VECTOR IDENTITIES AND THEOREMS

1 BVECTORIDENTITIESANDTHEOREMSA=X Ax+YAy+ZAzA+B=X(Ax+Bx)+Y(Ay+By)+Z (Az+Bz) +AyBy+AzBzAAAXYzAxB=detIAxAyAzBxByBz=X (AyBz-AzBy)+y(A~Bx-AxBz)+Z (AxBy-AyBx)A.(Bx C)=B. (CxA)= C.(AxB)Ax(Bx C)=(A. C)B-( )C(AxB). (CxD)=( )(B. D)-( )(B. C)V'x V'\I1=0V'. (V'xA)=0V'x(V'xA)=V'(V'.A)-V'2A----1--(V 'xA)xA=(A. V')A- -V'( )2V'(\11<1 =\11V'<I>+<I>V'\11V'. (\11A)=A. V'\I1+\I1V'. 'x('11A)=V"I1xA+ 'I1V'xAV'( )=( ')B+( ')A+Ax(V'xB)+Bx (V'xA)V'.(AxB)=B. (V'xA)-A. (V'xB)V'x(AxB)=A(V'.B)-B(V'.A)+(B. V')11- (:4. V')BGauss'sDivergenceTheorem:IvV'.Gdv= 'sTheorem:L(V'x G).it da= EXPLICITFORMSOFVECTOROPERATORSC artesian(x,y,z):Aa'l1Aa'l1Aa'l1V"I1=x-+y -+z-axayazAaAxaAyaAzV'.)

2 =-+-+ (aAzaAy)A(aAxaAz)A(aAyaAx)V'xA=x---+y--- + '11a2'11a2'11V'2'11= -+ -+-ax2ay2az2 Cylindrical(p,</J,z):Aa'l1A1a'l1Aa'l1V"I1=P-+</J- -+ z -appa</Jaz-1a(pAp) '.A=-+--+-Pappa</Jaz0-A( )A(aApaAz)A1(a( )aAp)vxA=p+</J---+z---pa</Jazazappapa</JV'2'11= ! ~(p a'l1)+2-a2'11+a2'11p apapp2a</J2az2 VectorIdentitiesandTheorems539 Spherical(r,e, <jJ):Aa\l1A1a\l1A1a\l1V\I1=r-+e--+<jJ--arraersinea<jJ-1a(r2Ar)1a(sineAo)1aAt/> +-+--r2arrsineaersinea<jJv xA=r~(a (sineAt/ -aAo)+0(~aAr-~a(rAt/ )rsineaea<jJrsinea<jJrarA1(a(rAo)aAr)+<j J----rarae21a(2a\l1)1a(.a\l1)1a2\11V\I1= --r-+--sme-+-r2ararr2sineaeaer2sin2e a<jJ2))