We dot our favorite vector field with it.
用我们喜欢的向量场来点乘它。
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。
Actually, they have a vector field is still pointing perpendicular to the level curves that we have seen, just to remind you.
实际上，向量场，还是垂直于水平线，就像之前看到的那样，我只是想提醒一下大家。
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。
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 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。
And we used a vector field that represents the flow of whatever the substance is whose diffusion we are studying.
我们用向量场表示物质的扩散，而其扩散就是我们要研究的内容。
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。
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'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。
If we have a vector field that cannot possibly be a gradient then we shouldn't try to look for a potential.
如果一个向量场不可能是梯度场，那我们就不应该尝试去找势函数。
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。
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。
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。
One is the vector field whose flux you are taking.
一个是要取通量的向量场。
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。
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。
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。
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。
That is called the curl of a vector field.
这个量叫向量场的旋度。
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。
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。
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