1-20 已知直角坐标系中的矢量A?aex?bey?cez，式中a, b, c均为常数，A是常矢量吗？试求该矢量在圆柱坐标系及圆球坐标系中的表示式。
解 由于A的大小及方向均与空间坐标无关，故是常矢量。
已知直角坐标系和圆柱坐标系坐标变量之间的转换关系为
r?x2?y2；??arctanyx； z?z
求得 r?a2?b2；??arctanba； z?c
sin??ba2?b2；cos??aa2?b2
又知矢量A在直角坐标系和圆柱坐标系中各个坐标分量之间的转换关系为
??Ar??cos?sin?0??Ax??A???????sin?cos?0????A??Az????001?y?
????Az??将上述结果代入，求得
?ab0??A?22r??a?a2?b2????a??a2?b2??A?a????bb0????????b????0? ?A???a2?b2a2?b2z??001?????????c???c????即该矢量在圆柱坐标下的表达式为
A?e22ra?b?ezc
直角坐标系和球坐标系的坐标变量之间的转换关系为
?x2?y2?z2；??arctan??x2?y2r????z?；??arctan??y????x?
由此求得
r?a2?b2?c2；??arctan??a2?b2????c?；??arctan??b???a?? 矢量A在直角坐标系和球坐标系中各个坐标分量之间的转换关系为
??Ar??sin?cos?sin?sin?cos????A??????cos?cos?cos?sin??sin???Ax?A????y?
?A??????sin?cos?0????Az??求得
?Ar??sin???sin?sin?cos???a?a2?b2?c2??A??cos?sin??sin??????cos??cos?cos????b??????0? ?A??????sin?cos?0????c?????0???
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即该矢量在球坐标下的表达式为
A?era2?b2?c2。
1-21 已知圆柱坐标系中的矢量A?aer?be??cez，式中a, b, c均为常数，A是常矢量吗？试求??A及??A以及A在相应的直角坐标系及圆球坐标系中的表示式。
解 因为虽然a, b, c均为常数，但是单位矢量er和e?均为变矢，所以A不是常矢量。
已知圆柱坐标系中，矢量A的散度为
??A?1??rA?AAzr?rr??1?r?????z 将A?aer?be??cez代入，得 ??A?1?r?r?ar??0?0?ar 矢量A的旋度为 erezerre?reez???A????r?r?????r??z?r???z?brez ArrA?Azarbc已知直角坐标系和圆柱坐标系坐标变量之间的转换关系为
x?rcos?;
y?rsin?;
z?z
cos??xxyx2?y2?; sin?ya?x2?y2?a 又知矢量A在直角坐标系和圆柱坐标系中各个坐标分量之间的转换关系为
??Ax??cos??sin?0??Ar?A??y???sin?cos?0??????
?Az????001??A?????Az??将上述接结果代入，得
?x?y?0??b???x?ay??Ax??A??a?yxay????a0??a????????A?a????b???y?bx? z?001a?????c?????c????????即该矢量在直角坐标下的表达式为
A???x?by???e?b??ax???y?ax??ey?cez，其中x2?y2?a2。
矢量A在圆柱坐标系和球坐标系中各个坐标分量之间的转换关系
??Ar??sin0cos???Ar??A???????cos?0?sin??????A?????010??A??
????Az??
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以及sin??a，cos??r?a0??Ar?r?c???A?0???r?A???01?????c，求得 rc??a2?c2?r??a??r??r????a???????b???0???0? r???b?0???c???b??????????即该矢量在球坐标下的表达式为A?rer?be?。
1-22 已知圆球坐标系中矢量A?aer?be??ce?，式中a, b, c均为常数，A是常矢量吗？试求??A及??A，以及A在直角坐标系及圆柱坐标系中的表示式。
解 因为虽然a, b, c均为常数，但是单位矢量er，e?，e?均为变矢，所以A不是常矢量。
在球坐标系中，矢量A的散度为
1?21??sin?A???1??A?2rAr?r?rrsin???rsin?????A????????? ?将矢量A的各个分量代入，求得??A?矢量A的旋度为
err2sin????A??rAre?rsin????rA?e?r? ??rsin?A?2ab?cot?。 rrerr2sin????rae?rsin????rbe?r?b?e? ??rrsin?c利用矢量A在直角坐标系和球坐标系中各个坐标分量之间的转换关系
?Ax??sin?cos?????Ay???sin?sin??A???z??cos?cos?cos?cos?sin??sin??sin???Ar???cos????A??
