It is proportional to this vector field.
它与向量场成比例。
We dot our favorite vector field with it.
用我们喜欢的向量场来点乘它。
That is called the curl of a vector field.
这个量叫向量场的旋度。
At the origin, the vector field is not defined.
在原点，向量场是没有意义的。
One is the vector field whose flux you are taking.
一个是要取通量的向量场。
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。
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。
We have a vector field that gives us a vector at every point.
有一个向量场来描述每一个点上的向量。
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。
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。
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。
And that measured how much the vector field was going across the curve.
它度量了向量场穿过曲线的量。
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。
In fact, our vector field and our normal vector are parallel to each other.
事实上，给定的向量场与法向量是相互平行的。
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。
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。
One is to say that the vector field is a gradient in a certain region of a plane.
一个是向量场,在给定平面区域内是梯度场。
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。
If you take the gradient of this, you should get again this vector field over there.
对这个函数取梯度，你应该又得到这个向量场。
And, of course, it is not a coincidence because this vector field is a gradient field.
当然这并非巧合，因为这个向量场是有势场。
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。
So, if the gradient of a function is a vector, the divergence of a vector field is a function.
如果说函数的梯度是向量，那么向量场的散度就是函数。
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。
And we looked at the component of a vector field in the direction that was normal to the curve.
我们研究的是，向量场在曲线法向量方向的情况。
One place where it comes up is when we try to understand whether a vector field is conservative.
当需要判断一个向量场是否保守向量场时,旋度也会派上用场的。
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。
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。
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。
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。
That means you need your vector field to be continuous and differentiable everywhere on the surface s.
也就是，向量场是连续的，且在曲面s上处处可微。
应用推荐