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.
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?
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。
那意味着法向量指向下。
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.
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.
就会得到一个被逆时针曲线包围的圆盘，法向量向上。
Then, if you had a flat disk with the curve going counterclockwise, the normal vector would go up.
我们已经知道，如何根据系数找到这个平面的法向量。
And, we know how to find a normal vector to this plane just by looking at the coefficients.
对于圆柱体，法向量是，除以圆柱体的半径。
For a cylinder, it's, a divided by the radius of a cylinder.
但是在寻找法向量时，可以确定法向量与水平面垂直。
But, when it comes to looking for the normal vector, be rest assured that the normal vector to a horizontal plane is just vertical.
如果给你的是，一条很复杂的曲线，而又找不到法向量？
What if I give you a really complicated curve and then you have trouble finding the normal vector?
看到了吧，系数变为0了，当直线垂直于法向量的时候。
So, see, this coefficient becomes zero exactly when the line is perpendicular to the normal vector.
我们研究的是，向量场在曲线法向量方向的情况。
And we looked at the component of a vector field in the direction that was normal to the curve.
你也可以这样记，如果曲面在你的右边,那么法向量就向下。
You can remember, if it helps you, that if a surface is to your right then the normal vector will go down.
如果你注意到了方向的约定，你会发现它的法向量是向上的。
And, if you pay attention to the orientation conventions, you'll see that you need to take it with normal vector pointing up.
记住两个平面之间的夹角和，两个法向量之间的夹角是一样的。
Remember, the Angle between two planes is the same thing as the Angle between the normal vectors.
我们要做的是，沿着曲线的每一点上，取向量场和法向量的点积。
What we will do is just, at every point along the curve, the dot product between the vector field and the normal vector.
如果想用单位法向量来计算通量，还得先把它标准化到单位长度？
Now, if you wanted the unit normal vector to compute flux, then you would just scale this guy down to unit length, OK?
这个值可以为负，从而在Face对象的法向量的反方向上挤压。
This value can be negative to extrude in the direction opposite the Face object's normal vector.
我们容易知道,法向量为，正负号取决于它是指向外面还是里面。
Namely, we will be saying the /a normal vector is x, y, z over a, plus or minus depending on whether we want it pointing in or out.
法向量或者指向我们，或者背向我们,这要根据我们选的方向来定。
Well, the normal vector is either coming straight at us, or it's maybe going back away from us depending on which orientation we've chosen.
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