So, in fact, it's a vector field.

事实上，是一个向量场。

We have this vector field.

这是向量场。

That was a vector field in the plane.

它是一个在平面上的向量场。

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。

And my vector field.

向量场。

So, this vector field is not conservative.

所以，这个向量场不是保守场。

That is called the curl of a vector field.

这个量叫向量场的旋度。

Well, our vector field, is actually vertical.

向量场是竖直的。

At the origin, the vector field is not defined.

在原点，向量场是没有意义的。

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。

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, so you take the divergence of a vector field.

取一个向量场的散度。

One is the vector field whose flux you are taking.

一个是要取通量的向量场。

I want to find the potential for this vector field.

我想找出这个向量场的势函数。

Let's say that our vector field has two components.

假设我们的向量场有两个分量。

The problem is not every vector field is a gradient.

问题是，不是所有向量场都是梯度。

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 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。

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。

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。

OK, so my vector field does something like this everywhere.

这个向量场处处都是这样。

Let's say that we had the same c, but now the vector field.

假设c还是一样的，但现在的向量场变为。

That actually is what we will call later a vector field.

这就是后面我们要讲的向量空间。

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。

We need, actually, a vector field that is well-defined everywhere.

实际上我们需要，一个处处有定义的向量场。

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。