电磁场与电磁波基础知识总结.doc

电磁场与电磁波基础知识总结
第一章
一、矢量代数 A?B=ABcos?
A?B
=
eAB
ABsin? A?(B?C) = B?(C?A) = C?(A?B)
A??B?C??B?A?C??C?A?C?
二、三种正交坐标系 1. 直角坐标系矢量线元dl
矢量面元dS?exdxdy?eydzdx?ezdxdy ?exx?eyy?ezz???
体积元dV = dx dy dz 单位矢量的关系ex?ey2. 圆柱形坐标系矢量线元dl体积元dV
?ez ey?ez?ex ez?ex?ey
?e?d??e??d??ezdzl 矢量面元dS?e??d?dz?ez?d?d?
e??ez=e?
ez?e??e?
??d?d?dz 单位矢量的关系e??e??ez
3. 球坐标系
矢量线元dl = erdr? + e?? rd? ? e?? rsin? d?? 矢量面元dS = er ? r2sin? d? d? 体积元
dV?r2sin?dr?d? 单位矢量的关系e??e?=er
e??er?e?
er?e??e?
三、矢量场的散度和旋度 1. 通量与散度
???
2. 环流量与旋度
S
?A?dS divA???A?lim
?v?0
S
A?dS?v
????lA?dl rotA=en
3. 计算公式
?S?0
A?dl??lim
l
max
?S
??A?
?Ax?Ay?Az
?x?y?z
??A?
1?1?A??Az
(?A?)?? ???????z
??A?
1?21?1?A?
(rA)?(si?nA? r?
r2?rrsin???rsi?n??
ex
?
??A?
?xAxey??yAyez??zAz
e?1?
??A?
???A?e?????A?ez?
?zAz
- 1 -
er
1???A?2rsin??r
Ar e?? ?? rA?e?? ??rsin?Az
??AdV 4. 矢量场的高斯定理与斯托克斯定理??A?dS??SV??A?dl????A?dS lS
四、标量场的梯度
1. 方向导数与梯度
?u
?l?lim
P0u(M)?u(M0)?u ?l?0?l?l?P0?u?u?ucos??cos??cos? ?x?y?z
?u?el??ucos? gradu?
2. 计算公式
?u?u?u?uen?ex?ey+ez ?n?x?y?z?u?ex?u?u?u?ey?ez?x?y?z ?u?e??u1?u?u?e??ez ??????z?u?er?u1?u1?u ?e??e??rr??rsin??z
五、无散场与无旋场
??(??A)?0 F???A
2. 无旋场??(?u)?0 F?-?u 1. 无散场
六、拉普拉斯运算算子
1. 直角坐标系
?2u?2u?2u?u?2?2?2?x?y?z2
2?2A?ex?2Ax?ey?2Ay?ez?2Az?2Ay?2Ay?2Ay?2Ax?2Ax?2Ax?2Az?2Az?2Az22?Ax???2 , ?Ay??? , ?Az???22222222?x?y?z?x?y?z?x?y?z
2. 圆柱坐标系
1???u?1?2u?2u?u????22?2?????z???????2
??212?A??12?A??2?2A?e???2A??2A??2?e?A?A??????ez?Az?2?2????????????
3. 球坐标系
1??2?u?1???u?1?2u ?u?2?r?sin?????r?r??r?r2sin???????r2sin2???22
- 2 -
?A??222cot?2?A?2
?2A?er??A?A?A??rr??r2r2r2??r2sin????
?22?Ar12cos??A??
??e???A??A???22222??????rrsin?rsin???
?
???
?2?Ar212cos??A??
??e???A??A?????22222
????rsin?rsin?rsin???
七、亥姆霍兹定理
如果矢量场F在无限区域中处处是单值的,且其导数连续有界,则当矢量场的散度、旋度和
边界条件(即矢量场在有限区域V’边界上的分布)给定后,该矢量场F唯一确定为
F(r)????(r)???A(r)
其中
?(r)?
1
4????F(r?)1
? dVA(r)??Vr?r?4????F(r?)