it measures failure of a vector field to be conservative.
它用来衡量向量场是不是保守场的。
we need, actually, a vector field that is well-defined everywhere.
实际上我们需要,一个处处有定义的向量场。
that was a vector field in the plane.
它是一个在平面上的向量场。
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。
actually, they have a vector field is still pointing perpendicular to the level curves that we have seen, just to remind you.
实际上,向量场,还是垂直于水平线,就像之前看到的那样,我只是想提醒一下大家。
the problem is not every vector field is a gradient.
问题是,不是所有向量场都是梯度。
my vector field is really sticking out everywhere away from the origin.
即给定的向量场是以原点为心向外延伸的。
and we used a vector field that represents the flow of whatever the substance is whose diffusion we are studying.
我们用向量场表示物质的扩散,而其扩散就是我们要研究的内容。
what changes is, of course, my vector field is no longer sticking straight out so i cannot use this easy geometric argument.
有所变化的是,向量场不是由原点直接向外辐射,从而我们不能再用一些简单的几何直观去解释。
so, the divergence theorem gives us a way to compute the flux of a vector field for a closed surface.
散度定理为我们提供了一种,计算向量场通过闭曲面的通量的方法。
at the origin, the vector field is not defined.
在原点,向量场是没有意义的。
i need to have a vector field that is defined and differentiable -- -- everywhere in d, so same instructions as usual.
我需要一个确定的向量场,而且它在,d,上是处处可微的,然后和平时一样的做法。
well, i want to figure out how much my vector field is going across that surface.
下面我们要搞清楚,这个向量场是如何穿过曲面的。
we have a vector field that gives us a vector at every point.
有一个向量场来描述每一个点上的向量。
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。
well, we「ve seen this criterion that if a curl of the vector field is zero and it」s defined in the entire plane, then the vector field is conservative, and it's a gradient field.
我们已经知道了一个準则,如果向量场的旋度为零,而且它在整个平面上有定义,那么这个向量场是保守的,而且它是个梯度场。
the curl of a vector field in space is actually a vector field, not a scalar function. i have delayed the inevitable.
空间中的向量场的旋度,是一个向量场,而不是一个标量函数,我必须告诉你们。
one place where it comes up is when we try to understand whether a vector field is conservative.
当需要判断一个向量场是否保守向量场时,旋度也会派上用场的。
that would be given by a vector field that points toward the origin and whose magnitude is inversely proportional to the square of a distance from the origin.
力场由一个指向原点的向量场给出,并且此向量场的大小,与其到原点的距离的平方成反比。
so, this vector field is not conservative.
所以,这个向量场不是保守场。
and my vector field represents how things are moving at every point of the plane.
那向量场代表的是,流体在平面上的每一点的流动情况。