发布时间 : 星期一 文章电磁场与电磁波理论(第二版)(徐立勤,曹伟)第1章习题解答更新完毕开始阅读255fc488680203d8ce2f24bc
第1章习题解答
1.4 计算下列标量场u的梯度?u:
(1)u?x2y3z4; (2)u?xy?yz?zx; (3)u?3x2?2y2?3z2。
????u??u??u???解:(1) ?u?ex?ey?ez?ex2xy3z4?ey3x2y2z4?ez4x2y3z3
?x?y?z???u??u??u???(2)?u?ex?ey?ez?ex?y?z??ey?x?z??ez?y?x?
?x?y?z???u??u??u???(3)?u?ex?ey?ez?ex6x?ey4y?ez6z
?x?y?z1.6 设f?x,y,z??x2y?y2z?1。试求在点A?2,1,3?处f的方向导数最大的方向的单位矢量及其方向导
数。方向导数最小值是多少?它在什么方向?
???f??f??f????ey?ez?ex2xy?ey?x2?2yz??ezy2 解: ?f?ex?x?y?z?因为?f??f??f??f????ex?ey?ez?ex4?ey10?ez
?x?y?zA所以
?f?l?f?l?117max????el??ex4?ey10?ez?117
??117min????el???ex4?ey10?ez?117 1.10 求下列矢量场在给定点的散度值:
????(1)A?xyz?exx?eyy?ezz?在M?1,3,2?处;
????(2)A?ex4x?ey2xy?ezz2在M?1,1,3?处;
????22212(3)A??exx?eyy?ezz??x?y?z?在M?1,1,1?处。
???Ax?Ay?Az??解:(1) ??A????2xyz?2xyz?2xyz?6xyz??A?36
M?x?y?z(2)??A??????Ax?Ay?Az???4?2x?2z??A?12
M?x?y?z???Ax?Ay?Az(3)??A? ???x?y?z?x2?y2?z2?x2??x2?y2?z2??3?x2?y2?z2?y2x2?y2?z23???x2?y2?z2?z2x2?y2?z2?3?2x?y?z222
????A
M2 31
1.15 求下列矢量场在给定点的旋度:
??2?2?2(1)A?exx?eyy?ezz在M?1,0,?1?处; ????(2)A?exyz?eyzx?ezxy在M?2,1,3?处;
????(3)A??ex?ey?ez?xyz在M?1,1,?1?处。
e?x解:(1)???A????xAxe?x(2)???A????xAxe?x(3)???A????xAx
e?ye?z???y?z?0???A??0
MAyAze?ye?z?????y?z?0??A?0
MAyAze?ye?z???y?z?e?xz?xy??e??x?y?xy?yz??ez?yz?xz?AyAz???A?M?02