求以下各向量的散度及旋度([tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]为常向量)：[br][/br][tex=2.214x1.071]UufID/HmohTQ8WZj5Kfllg==[/tex]；

2022-06-29

求以下各向量的散度及旋度([tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]为常向量)：[br][/br][tex=2.214x1.071]UufID/HmohTQ8WZj5Kfllg==[/tex]；

答案：

解  因[tex=11.429x1.357]QFDMS6QulI3cZwwdG1qBWzUHV669BhH792Fx7chiybRdL8eKS7lSlu1BithwXKoda0im5Nn71GDYd/lhhXGQGQ==[/tex][tex=12.571x1.357]tL2b+dPrXNwBIwAs3i4bPuDALxnRgvinG5VNtwpftsdN5EkBSrUT9Mb9l49zCwJP4VqVGfjxzA9jFe5Sj86OmQ==[/tex]则[tex=11.071x2.429]NovbxKl63Ey/milqTcbe/0nek9KfvUZ0j9heLVnXHVCl3CRlzVG2i8Og0P1CNzzr+r01y31WzHq7CJKJD/Woyi+D2kYvbDywPair23bD6Pf1kk/LD79u1ibJX7IxEYa6[/tex][tex=15.143x2.643]B+0BUeIRKMBYYBEEkL2q3W5msm+tImuuT6uVjeCB8GamWEch/ThdRLPd+bHGWbMNs4RkEODAnZ9aaZ0W570+O2hcjw4SWhP8Daml+eaEijLbTh5gwlGFfmEmvUCwdA4wu1qvM3nJRT97T9JMCOQ8RDw7O/eQXXS68aSJ/5dPZ0M=[/tex][tex=21.429x5.786]uqh+oOvD2P9iqZ7dD7XO1IDLXKX1enJczsXWAtGbsrqMy2bzu5zpmJGjK5p3fzJBtk9QYwaSpwLVREQ4mRvE4rbfKF/DrrdHPRCF2w8bcd4eV6ckcW6v9trzLWiJaayMr0zpEAlLz293NFhxemMhYhLOovg9gPlkvple5umgKHDkvKhyLOS5pxqFLTfhe7vxGbwVWopmav4iI+NKQhNBNDpym1I1bVy3xNO5WI+S9WsIvccrdUnwA8DJ3QPUWRxS2c27dh533DxXGTVv7/OAyT9Kg2pDoAwyRcqwNys6om/d3NiJpwUbuPoyMmgLVSUj[/tex]。

举一反三

内容

0
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]HtShgpkNmCs66SDQIt6cYg==[/tex],使[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex][img=203x220]17e1aa8f4234a38.png[/img]
1
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]NeILClIc8twxeO5vjYm8Dw==[/tex],将[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的外表面接地。[img=203x220]17e1aa8f4234a38.png[/img]
2
设[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex],[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]是整数且不全为0，而[tex=9.857x1.214]hhHzRVDsWGXE+Yltfe39hDUdsl3Yzf9jGRPDg4wYEoJYR6eBGAfms1GUG8a2PN1l[/tex]，证明[tex=0.571x1.0]TcM6B5Wrs5vy9dWrxRPSdg==[/tex]是[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]与[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的一个最大公因数当且仅当[tex=4.214x1.357]jI1oqbiyUHYU1xbNvvBdDK5ib01K7Vb7AmVkL7RKEyk=[/tex]
3
图9-26所示，金属球[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]和金属球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]同心放置，它们原先都不带电。设球为[tex=0.857x1.0]6WwbFXETRyeyXlvAruSoNg==[/tex]，球壳[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的内、外半径分别为[tex=0.857x1.0]BNzznGkXRFuGyw2vMy6rWw==[/tex]和[tex=0.857x1.0]Fz01PbYkU0SRGm3tB5KjiA==[/tex]。求在下列情况下[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]、[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]的电势差：使[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]带[tex=1.286x1.143]HtShgpkNmCs66SDQIt6cYg==[/tex][img=191x199]17e1ac6b01d9ad2.png[/img]
4
设 [tex=3.286x1.571]iZ5ZnjEFgki3+QaUvvTZiw==[/tex], [tex=2.286x1.357]tJpS7BJBKcBsyfJJa+KKng==[/tex], [tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex], [tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex] 的夹角为 [tex=0.857x2.143]GnDK4PNA+fAzobGq0aj/Jw==[/tex], 求向量 [tex=1.786x1.143]S9ildicJrv0Uvz/I1XnOaA==[/tex] 与 [tex=1.786x1.143]+JWM/sEBO49/oaEmZ4MdCQ==[/tex] 的夹角.