第二型曲面积分课件.ppt

2022/11/29,1,22.2 第二型曲面积分,2022/11/29,2,一、曲面的侧,观察以下曲面的侧 (假设曲面是光滑的),曲面分上侧和下侧,曲面分内侧和外侧,2022/11/29,3,曲面的分类:,1.双侧曲面;,2.单侧曲面.,典型双侧曲面,2022/11/29,4,莫比乌斯带,典型单侧曲面:,2022/11/29,5,莫比乌斯带,典型单侧曲面:,2022/11/29,6,典型单侧曲面:,莫比乌斯带,2022/11/29,7,典型单侧曲面:,莫比乌斯带,2022/11/29,8,典型单侧曲面:,莫比乌斯带,2022/11/29,9,典型单侧曲面:,莫比乌斯带,2022/11/29,10,典型单侧曲面:,莫比乌斯带,2022/11/29,11,典型单侧曲面:,莫比乌斯带,2022/11/29,12,典型单侧曲面:,莫比乌斯带,2022/11/29,13,典型单侧曲面:,莫比乌斯带,2022/11/29,14,典型单侧曲面:,莫比乌斯带,2022/11/29,15,典型单侧曲面:,莫比乌斯带,2022/11/29,16,典型单侧曲面:,莫比乌斯带,2022/11/29,17,典型单侧曲面:,莫比乌斯带,2022/11/29,18,典型单侧曲面:,莫比乌斯带,2022/11/29,19,典型单侧曲面:,莫比乌斯带,2022/11/29,20,典型单侧曲面:,莫比乌斯带,2022/11/29,21,曲面法向量的指向决定曲面的侧.,决定了侧的曲面称为有向曲面.,曲面的投影问题:,2022/11/29,22,二、概念的引入,实例: 流向曲面一侧的流量.,2022/11/29,23,2022/11/29,24,1. 分割,则该点流速为,法向量为,2022/11/29,25,2. 求和,2022/11/29,26,3.取极限,2022/11/29,27,三、概念及性质,2022/11/29,28,被积函数,积分曲面,类似可定义,2022/11/29,29,存在条件:,组合形式:,物理意义:,2022/11/29,30,性质:,2022/11/29,31,四、计算法,2022/11/29,32,2022/11/29,33,注意:对坐标的曲面积分,必须注意曲面所取的侧.,2022/11/29,34,2022/11/29,35,2022/11/29,36,例2：计算,其中,取外侧,解：,考虑,的计算,在 xoy 面上的投影均为 线段,而在,上，,取下侧,所以,2022/11/29,37,例2：计算,其中,取外侧,解：,取上侧,由坐标轮换性可得,所以,2022/11/29,38,2022/11/29,39,2022/11/29,40,2022/11/29,41,五、两类曲面积分之间的联系,2022/11/29,42,2022/11/29,43,两类曲面积分之间的联系,2022/11/29,44,向量形式,2022/11/29,45,解,2022/11/29,46,2022/11/29,47,提示：,48,谢谢！,