定理18.1（隐函数存在唯一性定理）设方程<img src="https://static.zxtiku.com/wangke/imgs/topic/4b48f9b841a829b67ba74f8ac61638e0.jpg"/>中的函数<img src="https://static.zxtiku.com/wangke/imgs/topic/0bfd27d00d097d126c2da3451f3c4af3.jpg"/>满足以下四个条件：<br /> (i) 在以<img src="https://static.zxtiku.com/wangke/imgs/topic/db3632ddb478914c0a11248635524e9b.jpg"/>为内点的某区域<img src="https://static.zxtiku.com/wangke/imgs/topic/20dbfc6cfb295b257495654043f1ba80.jpg"/>上连续；<br /> (ii)<img src="https://static.zxtiku.com/wangke/imgs/topic/b6f2f3fff13300cbba86412ad05a16df.jpg"/>(初始条件),<br /> (iii)在<img src="https://static.zxtiku.com/wangke/imgs/topic/f6f7f22c5706e9e911bb53ee2168b172.jpg"/>内存在连续的偏导数<img src="https://static.zxtiku.com/wangke/imgs/topic/6cd27d19524b4c99501a9b159a6a36dd.jpg"/>,<br /> (iv)<img src="https://static.zxtiku.com/wangke/imgs/topic/7822a2d29848abad1ebc48e94e10d7d2.jpg"/>.<br /> 则有如下结论成立：<br /> 1、存在某邻域<img src="https://static.zxtiku.com/wangke/imgs/topic/5b854f81664b872afd5a9c8556b63f46.jpg"/>，在<img src="https://static.zxtiku.com/wangke/imgs/topic/aaedf0d1c1b8977898e3516d501fad18.jpg"/>上<img src="https://static.zxtiku.com/wangke/imgs/topic/32541338023fafa788f5d1fdca7d45d0.jpg"/>惟一地确定了一个隐函数<img src="https://static.zxtiku.com/wangke/imgs/topic/b946928b4f9921b1eb68a83ec1ea19fa.jpg"/>,<br /> 它满足：<img src="https://static.zxtiku.com/wangke/imgs/topic/fe94f5b748636184fac574eb14363347.jpg"/>且当<img src="https://static.zxtiku.com/wangke/imgs/topic/3ec950362862026c29b52801ebf776f7.jpg"/>时, 有<img src="https://static.zxtiku.com/wangke/imgs/topic/f945350c42fcf00bf7765e98ceaf6177.jpg"/><br /> 2、<img src="https://static.zxtiku.com/wangke/imgs/topic/26da8acabb220eed1145d8d3b93e9b49.jpg"/>在<img src="https://static.zxtiku.com/wangke/imgs/topic/ebca93f8a11050995fa2325cc7cb6569.jpg"/>上 .