[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.52,0:00:03.24,Default,,0000,0000,0000,,在上一个视频中，我们开始学习斯托克斯定理，
Dialogue: 0,0:00:03.24,0:00:04.70,Default,,0000,0000,0000,,在这个视频中，
Dialogue: 0,0:00:04.70,0:00:07.06,Default,,0000,0000,0000,,我想来看看，它与我们
Dialogue: 0,0:00:07.06,0:00:09.05,Default,,0000,0000,0000,,已经学习过的是不是一致。
Dialogue: 0,0:00:09.05,0:00:12.19,Default,,0000,0000,0000,,为了这个目的，我们想象--我先画出数轴，
Dialogue: 0,0:00:12.19,0:00:14.34,Default,,0000,0000,0000,,这是我的 z 轴，
Dialogue: 0,0:00:14.34,0:00:16.68,Default,,0000,0000,0000,,这是我的 x 轴，
Dialogue: 0,0:00:16.68,0:00:19.61,Default,,0000,0000,0000,,这是我的 y 轴，
Dialogue: 0,0:00:19.61,0:00:23.43,Default,,0000,0000,0000,,我们想象在 xy 平面有一个区域，
Dialogue: 0,0:00:23.43,0:00:25.77,Default,,0000,0000,0000,,我把它画出来，
Dialogue: 0,0:00:25.77,0:00:30.67,Default,,0000,0000,0000,,我们说，这是我在 xy平面的区域，
Dialogue: 0,0:00:30.67,0:00:34.85,Default,,0000,0000,0000,,我叫它 区域 R，
Dialogue: 0,0:00:34.85,0:00:36.58,Default,,0000,0000,0000,,我还有这个区域的边界，
Dialogue: 0,0:00:36.58,0:00:39.47,Default,,0000,0000,0000,,我们关心
Dialogue: 0,0:00:39.47,0:00:40.72,Default,,0000,0000,0000,,我们沿边界移动的方向，
Dialogue: 0,0:00:40.72,0:00:41.65,Default,,0000,0000,0000,,我们是
Dialogue: 0,0:00:41.65,0:00:43.18,Default,,0000,0000,0000,,沿边界逆时针移动，
Dialogue: 0,0:00:43.18,0:00:47.15,Default,,0000,0000,0000,,这样，我们就有一个环绕这个区域的路径，
Dialogue: 0,0:00:47.15,0:00:49.89,Default,,0000,0000,0000,,我们可以叫它 c ，
Dialogue: 0,0:00:49.89,0:00:51.95,Default,,0000,0000,0000,,我们叫它 c ，我们
Dialogue: 0,0:00:51.95,0:00:57.01,Default,,0000,0000,0000,,要在它上面逆时针移动，
Dialogue: 0,0:00:57.01,0:01:02.31,Default,,0000,0000,0000,,我们还有一个矢量场，
Dialogue: 0,0:01:02.31,0:01:05.36,Default,,0000,0000,0000,,实质上，它的 i 分量只是
Dialogue: 0,0:01:05.36,0:01:08.03,Default,,0000,0000,0000,,x 和 y的函数，
Dialogue: 0,0:01:08.03,0:01:10.31,Default,,0000,0000,0000,,它的 j 分量
Dialogue: 0,0:01:10.31,0:01:12.53,Default,,0000,0000,0000,,只是 x 和 y 的函数，
Dialogue: 0,0:01:12.53,0:01:14.78,Default,,0000,0000,0000,,我们说，它没有 k 分量，
Dialogue: 0,0:01:14.78,0:01:17.23,Default,,0000,0000,0000,,这样，这个区域上的 矢量场，
Dialogue: 0,0:01:17.23,0:01:18.75,Default,,0000,0000,0000,,它就会是像这样的。
Dialogue: 0,0:01:18.75,0:01:20.42,Default,,0000,0000,0000,,我只是随机地画一些矢量，
Dialogue: 0,0:01:20.42,0:01:21.88,Default,,0000,0000,0000,,如果我离开这个区域，
Dialogue: 0,0:01:21.88,0:01:23.35,Default,,0000,0000,0000,,如果你沿 z 方向走，
Dialogue: 0,0:01:23.35,0:01:25.70,Default,,0000,0000,0000,,这只是越走越高，
