第1章 矢量分析

1、電路理論處理電路理論處理“集中參數(shù)集中參數(shù)”系統(tǒng)系統(tǒng)( (注注1)1)直流電路中：系統(tǒng)變量是常量，數(shù)學(xué)工具：代數(shù)方程直流電路中：系統(tǒng)變量是常量，數(shù)學(xué)工具：代數(shù)方程交流電路中：交流電路中： 標(biāo)量，標(biāo)量， 微分方程微分方程系統(tǒng)參數(shù)：大小、方向、分布系統(tǒng)參數(shù)：大小、方向、分布 “空間空間”、“矢量矢量” “矢量代數(shù)矢量代數(shù)”、“矢量偏微分方程矢量偏微分方程”電磁場理論主要處理電磁場理論主要處理“分布參數(shù)分布參數(shù)”系統(tǒng)系統(tǒng).)()(zzyyxxzzyyxxaBaBaBaAaAaABAA+BBA0 BAAAAABBABAcosBABAABCcos222BABACCCCCBACcos222BABACCCC

2、CBACBABBAABABACC2)()()cos(BABAABCABBABAsinBAABABAxBABa四邊形面積=)sin(ABBAzyxzyxzyxABABBBBAAAeeeBAaBA)sin(ABBACABACBA)(CBACBA)()()()()(BACACBCBA)()()(BACACBCBA)()()(BACACBCBABCBCBxCBCa四邊形面積=)sin(BCCBBCBxCBCa底面積=)sin(BCCBAABBCA,高=BCAA,cosBCBxCBCa底面積=)sin(BCCBAABBCA,高=BCAA,cosABDDaCABDC)()()(BACACBCBA),;,(

3、zyxeeezyx),;,(zreeezr),;,(eeerrzzyyxxaAaAaAAzxyz平平面面x平平面面y平平面面zzyyxxBABABAABBA?zyxzyxzyxBBBAAAaaaBA)()()(?)(?)xyyxzzxxzyyzzyxBABAaBABAaBABAaBAzyxaaazxyz平平面面x平平面面y平平面面dzadyadxal dzyxSddydzaSdxxdzdxaSdyydxdyaSdzzdxdydzdV zzrraAaAaAAz z平平面面r柱柱面面平平面面xyzz平面z平面r柱面柱面xyzzr,zyx,xyrayaxaaOz平面z平面r柱面柱面xyzl drdd

4、zadradral dzr)(SddzdraSdrr)(drdzaSddrdraSdzz)(xyzddrdrdzdzrddrdV)(xyzRaAaAaAARRxyzR)sin()(dRadRadRal dRSd)()sin(RddRaSdRRdRdRaSd)sin(dRRdaSd)(dRdRRddV)sin()(xyzddR dRRdRsinsinRd)sin()(dRadRadRal dRxyzddR dRRdRsinsinRdSd)()sin(RddRaSdRRdRdRaSd)sin(dRRdaSd)()sin()(dRadRadRaRxyzddR dRRdRsinsinRddRdRRdd

5、V)sin()()sin()(dRadRadRaR等值面等值面bacdodldll duuuldldUlUdndUagradUUnl dul dzuayuaxuadzzudyyudxxududzadyadxal dzyxzyx lnlaUnUlnnUlU)(cos,l dUdU)(zayaxazyxsin11RaRaRaRzararazr1333222111111uhauhauhauuudzadyadxal dzyxzayaxazyxdzadradral dzr)(zararazr1)sin()(dRadRadRal dRsin11RaRaRaRcos),(0RVRVVVE?Esin11RaR

6、aRaR?VE0)sincos(VaaVERxyzRzVRVRVV00cos),(zayaxazyx0VaVEz0)sincos(VaaVER0VaVEzASASsSdAdSaSdnSCAVsdAVAdivS0limzAyAxAAzyxARARARRRARsin1)sin(sin1)(122zAArArrrAzr1)(1333222111111uhauhauhauuu?)(?zayaxazyxzayaxazyxzAyAxAAzyxzzyyxxaAaAaAAAVsdAVAdivS0limSVsdAdvA)(AAdiv散度定理散度定理ACl dACdlaldlCAASl dASAcurlArotC0limAzyxzyxzyxABABBBBAAAaaaBAaBA)sin(zyxzyxBBBzyxaaaBzayaxazyxzzyyxxAaAaAaACSl dASdA)(ASl dASAcurlArotC0limvvfuufvufvuvuCuCCuCC),()(是常數(shù),)(是常數(shù), 0是標(biāo)量函數(shù)是常數(shù)是常矢量uuFFuFuGFGFCFCFCCC,)()(,)(, 0GFFGGFuFuFuFuGFGFCFCFCCC)(,)()(,)(, 0是標(biāo)量函數(shù)是常數(shù)是常矢量先看兩個恒等式先看兩個恒等式(可逆可