?0????A??2222?x?x?yx?y??sin???cos??22222x?ya?x?y?z?以及?，?，求得该矢量在直角坐标下的表达式为
yzz?sin???cos???22??222x?yax?y?z?? 13
A???bxzcy?cx??x????ax2?y2?x2?y2?ex???y?byz??ax2?y2?x2?y2?ey??
22 ???bx?y??z??e?a?z?利用矢量A在圆柱坐标系和球坐标系中各个坐标分量之间的转换关系 ??Ar?cos?0??A?rzr??a0??b???a??r?az??A??sin??01???01??????????0??A????a?Az????cos??sin?0????A???0??z??a?r??b??c?? a0?????c??????z?bar??求得其在圆柱坐标下的表达式为
A????r?baz???e?b?r?ce????z?ar??ez。
1-23 若标量函数?1(x,y,z)?xy2z，?2(x,?,z)?rzsin?，?sin?3(r,?,?)?r2，试求?2?1，?2?3。
解 ?2??2?1?2?1?21??x2??y2??1?z2?0?2xz?0?2xz ?2??1????2?1??2?2r?r??r?r?????2?22r2?????2?????z2 ?1?r?r?rzsin???1r2??rzsin???0 ?2?1????3?1????3?1?3?r2?r??r2?r???r2sin?????sin??????r2sin2????2?3????2???
?1??2?2???1??sin?cos??r2?r??rsin?r3?r2sin?????r2???0 ?2sin?r4?cos2??sin2?1r4sin??r4sin? 1-24 若 A(x,y,z)?xy2z3ex?x3zey?x2y2ez
A(r,?,z)?e2rrcos??ezr3sin?
A(r,?,?)?errsin??e1?rsin??e1?r2cos? 试求??A，??A及?2A。
?2?2及14
解 ①??A??Ax??Ay??Az?y2?x?y?zz3?0?0?y2z3； exeyezexeyez??A????????x?y?z??x?y?z AxAyAzxy2z3x3zx2y2??2x2y?x3?ex??3xy2z2?2xy2?e?2y?3xz?2xyz3?ez； ?2A?ex?2Ax?e2y?Ay?ez?2Az
??2xz3?6xy2z?ex?6xzey??2y2?2x2?ez；
② ??A?1??rA?A?1?r?rr??1r????Az?z?r?r?r3cos???0?3rcos? erree?zeree?z??A???r?r??r??r???z??r???z ArrA?Azr2cos?0r2sin??err?r2cos???e??ez???2rsin?r?r2sin?? ?errcos??2e?rsin??ezrsin??2A?e?????2AAr2?A???2A?2?Ar?rr?r2?r2??????e?????A??r2?r2??????e2z?Az ?2ercos??2e?sin??3ezsin?；（此处利用了习题26中的公式） ③ ??A?1?r2?r?r2Ar??1??rsin?sin?A???1??A????rsin????????? ?1?r2?r?r3sin???1?rsin????r?1sin2???0 ?3sin??2cos?r2； ere?e?ere?e?r2sinrsinrsin??A?????r?r2sin????r??r??????r????
ArrA?rsin?A?rsin?sin?r?1sin?cos? ?e?sin????2?r???r3??ecos????r3???e?sin??????r2?cos???
15
??esin?r3?e2cos?r?r3?e????cos??sin??r2??； ?2A?e?222?r???Ar?r2Ar?r2sin????sin?A2?A?????r2sin?????
?e??2AA2?Ar2cos??A????????r2sin2??r2???r2sin2?????
?e?2A?2?A??r2cos??A????A??r2sin2??r2sin????r2sin2?????
将矢量A的各个坐标分量代入上式，求得
?2A?e??cos2?4cos???2cos?2sin??cos?r?rsin??r3???e???r?r3???e?r4sin2? 1-25 若矢量A?ecos2?rr3, 1?r?2，试求? V ??AdV，式中V为A所在的区域。解 在球坐标系中，dV?r2sin?drd?d?，
??A?1?21r2?r?rAr????rsin????sin?A???1rsin???A????????? 将矢量A的坐标分量代入，求得
?V??AdV???V????cos2??r4?2??2?cos2?2?dV???0d??0d??1r4rsin?dr
???2?d??cos2?0?02sin?d????2?0cos2?d????
1-26 试求
? S(er3sin?)?dS，式中S为球心位于原点，半径为5的球面。
解 利用高斯定理，?SA?dS??V??AdV，则
?A?dS????AdV??2??56sin?SV0d??0d??0rr2sin?dr?75?2 16
电磁场与电磁波（杨儒贵第二版）课后答案-1
