err2sin????A??rAre?rsin????rA?e?r? ??rsin?A?err2sin????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?ys???sin???co?22222ax?y?x?y?z?以及?，?，求得该矢量在直角坐标
yzz?cos???sin????222?ax2?y2x?y?z??下的表达式为
?bxzA??x??22?ax?y??bx?y???e ?z???za??22??byz?e??y??x22?22?x?y?ax?y?cy??ey22?x?y?cx
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??sin?????A????0?A???z??cos?cos?0?sin??r0??Ar?????a?1??A???0?z??0???A????a?za0r?a?b??0??a??r?az??????1?b???c? ???0???c???z?br????a?求得其在圆柱坐标下的表达式为
b?b???A??r?z?er?ce???z?r?ez。
a?a???
16
1-23 若标量函数?1(x,y,z)?xy2z，?2(x,?,z)?rzsin?，?3(r,?,?)?sin?，试求2r?2?1，?2?22及??3。
2?2?2解 ??1?2?1??11??x2??y2??z2?0?2xz?0?2xz
?2?1????2?1??2?2?2?r?r??r?r???r2??????2????2?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?2sin??1r2?r??rr3?????r2sin????sin?cos???r2???0 ?2sin?cos2??sin2?1r4?r4sin??r4sin?
1-24 若 A(x,y,z)?xy2z3ex?x3ze2y?xy2ez A(r,?,z)?e23rrcos??ezrsin?
A(r,?,?)?errsin??e1?rsin??e1?r2cos? 试求??A，??A及?2A。
解 ①??A??Ax?Ay?Az?x??y??z?y2z3?0?0?y2z3；
exeyezexeyez??A????????x?y?z??x?y?z AxAyAzxy2z3x3zx2y2??2x2y?x3?e?2x?3xyz2?2xy2?ey??3x2z?2xyz3?ez； ?2A?e2x?Ax?e22y?Ay?ez?Az
??2xz3?6xy2z?ex?6xzey??2y2?2x2?ez；
17
② ??A?1?r?r?rA?Ar??1?r????Az?z?1?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?2Ar2?A???2A?2?Ar?2r????Ar?r2?r2??????e?????A??r2?r2??????ez?Az
?2ercos??2e?sin??3ezsin?；
（此处利用了习题26中的公式） ③ ??A?1?r2A?1??sin?A?1?r2?r?r?rsin?????rsin???A????????? ?1?r2?r?r3sin???1?rsin????r?1sin2???0 ?3sin??2cos?r2；
ere?e?ere?e?r2sinrsin??A?????r?r2sin??rsin??r??r??????r????ArrA?rsin?A?rsin?sin?r?1sin?cos? ?e????sin???2cos???sin??rr3???e???r3???e????r2?cos??? ??esin?2cos??sin??rr3?e?r3?e???cos??r2??； ?2A?e??22???2A?sin?A2?A??rr?r2Ar?r2sin??????r2sin?????
18
?A2?A2cos??A???e???2A??2?2?2r?22?
rsin?r??rsin?????A??22?Ar2cos??A???e???A??22?2?22?
rsin?rsin???rsin?????将矢量A的各个坐标分量代入上式，求得
cos??cos2?4cos???2cos?2sin?? ?2A?er???e??e???4332??rsin?rrrrsin?????cos2?, 1?r?2，试求? ??AdV，式中V为A所在的区域。1-25 若矢量A?er 3 Vr解 在球坐标系中，dV?r2sin?drd?d?，
1?21??sin?A???1??A?2rAr?r?rrsin???rsin?????A????????? ?将矢量A的坐标分量代入，求得
22??2cos??cos2??2????AdV??dV??d?d?rsin?dr 44?V?V?????001r?r?2?
???d??0?02?cos2?sin?d????cos2?d????
021-26 试求
? S(er3sin?)?dS，式中S为球心位于原点，半径为5的球面。
SV解 利用高斯定理，?A?dS????AdV，则
?A?dS????AdV??d??d??SV002??506sin?2rsin?dr?75?2 r 19