Dialogue: 0,0:01:25.70,0:01:27.93,Default,,0000,0000,0000,,而那个矢量
Dialogue: 0,0:01:27.93,0:01:29.66,Default,,0000,0000,0000,,在你的 z 分量变化时，不会变化。
Dialogue: 0,0:01:29.66,0:01:31.45,Default,,0000,0000,0000,,所有的矢量实际上
Dialogue: 0,0:01:31.45,0:01:35.79,Default,,0000,0000,0000,,都平行于--当 z 等于 0 时--
Dialogue: 0,0:01:35.79,0:01:39.10,Default,,0000,0000,0000,,都在 xy 平面上，
Dialogue: 0,0:01:39.10,0:01:41.48,Default,,0000,0000,0000,,这样，我们来思考一下，
Dialogue: 0,0:01:41.48,0:01:46.01,Default,,0000,0000,0000,,根据斯托克斯定理
Dialogue: 0,0:01:46.01,0:01:48.98,Default,,0000,0000,0000,,在这个路径上的线积分值是什么？
Dialogue: 0,0:01:48.98,0:01:51.47,Default,,0000,0000,0000,,我画得更好一点，
Dialogue: 0,0:01:51.47,0:02:00.96,Default,,0000,0000,0000,,f 点 dr 在路径 c 上的线积分，
Dialogue: 0,0:02:00.96,0:02:05.96,Default,,0000,0000,0000,,f 点 小写 dr，这里很明显 dr
Dialogue: 0,0:02:05.96,0:02:08.28,Default,,0000,0000,0000,,沿着这个路径。
Dialogue: 0,0:02:08.28,0:02:11.47,Default,,0000,0000,0000,,我们使用斯托克斯定理，
Dialogue: 0,0:02:11.47,0:02:13.85,Default,,0000,0000,0000,,这个量应该是
Dialogue: 0,0:02:13.85,0:02:14.61,Default,,0000,0000,0000,,等于这个量，
Dialogue: 0,0:02:14.61,0:02:18.85,Default,,0000,0000,0000,,它应该等于这个表面的双重积分，
Dialogue: 0,0:02:18.85,0:02:21.27,Default,,0000,0000,0000,,这个区域其实只是一个
Dialogue: 0,0:02:21.27,0:02:23.45,Default,,0000,0000,0000,,位于 xy 平面上的一个表面。
Dialogue: 0,0:02:23.45,0:02:26.08,Default,,0000,0000,0000,,它就应该是双重积分--
Dialogue: 0,0:02:26.08,0:02:27.66,Default,,0000,0000,0000,,我来写成相同的 --
Dialogue: 0,0:02:27.66,0:02:31.31,Default,,0000,0000,0000,,它会是这个区域
Dialogue: 0,0:02:31.31,0:02:35.11,Default,,0000,0000,0000,,也就是我们的这个表面
Dialogue: 0,0:02:35.11,0:02:37.84,Default,,0000,0000,0000,,f 点 n 的旋度的双重积分，
Dialogue: 0,0:02:37.84,0:02:40.44,Default,,0000,0000,0000,,所以，我们就需要考虑 f 点 n 的旋度是什么，
Dialogue: 0,0:02:40.44,0:02:42.27,Default,,0000,0000,0000,,
Dialogue: 0,0:02:42.27,0:02:45.51,Default,,0000,0000,0000,,
Dialogue: 0,0:02:45.51,0:02:46.22,Default,,0000,0000,0000,,
Dialogue: 0,0:02:46.22,0:02:50.18,Default,,0000,0000,0000,,
Dialogue: 0,0:02:50.18,0:02:54.00,Default,,0000,0000,0000,,
Dialogue: 0,0:02:54.00,0:02:56.06,Default,,0000,0000,0000,,
Dialogue: 0,0:02:56.06,0:02:58.91,Default,,0000,0000,0000,,
Dialogue: 0,0:02:58.91,0:03:00.81,Default,,0000,0000,0000,,
Dialogue: 0,0:03:00.81,0:03:06.85,Default,,0000,0000,0000,,
Dialogue: 0,0:03:06.85,0:03:10.82,Default,,0000,0000,0000,,
Dialogue: 0,0:03:10.82,0:03:12.33,Default,,0000,0000,0000,,
