That is my curve and my vector field.
那就是曲线和向量场。
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。
xi+yj And let's say that our vector field is xi+yj.
假设向量场是。
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。
It measures how much a vector field goes across the curve.
它度量有多少向量场穿过了曲线。
It is a vector field in some of the flux things and so on.
也可以是一个向量场的通量，等等。
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。
But that assumes that your vector field is well-defined there.
那是假定了向量场是有定义的。
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。
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。
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