Tuesday, 13 October 2015

homework and exercises - Passing from curl to vector product


I don't understand how to obtain second equation with first part in the equation ×A0ejkr=jk×A0ejkr.

Can you show me how to derive it?



Answer



[x1x1x1x2x1x3],A=[A1A2A3],kx=k1x1+k2x2+k3x3


×(Aeikx)=[e1e2e3x1x1x1x2x1x3A1eikxA2eikxA3eikx]=[A3eikxx2A2eikxx3A1eikxx3A3eikxx1A2eikxx1A1eikxx2]=ieikx[k2A3k3A2k3A1k1A3k1A2k2A1]=ieikx[e1e2e3k1k2k3A1A2A3]k×A

so ×(Aeikx)=ieikx(k×A)121212




More generally :



If ψ(x1,x2,x3) and A(x1,x2,x3) are scalar and vector functions respectively of the coordinates in R3 then ×(ψA)=ψ×A+ψ×A121212

Equation (03) is a special case of (04) with ψ(x)eikx,A(x)=constant
and the fact that ψ=eikx=ieikxk




Proof of identity (04): ×(ψA)=[e1e2e3x1x1x1x2x1x3ψA1ψA2ψA3]=[(ψA3)x2(ψA2)x3(ψA1)x3(ψA3)x1(ψA2)x1(ψA1)x2]=[A3ψx2+ψA3x2A2ψx3ψA2x3A1ψx3+ψA1x3A3ψx1ψA3x1A2ψx1+ψA2x1A1ψx2ψA1x2]=[A3ψx2A2ψx3A1ψx3A3ψx1A2ψx1A1ψx2]+[ψA3x2ψA2x3ψA1x3ψA3x1ψA2x1ψA1x2]=[e1e2e3ψx1ψx2ψx3A1A2A3]ψ×A+ψ[e1e2e3x1x1x1x2x1x3A1A2A3]×A=ψ×A+ψ×A


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...