电动力学课件01590new

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1、课程学习网页：http://science.nuaa.edu.cn/diandongEmail：pwu2000@nuaa.edu.cn§0-1矢量代数F两矢量相等时，它们对应分量一定相等vv1、矢量avz即若a=bvav=aevaza则a=ba=ba=baxxyyzzvveaF位置矢量ya=a----大小avvvaxyr=reP(x,y,z)varvea=----单位矢量xyvvvvveyyavvvvr=xex+yey+zezezvF在直角坐标系中a=axex+ayey+azezvzOexzx222xv222r=r=x+y+za=a=a+a+axyzvvv2、矢量运算(2)标积(

2、点积)a⋅b=abcosθezvvvvvvvv(1)相加a+b=cex⋅ey=0ey⋅ez=0eveyvvvvvvvvvvxQa+b=(axex+ayey+azez)+(bxex+byey+bzez)ex⋅ez=0vvvvvvvvv=(ax+bx)ex+(ay+by)ey+(az+bz)ezex⋅ex=1ey⋅ey=1ez⋅ez=1∴cx=ax+bxcy=ay+bycz=az+bzav⋅bv=(aev+aev+aev)⋅(bev+bev+bev)xxyyzzxxyyzzvvvvvbccb=axbx+ayby+azbzbavavvvvvθF满足交换性a⋅b=b⋅av平行四边形法则三

3、角形法则a1vvvvv(3)矢积(叉积)av×bv=(absinθ)eva×b=(axex+ayey+azez)vcvvvcv×(bxex+byey+bzez)b方向：满足右手螺旋法则vvvvv=(aybz−azby)ex+(azbx−axbz)eyθva×b=absinθezvvavv+(axby−aybx)ezezeeyvvvvvvvvvxexeyezveex×ex=0ex×ey=ezvvexyev×ev=0ev×ev=eva×b=axayazevx×evx=0evy×evy=0evz×evz=0yyyzxvvvvvvvvvvvvvvbxbybzex×ey=ezey×ez=e

4、xez×ex=eyez×ez=0ez×ex=eyvvvvF反交换性a×b=−b×a→→vvvvvvT=abee+abee+abee33111112121313(4)并矢→→vv+abevev+abevev+abevevvvvvvvvvF一般T=Tee212122222323a=ae+ae+ae=++∑∑ijij+abevev+abevev+abevev设有112233bb1e1b2e2b3e3ij==11313132323333→→vvvvvvvvvv并矢T=ab=a1b1e1e1+a1b2e1e2+a1b3e1e3eiej：张量的基Tij：张量在基上的分量vvvvvv+abee

5、+abee+abee21v2v122v2v223v2v3Tij=Tji----对称张量⎡T11T12T13⎤⎢⎥TTT+abee+abee+abee⎢212223⎥313132323333⎢TTT⎥Tij=−Tji----反对称张量⎣313233⎦F在三维空间有9个分量i=ji≠j⎡100⎤⎢⎥a1b1a1b2a1b3矩阵形式⎡T11T12T13⎤当Tij=1Tij=0⎢010⎥ababab⎢⎥⎢⎣001⎥⎦212223⎢T21T22T23⎥→→vvvvvva3b1a3b2a3b3⎢⎣T31T32T33⎥⎦则I=e1e1+e2e2+e3e3----单位张量→→33vvT=∑∑Ti

6、jeiejF张量运算规则ij==11说明：→→33vvnϕT=∑∑ϕTijeiej----标量与各分量相乘F张量：N维空间内，有N个分量的一种量；ij==11坐标变换时，这些分量作线性变换v→→vvvvvvf⋅T=f⋅(ab)=(f⋅a)b⎫n：张量阶数v→→vvvvvv⎪相邻成f×T=f×(ab)=(f×a)b⎬在三维空间：→→→→vvvvvvvv⎪分作用2T⋅T'=(ab)⋅(cd)=(b⋅c)ad⎭二阶张量的分量3=9→→→→vvvvvvvv1双点乘T:T'=(ab):(cd)=(b⋅c)(a⋅d)一阶张量的分量3=3----矢量0----相邻先点乘，余下的再点乘零阶张量的

7、分量3=1----标量2vvvvvv→→vv→→v[例1]证明：I⋅f=f⋅I=f(5)混合积c⋅(a×b)=ca×bcosθvvvv→→vvvvvvvvvva×ba×b=absinϕ证：I⋅f=(e1e1+e2e2+e3e3)⋅(f1e1+f2e2+f3e3)vvvvvvvvvvc----平行六面体的底面积=(fe⋅e)e+(fe⋅e)e+(fe⋅e)e111122223333vccosθ----平行六面体的高vvvθbvvv=fe+fe+fev112233ϕabsinϕvc⋅