So, this vector field is not conservative.
所以，这个向量场不是保守场。
We dot our favorite vector field with it.
用我们喜欢的向量场来点乘它。
I want to find the potential for this vector field.
我想找出这个向量场的势函数。
That was a vector field in the plane.
它是一个在平面上的向量场。
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。
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。
The problem is not every vector field is a gradient.
问题是，不是所有向量场都是梯度。
The curl of a vector field in space is actually a vector field, not a scalar function. I have delayed the inevitable.
空间中的向量场的旋度，是一个向量场，而不是一个标量函数，我必须告诉你们。
At the origin, the vector field is not defined.
在原点，向量场是没有意义的。
Well, we've seen this criterion that if a curl of the vector field is zero and it's defined in the entire plane, then the vector field is conservative, and it's a gradient field.
我们已经知道了一个准则，如果向量场的旋度为零，而且它在整个平面上有定义，那么这个向量场是保守的，而且它是个梯度场。
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。
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。
Let's say that our vector field has two components.
假设我们的向量场有两个分量。
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。
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。
Remember that was the vector field that looked like a rotation at the unit speed.
我们记得，这是个以单位速度旋转的向量场。
OK, so my vector field does something like this everywhere.
这个向量场处处都是这样。
Let's say I want to do it for this vector field.
比如说，我想对这个向量场来求解。
We had a curve in the plane and we had a vector field.
平面上有一曲线，且存在着向量场。
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.
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。
Now, an important difference between curl here and curl in the plane is that now the curl of a vector field is again a vector field.
和平面上的旋度的一个重要的不同点是，这里向量场的旋度，仍然是一个向量场。
But now, let's say that I have a general vector field.
但是现在，假设有一个一般向量场。
One is the vector field whose flux you are taking.
一个是要取通量的向量场。
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。
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。
So, in fact, it's a vector field.
事实上，是一个向量场。
That is called the curl of a vector field.
这个量叫向量场的旋度。
F I have my surface and I have my vector field f.
有一个曲面，还有一个向量场。
OK, and that surface integral, well, it's not for the same vector field.
这个曲面积分，不是同一个向量场。
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