矢量代数矢量代数

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1、§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ax=bxay=byaz=bzavveaF位置矢量ya=a----大小vayvvvaaxr=rervP(x,y,z)ea=----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+axyz2、矢量运算vvvvv(2)标积(点积)a⋅b=abcosθ(1)相加a+b=cvvvvvvQav+bv=(aev+aev+aev)+

2、(bev+bev+bev)ex⋅ey=0ey⋅ez=0ex⋅ez=0xxyyzzxxyyzzvvvvvvvvv=(ax+bx)ex+(ay+by)ey+(az+bz)ezex⋅ex=1ey⋅ey=1ez⋅ez=1vvvvvvvv∴cx=ax+bxcy=ay+bycz=az+bza⋅b=(axex+ayey+azez)⋅(bxex+byey+bzez)vvvvvbccb=axbx+ayby+azbzbavavvvvvθF满足交换性a⋅b=b⋅av平行四边形法则三角形法则a1vvvvv(3)矢积(叉积)av×bv=(absinθ)eva×b=(axex+

3、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=absinθezvvav+(axby−aybx)ezezeveyvvvvvvvvvxexeyezeveyex×ex=0ex×ey=ezvvxvvvvvvvvvvva×b=axayazex×ex=0ey×ey=0ez×ez=0e×e=0e×e=evvvvvvvvvyyyzxex×ey=ezey×ez=exez×ex=eyvvvvvbxbybzez×ez=0ez×ex=eyvvvvF反交换

4、性a×b=−b×a(4)并矢a1b1a1b2a1b3矩阵形式⎡T11T12T13⎤ababab⎢TTT⎥vvvv212223⎢212223⎥设有a=a1e1+a2e2+a3e3a3b1a3b2a3b3⎢⎣T31T32T33⎥⎦vvvv→→33vvb=b1e1+b2e2+b3e3F一般T=∑∑Tijeiej----二阶张量→→vvvvvvvvvvij==11并矢T=ab=a1b1e1e1+a1b2e1e2+a1b3e1e3eiej：张量的基Tij：张量在基上的分量vvvvvv+abee+abee+abeeT=T----对称张量T=−T----反对称张量

5、212122222323ijjiijjivvvvvvi=ji≠j+a3b1e3e1+a3b2e3e2+a3b3e3e3当Tij=1Tij=0→→vvvvvvF在三维空间有9个分量则I=ee+ee+ee----单位张量11223333tvvT=∑∑TijeiejF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

6、×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分量3=1----标量2vvvvvvvvvvvv→→vv→→vf⋅(ab)=(f⋅a)b[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)⋅(f1e

7、1+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⋅(a×b)a=f----平行六面体的体积vvvvvvvvv同理可证∴c⋅(a×b)=b⋅(c×a)=a⋅(b×c)v→→vf⋅I=f----三矢量按循环次序轮换，其积不变vvvf=b(ac+ac)−c(ab+ab)eeexxyyzzxyyzzvvvvvxyzvvvv(6)三矢量矢积a×(b×c)b×c=bxb

8、ybz=b(a⋅c−ac)−c(a⋅b−ab)vvvvvvxvvxvxvxxxccc设d=b×cf=a×dx