Normal vector points up. Imagine that you put your thumb along c, your index towards s and then your middle finger points up.
法向量向上，想象你把你的大拇指沿着C放，食指指向s，那么你的中指就会向上。
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。
OK, the best way to it, now that we have the gradient vector, is actually to directly say oh, we know the normal vector to this plane.
最好的方法是，既然我们已经有了梯度向量，就可以直接说,我们知道了切平面的法向量。
What we will do is just, at every point along the curve, the dot product between the vector field and the normal vector.
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。
And, we know how to find a normal vector to this plane just by looking at the coefficients.
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 the level set, namely the gradient vector is always perpendicular to the level set.
我们是知道如何去求水平集的法向量的，也就是垂直于水平集的法向量。
And, if you pay attention to the orientation conventions, you'll see that you need to take it with normal vector pointing up.
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。
Well, to get our conventions straight, we should take the normal vector pointing up for compatibility with our choice.
为了配合约定习俗,选择相容的、指向上的法向量。
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.
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。
OK, and that's going to be the normal vector to the surface or to the tangent plane.
这就是切平面的，或者说这个曲面的法向量。
It's the dot product between the normal vector of a plane and the vector along the line.
这是平面法向量，和沿直线向量的点积。
And, if I look carefully at the orientation convention, Stokes I have to take the normal vector pointing up again.
如果我仔细考虑了方向的约定，定理告诉我们，Stokes, theorem, tells, me, that，法向量必须再次指向上。
K instead of using the vertical vector k, you use the normal vector I.
你用了法向量i，而不是竖直向量。
That is our convention to get a unit normal vector that points to the right of the curve as we move along the curve.
这是约定的得到单位法向量的方法，这种做法使得，当沿着曲线行进时，法向量始终指向右手方向。
The normal vector pointing up, here we know what it means.
法向量指向上，这里我们知道它是什么意思。
When we know two vectors in a plane, it let us find the normal vector to the plane, and that is what we need to find the equation.
当我们已知平面上的两个向量，我们就可以找到，这个平面的法向量，然后我们就可以用法向量来找到方程。
It is this guy.If you continue to follow your normal vector, see, they are actually pointing up and into the paraboloid.
就是这它了，如果你继续跟着法向量看，会看到它们实际上，指向上并且指向抛物面里。
That corresponds to normal vector pointing up.
那相当于法向量指向上。
Tasklets from the high-priority vector are serviced first, followed by those on the normal vector.
来自高优先级矢量的微线程先得到服务，随后是来自正常优先级矢量的微线程。
That means your normal vector points down.
那意味着法向量指向下。
Hopefully, you can see that if I take a normal vector to the sphere it is actually pointing radially out away from the origin.
我希望大家能够了解，如果我将法向量平移至球面，那么它将以原点为心向外放射。
The notation suggests it is a normal vector, so what does that mean?
从记号来看，是一个法向量，那它有什么意义呢？
Just to reiterate what I said, positively here means, because we are going counterclockwise, the normal vector points out of the region.
再说一遍，这里“正的”意味着…，由于是逆时针走向，法向量指向区域外侧的。
Well, the question we have now is what is the area of this little piece of surface and what is its normal vector?
现在的问题是，曲面上这一小块儿的面积是什么，及其法向量是什么？
Let me show you a picture. The rule is if I walk along C with S to my left then the normal vector is pointing up for me.
给你们看张图片，“相容”就是,如果我沿着C走，而且S在左边，法向量就是朝上的。
What if I give you a really complicated curve and then you have trouble finding the normal vector?
如果给你的是，一条很复杂的曲线，而又找不到法向量？
You can just multiply a normal vector by any constant, you will still get the normal vector.
一个法向量与任意常数相乘，还是法向量。
And let's say I want to Orient my cylinder so that the normal vector sticks out.
我给这个圆柱确定了方向，即法向量指向圆柱之外。
You can remember, if it helps you, that if a surface is to your right then the normal vector will go down.
你也可以这样记，如果曲面在你的右边,那么法向量就向下。
The orientation that will work for this theorem is choosing the normal vector to point outwards.
对于这个定理，我选择的方向是,指向外的正方向。
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