试求一函数，在半径的园的内部是调和的，而且在圆周上取值：[tex=7.5x1.286]rjdZS4acQyGjFUUbF3NwgChFHpw/b6C69jiIorCHSnTRPtqnCjYc0hTYidhscHnR[/tex]，其中[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]都为常数 .
[b]解[/b] 利用泊松公式[tex=4.929x1.286]HD5M4FpOVIJMcmstzRn6+/FrOvVCTAFxq5tFLn4SQn1pKcfV2ejlHY4l7Z7vfe0e[/tex][tex=16.357x2.5]kqZaMbZlmuDy7GEAkzJUiz1ET/AsR8LPy6gHz2zuwFCfB30RGdXQzR6z4YJlqyVFd1SbqOi+m6TCW6/G1ysNcCwI3mG6xFiUDQ3K6Lj5sA+hw0GlYsh3y1YZGrdJLwiVOxTEH44sg+poWoSWIjt7THjR6Szf2ZJooNtlBBFcMinnzf6T9+gKA7Vn/5rExABFah+QcAbv/dP6s836gcoXGPMNilmdHq9nRiZDjqyOizQ=[/tex]，[tex=8.429x2.143]HD5M4FpOVIJMcmstzRn6+/FrOvVCTAFxq5tFLn4SQn0brXkvI1H42jyz80W4FMUxT7n8lF2v6vlnAJ3LW+Xnhf/dXpd2CEtlc5M0xgWWXvY=[/tex][tex=15.929x2.5]43DHJW5hNMMx+YjFVGcc+rTq2+Ox7NBp7K6RchiHf0j7c/mK6MOrmmW8yxGju2tcz4G/wEoTs378lUIML8sAZOP5y9EeFiOU7zPW4hmu0P97m8Ko9ZQ2OcHVoG03qD7N7UhGxVOu1yET65rYxPvxqkiz6EEDxnHymjTV6nTkTUCA0Fath0i1Cj4YXdmdHzuM[/tex][tex=2.714x2.0]wXh8KTZrmfwl0ov7zZIHF5H81ChxZ9VPrb+s9NvyL14=[/tex][tex=15.643x2.5]43DHJW5hNMMx+YjFVGcc+lVMmMT5tUMH7+DRVVVxuFUrJGyHZEE5VGsMItLTaB0CyyBgch6LWsc3BJ2UaEVnTawoLdYdhPO03VDjnsNavqTHdJFLFZgopzINpjAvJfNJUSo1nSr556MwC93tckhKbXMA4Gd5/QLRi2a/Cya2kjfWDoK0r3Msd8zticP3q1aj[/tex][tex=5.429x2.0]npQIkOzSjP9jyFnNUdj8wwinhodLACL4/vLDa6Fv3soZNxghkHKbLon6jqqEqxOR[/tex][tex=15.643x2.5]43DHJW5hNMMx+YjFVGcc+pJroWhbuKUrEHEysob7qcmX549IbyUlc7sxAO+Tnavx6DtwD5fvcux+TooXErSFUiE4LB8eFbj4HcvfX8ynhDG4StXBNZxLxBYAIuWM/M3q41hjMZ9sGlrmMZKrMekEmMT87BlzFOe2eeXB3QvxYg9s8rWdybYVzUSBC1TkwkvrU+VgQQNjH7Txs80TjM/4vLIDlMpAQPETwn4yc9DHcvOeQriDBF7ZumpV+hFVgU6O[/tex][tex=5.357x2.0]b/csyBv5QcOeB2qF74zxh9Lzuf9XIShwPGowwR5U8KId2UbQTTa2e/N+6Lm+Z93l[/tex][tex=17.571x2.5]43DHJW5hNMMx+YjFVGcc+pJroWhbuKUrEHEysob7qcmX549IbyUlc7sxAO+TnavxMhYvFkulw5NqzbIVGV62U0421ruKn6tQFVgcqI7dUe0l9bhVNaKcH70ChJUpLCvVvpeJhngepgCSuduw8lmRv6YnCbO+XUKXBAL810NjGLj4fYYaU73/QycIGPjLDTBApFALxLA1JxWQBvkncvSFZBj7g+LSlQRBwc3r6m6Q3kwGX8xq+CeFym6ZlvBh/iP6egLn7/++hoHaU0TyvRGiBA==[/tex]，即[tex=1.286x2.0]kqZaMbZlmuDy7GEAkzJUi9bRnliC9yupHx7upGmJV9E=[/tex][tex=16.643x2.5]43DHJW5hNMMx+YjFVGcc+pJroWhbuKUrEHEysob7qcmX549IbyUlc7sxAO+Tnavx6DtwD5fvcux+TooXErSFUiE4LB8eFbj4HcvfX8ynhDG4StXBNZxLxBYAIuWM/M3q41hjMZ9sGlrmMZKrMekEmMT87BlzFOe2eeXB3QvxYg9s8rWdybYVzUSBC1TkwkvrU+VgQQNjH7Txs80TjM/4vLIDlMpAQPETwn4yc9DHcvObqVOY2aFr8M9vpNplRMOX[/tex]，[tex=16.929x2.5]kqZaMbZlmuDy7GEAkzJUiz1ET/AsR8LPy6gHz2zuwFCfB30RGdXQzR6z4YJlqyVFd1SbqOi+m6TCW6/G1ysNcJ30CCtXqrSmEsnevEZJ51JylLCygD1McdjpWUjbZL+InFbAE0jAccvbqZL3X2G7+Lmr3ey8PGxKjfz5vZ9BhuFOcwsalYX8xwBkYF6RemW3B5tFrexNrMJp9hUJJXRUL/hHgQIwtsy5MgZUbSJYQAEF3WZqFN6a8AbNIrySFeUoZeZ5oRrxQqE86ddoYQ0+rg==[/tex][tex=1.143x1.786]fNoQVs+2SqS0AtEOZcz2raVmZYpnRy6ZzcUBhi2nBiI=[/tex]，所以[tex=4.929x1.286]HD5M4FpOVIJMcmstzRn6+/FrOvVCTAFxq5tFLn4SQn1pKcfV2ejlHY4l7Z7vfe0e[/tex][tex=6.429x2.0]s1BjkLg1fS4BxEOG3k0x0DI5y4CVGaWPTzuZo/fKkJaZS+gr/QcxqK9lhwvZTQj9[/tex]，即[tex=3.929x1.286]1iLx/tiMnuJRSCCmVG7M/BUPfgumcduilnPMzgsk+DQ=[/tex][tex=6.0x2.0]s1BjkLg1fS4BxEOG3k0x0LhqpU7cLrTkplKSlrJgPiyXxZVTCzGhpliPcqtpfxoH[/tex]，或按以下方法，将边值写成[tex=10.643x1.786]rjdZS4acQyGjFUUbF3NwgMAXDtRpfL7RQ7xp0e6B4X6mvHvPsXzjX/n/nb/FEl9BSTkcrxD1GtrC2jqOhHxXTuelMghOcYsWJx29qPN69x0=[/tex]，由于方程齐次的，利用可加性，则[tex=4.0x1.143]df9fkIjhmWw5B+S+INnTDQ==[/tex]，其中[tex=0.929x1.0]6ajasqZpuIbDhIXAyBtsFg==[/tex]满足边值 [tex=3.214x1.357]m4yUa360SKbDNwSJeh0AFHNfUtYXJ4r3ODvj8cNoqxw=[/tex]，[tex=0.929x1.0]JVQwv7R0g2cuiqhypom87g==[/tex]满足边值[tex=8.5x2.214]8hdol2pNHzpwNEbEhDZSp/VRlL++vczba4o/+vteAfFcjfu5WHxb0P2LS3Sfh7qqHU655ZdrnBLJ1goZTCspQe6ejwI964Vx0IDH50lrRBc=[/tex]，则显然[tex=2.5x1.214]r6OgLv2BHae3j5XhN4PP/w==[/tex]，且知[tex=9.571x2.429]O/TMUWtqkY3jBtEwqEkP3Vr2KtfAfDCEsoRjleIj0mfq0s2oUwKHeoIc89mT/bxwYQ36dmPpRa3U9ycARj1k+mRjM/xWOyTVcvpKIFfL5Gk=[/tex]，则显然[tex=2.5x1.214]gTr8zN08BF0JKo68TcGF2g==[/tex]是知[tex=8.857x2.429]IQt5kMyIZgsaxWq4RIFMoCP1CYbqHMeMoBxE/vhZle382quXuiwQL+2KmQ7AUprhiR39a9IOpG7DQ8Yuv9jDAQ==[/tex]，故[tex=3.786x1.286]amDEW4hGDj+x0COA3F3zvA==[/tex][tex=8.0x2.0]VY8u7nBaWBo+FU+4osDE4gG5I556Lfvc2VQ471NXG6P3KaFArY/u7Ime3JReCNO/Zviprcw1n0Jwhs3YHjrnVw==[/tex][tex=5.643x2.0]s1BjkLg1fS4BxEOG3k0x0J9NbObYUUGCbPqjx1d80fKjOoJW8j/Xrkgf6QFdPWqu[/tex] .