Dialogue: 0,0:03:12.33,0:03:14.45,Default,,0000,0000,0000,,
Dialogue: 0,0:03:14.45,0:03:16.82,Default,,0000,0000,0000,,
Dialogue: 0,0:03:16.82,0:03:18.91,Default,,0000,0000,0000,,
Dialogue: 0,0:03:18.91,0:03:20.98,Default,,0000,0000,0000,,
Dialogue: 0,0:03:20.98,0:03:24.42,Default,,0000,0000,0000,,
Dialogue: 0,0:03:24.42,0:03:26.99,Default,,0000,0000,0000,,
Dialogue: 0,0:03:26.99,0:03:30.72,Default,,0000,0000,0000,,
Dialogue: 0,0:03:30.72,0:03:32.89,Default,,0000,0000,0000,,
Dialogue: 0,0:03:32.89,0:03:34.35,Default,,0000,0000,0000,,
Dialogue: 0,0:03:34.35,0:03:35.96,Default,,0000,0000,0000,,
Dialogue: 0,0:03:35.96,0:03:42.57,Default,,0000,0000,0000,,
Dialogue: 0,0:03:42.57,0:03:43.45,Default,,0000,0000,0000,,
Dialogue: 0,0:03:43.45,0:03:46.00,Default,,0000,0000,0000,,
Dialogue: 0,0:03:46.00,0:03:48.19,Default,,0000,0000,0000,,
Dialogue: 0,0:03:48.19,0:03:50.47,Default,,0000,0000,0000,,
Dialogue: 0,0:03:50.47,0:03:52.33,Default,,0000,0000,0000,,
Dialogue: 0,0:03:52.33,0:03:56.18,Default,,0000,0000,0000,,
Dialogue: 0,0:03:56.18,0:03:57.13,Default,,0000,0000,0000,,
Dialogue: 0,0:03:57.13,0:04:01.00,Default,,0000,0000,0000,,
Dialogue: 0,0:04:01.00,0:04:02.29,Default,,0000,0000,0000,,
Dialogue: 0,0:04:02.29,0:04:04.34,Default,,0000,0000,0000,,
Dialogue: 0,0:04:04.34,0:04:05.70,Default,,0000,0000,0000,,
Dialogue: 0,0:04:05.70,0:04:07.50,Default,,0000,0000,0000,,
Dialogue: 0,0:04:07.50,0:04:10.26,Default,,0000,0000,0000,,
Dialogue: 0,0:04:10.26,0:04:16.70,Default,,0000,0000,0000,,
Dialogue: 0,0:04:16.70,0:04:20.08,Default,,0000,0000,0000,,
Dialogue: 0,0:04:20.08,0:04:22.18,Default,,0000,0000,0000,,
Dialogue: 0,0:04:22.18,0:04:25.59,Default,,0000,0000,0000,,
Dialogue: 0,0:04:25.59,0:04:28.16,Default,,0000,0000,0000,,
Dialogue: 0,0:04:28.16,0:04:33.91,Default,,0000,0000,0000,,
Dialogue: 0,0:04:33.91,0:04:34.41,Default,,0000,0000,0000,,
Dialogue: 0,0:04:34.41,0:04:36.32,Default,,0000,0000,0000,,
Dialogue: 0,0:04:36.32,0:04:38.18,Default,,0000,0000,0000,,
Dialogue: 0,0:04:41.16,0:04:43.45,Default,,0000,0000,0000,,
Dialogue: 0,0:04:43.45,0:04:44.57,Default,,0000,0000,0000,,
Dialogue: 0,0:04:49.69,0:04:56.15,Default,,0000,0000,0000,,
Dialogue: 0,0:04:56.15,0:04:58.88,Default,,0000,0000,0000,,
Dialogue: 0,0:04:58.88,0:05:02.25,Default,,0000,0000,0000,,
Dialogue: 0,0:05:02.25,0:05:04.30,Default,,0000,0000,0000,,
