足这个条件，就会出问题。
Well, actually here it is not very hard to find a function whose gradient is this vector field.
实际上，找出一个函数，其梯度是这个向量场并不难。
OK, so the first property that I will have for a vector field is that it's conservative.
第一个性质是有一个向量场,它是保守的。
Well, it looks like positive should win because here the vector field is much larger than over there.
看上去正的会多一点,因为这里的向量场要大一些。
Just to remind you, a vector field in space is just the same thing as in the plane.
提醒大家，空间中的向量场与平面中的相同。
So, it's not completely obvious what it looks like, but here is a computer plot of that vector field.
这样说不太直观，但我们有电脑画出来的向量场。
So, to understand the explanation, and given what we've been doing, we should have a vector field in there.
为了理解这个解释，我们在这里设一个向量场。
s Let's say that we have a vector field and s, a surface in space.
假设空间中存在一个向量场和一个曲面。
If you remember what we did several times, well, along the circle that vector field now is tangent to the circle.
我们做过若干次了，沿着圆，向量场与圆相切。
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 that is indeed going to be the curl of F. in practice, if you have to compute the curl of a vector field, you know, don't try to remember this formula.
这就是F的旋度，在实际应用中，如果你必须计算一个向量场的旋度，不要尝试记忆这个公式。
And, in some cases, for example, if you know that the vector field is tangent to the curve or if a dot product is constant or things like that then that might actually give you a very easy answer.
例如，在某些情况下，如果已知向量场与曲线相切，或者内积是一个常数等等，那么结果将会很简单。
In CFD vector field visualization streamline is an important technique.
流线是CFD矢量场可视化的重要技术。
In CFD vector field visualization streamline is an important technique.
流线是CFD矢量场可视化的重要技术。
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