So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region. 其中一种说明了,在向量场上,沿逆时针方向,向量做的功等于,平面区域上旋度F的二重积分。
And so, when I take the line integral along a closed curve, 0 I don't have to compute it. I notes going to be zero. 所以,当沿一个封闭曲线做线积分的时候,不用去计算它,它一定会是。
Well, you just take a closed curve in the plane. 只需在区域中取一条封闭曲线。
It says that the work done by a vector field along a closed curve can be replaced by a double integral of curl F. 物体在向量场里面做功的路径是封闭曲线的话,它所做的功可以写成一个二重积分。
It just reminds you that you are doing it on a closed curve. 只是提醒你,是在封闭曲线上做积分。
Well, the line integral along C1 minus C2, well, C1-C2 let's just form a closed curve that is C1 minus C2. 沿着C1的线积分减去沿着C2的线积分,我们建立一个封闭曲线来表示。
OK, so in particular, if you have a vector field that's defined everywhere the plane, then you take any closed curve. 如果特殊一些,你有一条,处处有定义的向量场,做任何一条封闭曲线。
So, Green's theorem says that if I have a closed curve, then the line integral of F is equal to the double integral of curl on the region inside. 格林公式内容是如果有一条封闭曲线,那么对F的线积分就等于,区域内F旋度的二重积分。
You know it's automatically OK because if you have a closed curve, then the vector field is, I mean, if a vector field is defined on the curve it will also be defined inside. 它必然成立,因为如果给出一条闭合曲线,然后向量空间是。。。,我是指,向量空间在曲线上有定义,当然在区域内部也有定义。
If we have a closed curve then the line integral for work is just zero. 如果给定一条封闭曲线,那么求所做功的线积分为零。
If we apply Stokes'theorem to compute the work done by the electric field around a closed curve, that means you have a wire in there and you want to find the voltage along the wire. 如果用Stokes定理计算的话,电场对一条封闭曲线做的功,比如说,电厂中有一个金属线圈,想要知道金属线圈的电压是多少。
It says that the line integral is zero along any closed curve. 也就是,在任意封闭曲线上的线积分为。
If it is a closed curve, we should be able to replace it by a double integral. 如果是一条闭曲线,也可以用二重积分来代替的。
If the interior of any closed curve in R& is also contained in R. 如果R内部存在一条闭合曲线-,当然它必须得包含在R中。
If I am path independent, then if I take a closed curve, well, it has the same endpoints as just the curve that doesn't move at all. 如果有路径独立,找一条闭合曲线,它的起点和终点是一样的,就像曲线一点也没动。
If I move along this closed curve, I start at the origin. 沿这条闭曲线,从原点出发。
A closed curve bounding a plane area. Add-on library is using older than supported interface. 周界围绕一平面域的封闭性边界加载项库所使用的界面旧于支持的界面。
Adds a small break between the beginning and end of a closed curve line 在封闭曲线的开始和结束位置之间添加一个小间隔
To remind ourselves that we are doing it along a closed curve, very often we put just a circle for the integral to tell us this is a curve that closes on itself. 为了提醒我们是在封闭曲线上做积分,经常在积分符号上加个圆圈,告诉我们,这条曲线自我封闭。
Well, let's take my favorite closed curve on the surface of a sphere. 在球面上任取一个封闭曲线。
If I take any closed curve, the work will always be zero. 如果我取任意闭合曲线,所做功都全是。
Interpolation of Cubic B-spline Closed Curve Based on Periodic Extension 基于周期性延伸的三次B样条闭曲线插值
Both of them have to do with comparing a line integral along a closed curve to a double integral over the region inside enclosed by the curve. 它们要表达的都是,沿一条闭曲线作线积分,都可以表示成,平面区域上的二重积分。
And, I still want to compute the line integral along a closed curve. 但仍然想要沿着封闭曲线的线积分计算。
To compute things, Green's theorem, let's just compute, well, let us forget, sorry, find the value of a line integral along the closed curve by reducing it to double integral. 用格林公式计算。。。,只是计算。。。,让我们忘记。。。,应该是,算沿闭曲线的线积分值,可以通过二重积分来算。
I need to be on a closed curve to do it. 我需要在一条封闭曲线上来做。
That's a closed curve. So, I would like to use Green's theorem. 这是封闭曲线,所以我们可以用格林公式。
Green's theorem for flux says I have a closed curve that goes counterclockwise around some region. 通量的格林公式说,绕区域逆时针方向旋转的闭合曲线。
Let's us replace a line integral along a closed curve by a double integral Well, here it is the same. 用一个二重积分,来代替沿闭曲线的线积分,原理是一样的。
I know in advance that any closed curve, C so, C in particular, has to bound some surface. 我提前已经知道,任何的封闭曲线,特别的,都是某曲面的界。