其收缩是一个向量。

Its contraction is a covector.

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所以，这个向量场不是保守场。

So, this vector field is not conservative.

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我想找出这个向量场的势函数。

I want to find the potential for this vector field.

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方向基本和这里的单位切向量一致。

And, the direction is essentially that of a vector.

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然后得到平面上的两个给定向量。

Then you have two vectors given on the plane.

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这意味着这一项就是单位向量。

So, that means this actually should be the unit vector.

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我们平时怎么验证两个向量平行呢？

How do we decide if two vectors are parallel?

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问题是，不是所有向量场都是梯度。

The problem is not every vector field is a gradient.

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还有什么原因你会知道法向量？

Why else would you know a normal vector?

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但是你也可以继续只用法向量的。

But you can also just stay with a normal vector.

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现在我们拿这个法向量向上的圆盘。

Now take your disk with its normal vector sticking up.

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这就是后面我们要讲的向量空间。

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

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然后利用法向量得出切平面的方程。

And then get this equation for the plane using the normal vector.

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也有向量，不过在这里不作讨论。

There are also vectors, but I won't discuss them here.

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将访问点添加到位置向量中。

Add the access point to your location vector.

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它是一个在平面上的向量场。

That was a vector field in the plane.

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我从这个向量开始，它已经在这边了。

So I start with this vector, that’s already here.

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向量BP，我们知道关于它的两件事。

So, vector BP, we know two things about it.

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这通常不是一个单位法向量。

And that is, in general, not a unit normal vector.

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现在用单位法向量乘以那个。

Now, let's multiply that by the unit normal vector.

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那么当然知道它的法向量。

Well, you know the normal vector.

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从而法向量的方向是竖直向上或向下的。

The normal vector is just sticking straight up or straight down.

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那时候，它只是一个向量。

Well, at the time, it was just a vector.

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为什么我们会知道法向量？

Why would we ever know a normal vector?

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这个向量场处处都是这样。

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

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让我们试试更大的向量值。

And let's try a really big vector.

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那么如果我选择了任何位于切平面的向量。

Then, if I choose any vector in that tangent plane.

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那么法向量又是什么呢？

What is the normal vector?

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当然我们可以不用向量。

And, we don't really need vectors.

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即竖直方向的单位向量。

It's the unit vector in the vertical direction.

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