Dialogue: 0,0:05:04.30,0:05:05.93,Default,,0000,0000,0000,,
Dialogue: 0,0:05:05.93,0:05:07.94,Default,,0000,0000,0000,,
Dialogue: 0,0:05:07.94,0:05:10.39,Default,,0000,0000,0000,,
Dialogue: 0,0:05:10.39,0:05:12.45,Default,,0000,0000,0000,,
Dialogue: 0,0:05:12.45,0:05:14.66,Default,,0000,0000,0000,,
Dialogue: 0,0:05:14.66,0:05:18.49,Default,,0000,0000,0000,,
Dialogue: 0,0:05:18.49,0:05:21.88,Default,,0000,0000,0000,,
Dialogue: 0,0:05:21.88,0:05:24.51,Default,,0000,0000,0000,,
Dialogue: 0,0:05:24.51,0:05:26.92,Default,,0000,0000,0000,,
Dialogue: 0,0:05:26.92,0:05:28.23,Default,,0000,0000,0000,,
Dialogue: 0,0:05:28.23,0:05:31.16,Default,,0000,0000,0000,,
Dialogue: 0,0:05:31.16,0:05:34.03,Default,,0000,0000,0000,,
Dialogue: 0,0:05:34.03,0:05:36.08,Default,,0000,0000,0000,,
Dialogue: 0,0:05:36.08,0:05:39.73,Default,,0000,0000,0000,,
Dialogue: 0,0:05:39.73,0:05:43.93,Default,,0000,0000,0000,,
Dialogue: 0,0:05:43.93,0:05:45.40,Default,,0000,0000,0000,,
Dialogue: 0,0:05:45.40,0:05:49.26,Default,,0000,0000,0000,,
Dialogue: 0,0:05:49.26,0:05:54.98,Default,,0000,0000,0000,,
Dialogue: 0,0:05:54.98,0:05:57.94,Default,,0000,0000,0000,,
Dialogue: 0,0:05:57.94,0:05:59.61,Default,,0000,0000,0000,,
Dialogue: 0,0:05:59.61,0:06:03.03,Default,,0000,0000,0000,,
Dialogue: 0,0:06:03.03,0:06:07.96,Default,,0000,0000,0000,,
Dialogue: 0,0:06:07.96,0:06:12.03,Default,,0000,0000,0000,,
Dialogue: 0,0:06:12.03,0:06:15.92,Default,,0000,0000,0000,,
Dialogue: 0,0:06:15.92,0:06:17.84,Default,,0000,0000,0000,,
Dialogue: 0,0:06:17.84,0:06:20.39,Default,,0000,0000,0000,,
Dialogue: 0,0:06:20.39,0:06:22.80,Default,,0000,0000,0000,,
Dialogue: 0,0:06:22.80,0:06:27.36,Default,,0000,0000,0000,,
Dialogue: 0,0:06:27.36,0:06:30.14,Default,,0000,0000,0000,,
Dialogue: 0,0:06:30.14,0:06:32.24,Default,,0000,0000,0000,,
Dialogue: 0,0:06:32.24,0:06:34.53,Default,,0000,0000,0000,,
Dialogue: 0,0:06:34.53,0:06:36.78,Default,,0000,0000,0000,,
Dialogue: 0,0:06:36.78,0:06:39.43,Default,,0000,0000,0000,,
Dialogue: 0,0:06:39.43,0:06:40.81,Default,,0000,0000,0000,,
Dialogue: 0,0:06:40.81,0:06:41.38,Default,,0000,0000,0000,,
Dialogue: 0,0:06:41.38,0:06:42.56,Default,,0000,0000,0000,,
Dialogue: 0,0:06:42.56,0:06:44.19,Default,,0000,0000,0000,,
Dialogue: 0,0:06:44.19,0:06:47.92,Default,,0000,0000,0000,,
Dialogue: 0,0:06:47.92,0:06:50.84,Default,,0000,0000,0000,,
Dialogue: 0,0:06:50.84,0:06:54.09,Default,,0000,0000,0000,,