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。
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。
Doing area surface and volume integrals.
作面积分和体积分。
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。
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？
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。
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. 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。
Line integrals are used extensively in the theory of functions of a complex variable.
在复变量函数理论中广泛使用线积分。
We shall now show that the change of variable formula for integrals is just as unforgettable.
我们现在来说明积分的换元公式也一样好记。
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。
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。
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。
OK, if that's not completely clear to you, then I encourage you to go over how we set up double integrals again.
如果你还不是完全清楚，建议你去复习一下，如何建立二重积分。
That is the key to computing things in practice. It means, actually, you already know how to compute line integrals for flux.
这是计算过程的关键步骤，也就是说，你们已经知道如何用线积分去计算通量了。
For example, in a calculus course it's not terribly difficult to memorize the formula for integration by parts for integrals.
例如，在微积分课程中，记下分步积分法的公式并不是特别难。
When we do triple integrals in space, well, it is the same kind of story, except now we have, of course, more coordinate systems.
做三重积分，和二重一样，当然，我们会有更多的坐标系。
Longitudinal displacement measures myocardial motion amplitudes in systole and visualizes the segmental velocity - time integrals.
纵向位移是测量收缩期心肌运动幅度，显现为心肌节段的速度-时间积分。
After spending a lot of time telling you how to compute surface integrals, now I am going to try to tell you how to avoid computing them.
在我花了这么多时间来告诉你们，如何计算曲面积分之后，我打算告诉你们，如何避免计算它们。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
If the whole integrals on the great circle are used, the average of the function values for any point and its antipodal can be obtained.
如利用的是整个大圆的积分，得到的是球面上任何点及其对蹠点上函数的平均值，或者说得到的是球面上函数的偶对称部分。
If the whole integrals on the great circle are used, the average of the function values for any point and its antipodal can be obtained.
如利用的是整个大圆的积分，得到的是球面上任何点及其对蹠点上函数的平均值，或者说得到的是球面上函数的偶对称部分。
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