We can convert them to integrals.
我们可以把它变成积分。
In a few weeks, we will be triple integrals.
几个星期后，我们会学习三重积分。
Doing area surface and volume integrals.
作面积分和体积分。
So then we have integrals instead of sums.
那么我们用积分代替求和。
OK. Let's see more ways of taking flux integrals.
来看看更多的取通量积分的方法。
Well, remember we were trying to do triple integrals.
我们学过了三重积分。
Anyway, that is double integrals in polar coordinates.
这是极坐标系下的二重积分。
Other kinds of integrals we have seen are triple integrals.
我们学过的另一种积分是三重积分。
So now we're going to triple integrals in spherical coordinates.
现在，在球坐标中进行三重积分。
OK, so I proved that my two line integrals along C1 and C2 are equal.
我证出了分别沿着C1和C2的两个线积分是相等的。
We've learned about double integrals, and we've learned about line integrals.
我们已经学过，二重积分和线积分。
OK, any questions about how to set up double integrals in xy coordinates?
关于在xy坐标系里建立二重积分有问题吗？
It is not a surprise that you will get the same answer for both line integrals.
对它们两个做线积分，得到相同结果，就不令人感到吃惊了。
OK, so that should give you overview of various ways to compute line integrals.
这向我们展示了，计算线积分的办法。
OK. Let me move on a bit because we have a lot of other kinds of integrals to see.
我们继续吧，还有很多其他的积分类型要看。
OK, so triple integrals over a region in space, we integrate a scalar quantity, dV.
那么在一个空间上的区域的三重积分，对标量dV做积分。
So, Green's theorem is another way to avoid calculating line integrals if we don't want to.
格林公式是另一种可以,避免计算线积分的方法。
When you do single integrals it is usually not to find the area of some region of a plane.
当我们在做一重积分时，并不是为了去求平面某区域的面积。
Calculus, developed by Newton and Leibniz, is based on derivatives and integrals of curves.
演算，由牛顿和莱布尼茨的，是基于对衍生工具和积分的曲线。
OK, so I'm going to divide my blackboard into three pieces, and here I will write triple integrals.
我要把我的黑板分成三部分，在这我将要写三重积分。
So, these are special cases of what's called the fundamental theorem of calculus for line integrals.
这些都是反映了,线积分的微积分基本定理的特例。
And so that was stuff about double and triple integrals and vector calculus in the plane and in space.
也就是二重和三重积分的内容，以及平面和空间中的向量积分。
Let me just switch gears completely and switch to today's topic, which is line integrals and work in 3D.
我们换一个话题,开始今天的内容，线积分和3维空间中的“功”
OK, now another thing we've seen with double integrals is how to do more complicated changes of variables.
另一件关于二重积分的是，我们已经讲过了，如何做更复杂的变量变换。
The first of these integrals is seen to be a logarithmic form and the second the inverse tangent form.
这些积分中的第一个被看成是对数形式，而第二个被看成是反正切形式。
You have three line integrals to compute instead of two, but conceptually it remains exactly the same idea.
要计算三个线积分，而不是两个，但概念上是一样的。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
也就是最终要摆脱曲面积分，回到常规的二重积分。
OK, so these are just formulas to remember for examples of triple integrals It doesn't change conceptually.
这些都是三重积分的例子，从概念上讲是相同的。
We have been working with triple integrals and seeing how to set them up in all sorts of coordinate systems.
我们目前已经学习了三重积分，以及如何在各种坐标系中建立它们。
A calculating formula for surface integrals under orthogonal transformation of space coordinates is given.
给出曲面积分在空间坐标的正交变换下的一个计算公式。
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