现在我们拿这个法向量向上的圆盘。
每一个平面都有无数条法向量。
你用了法向量i，而不是竖直向量。
K instead of using the vertical vector k, you use the normal vector I.
这通常不是一个单位法向量。
现在用单位法向量乘以那个。
关于球体，法向量是,除以球的半径。
So, for a sphere, the normal vector is divided by the radius of the sphere.
先思考一下圆柱面的法向量？
First of all, what does the normal vector to a cylinder look like?
那么当然知道它的法向量。
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.
它和曲面有相同的法向量。
这是平面法向量，和沿直线向量的点积。
It's the dot product between the normal vector of a plane and the vector along the line.
那相当于法向量指向上。
那么法向量又是什么呢？
那意味着法向量指向下。
它可以是曲面上想要的任意长度的法向量。
It can be a normal vector of any length you want to the surfaces.
一个法向量与任意常数相乘，还是法向量。
You can just multiply a normal vector by any constant, you will still get the normal vector.
法向量指向上，这里我们知道它是什么意思。
这是与N长度相同的法向量，只是符号不定。
This is a normal vector of the same length as n, well, up to sign.
法向量就是梯度向量。
这就是切平面的，或者说这个曲面的法向量。
OK, and that's going to be the normal vector to the surface or to the tangent plane.
比方说我打算让它法向量指向向上。
Let's say that I am going to do it with the normal pointing upwards.
事实上，给定的向量场与法向量是相互平行的。
In fact, our vector field and our normal vector are parallel to each other.
图上有个东东我没画,就是这个平面的法向量。
One thing that's on this picture that I haven't drawn down is the normal vector to the plane.
事实上，如果你随着你的法向量，一直朝上走。
And, in fact, if you try to follow your normal vector that's pointing up, it's pointing up, up, up.
从记号来看，是一个法向量，那它有什么意义呢？
The notation suggests it is a normal vector, so what does that mean?
我来再次解释一下如何得出这个圆柱面的法向量。
Let me re-explain quickly how I got a normal vector for the cylinder.
第一单元学得叉积、法向量这些,现在就有用了。
The first unit about cross-products,normal vectors and so on is actually useful now.
法向量方向上，分量为零。
我给这个圆柱确定了方向，即法向量指向圆柱之外。
And let's say I want to Orient my cylinder so that the normal vector sticks out.
为了配合约定习俗,选择相容的、指向上的法向量。
Well, to get our conventions straight, we should take the normal vector pointing up for compatibility with our choice.
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