举一反三
- 试求一函数，在半径的园的内部是调和的，而且在圆周上取值：[tex=5.643x1.286]rjdZS4acQyGjFUUbF3NwgIG2d6mXOo8am2QUgzSdtsBcQLU3I40H8m9DZ5Kg1au/[/tex]，其中[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]都为常数 .
- 同阶实对称矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]合同的充分必要条件是[input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=5.786x1.286]g8SwOoc3mjjpDvp+UxwKlmKIyJYBt/w89p9ioRDdWtw=[/tex]', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为同型矩阵', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]的秩与正惯性指数都对应相等', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]的正惯性指数相等'], 'type': 102}
- 试求在单位圆[tex=0.857x1.0]FfIhW8W8Jb8XV2jfmtoNZA==[/tex] 内调和、在闭圆 [tex=0.857x1.143]eejPnmEu5LFY6GWX3u2XjQ==[/tex]上(除去其上两点[tex=1.714x1.214]jsM/Lg33JMLvoOCckk59rQ==[/tex] 外）连续的函数,这个函数在圆弧 上取值 1, 在单位圆周的其余部分上取值0 .
- 试求在单位圆[tex=0.857x1.0]FfIhW8W8Jb8XV2jfmtoNZA==[/tex]内调和、在闭圆[tex=0.857x1.143]eejPnmEu5LFY6GWX3u2XjQ==[/tex]上(除去其上两点[tex=1.5x1.214]jsM/Lg33JMLvoOCckk59rQ==[/tex]外)连续的函数，这个函数在圆弧[tex=1.214x1.571]DpUSZIabFhsiIKmG9s1M77y7ryDmN9/dhJimuTt0NlU=[/tex]上取值1，单位圆周的其余部分上取值0。
- [tex=3.143x1.286]REaUoNxha/GBn3DE8cgfDA==[/tex]为平行四边形，已知[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]及对角线交点的坐标分别为[tex=12.357x1.286]84XE/GmE6kpq9IAQaboLvGv2mG+w96Kdsumf86zGrZg=[/tex]。试确定点[tex=2.0x1.286]pztp19HoiX/t3GEAQHtMyg==[/tex]的坐标。
内容
- 0
设三阶矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]满足[tex=6.5x1.286]fKMuxXvMmkhZ6KBFJApU0RzCv0uzUPjL4nc3K7Aig7k=[/tex]，证明：[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆 .
- 1
设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]均为正定矩阵，证明[tex=5.0x2.786]075gCzZzsMRb6HYXYk9X92zk8W4u1qJBIO8aFf+ZsZxwp/haXQ2S0bij0nON3lddoX4sG6nvdaxHgFoCKqduPw==[/tex]为正定矩阵 .
- 2
已知线段[tex=1.571x1.286]cHJ4KDAad01mWuGaiQQpfA==[/tex]被点[tex=3.929x1.357]T+HHH1fotIq9eBc3izyp0g==[/tex]和[tex=4.786x1.357]v5sVj2WR3ZZ7vtIXPlLtVQ==[/tex]三等分，试求出这线段的两个端点[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]的坐标。
- 3
有[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]两个整数数列,从[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]中删去在[tex=0.786x1.286]q1djlrfSWHAqH21hBgtrSw==[/tex]中出现的那些数。
- 4
设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]是[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]阶方阵，且[tex=5.786x1.286]g8SwOoc3mjjpDvp+UxwKlmKIyJYBt/w89p9ioRDdWtw=[/tex]，则正确的是[input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=6.0x1.286]uT60fd3iTh3Cb0usaDNoWpVPAEKezd55b2DG5kI6kS4=[/tex]', '[tex=8.214x1.286]UjPR5vmYvABrZ+plCC6qBWfhfHN/toiy7hP5ZybHvQU=[/tex]', '[tex=8.214x1.286]IkA4G4ihMbb2iNZKyn/eihJfGfrq0WYtzVJVmzFbYws=[/tex]', '[tex=11.143x1.286]8yg3SxXjDJ8EDUlBLqEszERGAGhJRJw8wz16BkNIjmN17SoyLMSIbTt5Szsj6w+r[/tex]'], 'type': 102}