Irrotational and divergenceless?

[jlmac2001]

I'm not really sure what I'm being asked to do with the following questions. Will someone help me?

How can you show that E(r)=-gradV(r) has zero curl an is irrotational, i.e. the quantity (grad x E) =0


How can you show that the magnetic field ,B(r)=grad x A(r), is divergenceless i.e. that the quantity grad dot B(r) =0?

 

[HallsofIvy]

Basically, you just go ahead and do it!

That is, start by assuming that you have V(x,y,z) so that E(r)=-gradV(r)= -V_xi- V_y-V_z and calculate curl(E)= curl(grad V). See what happens!

 

[Norman]

if B=curl A, then div B = div (curl A), the divergence of a curl is always zero. Actually this proof is usually done in the opposite order since the div B=0 is one of Maxwell's equations, valid even for time dependent phenomenon (the old there are no magnetic monopoles theorem). If div B=0 then B can be expressed as the curl of another vector, which gets labeled A and we call it the magentic vector potential.
Hope this helps.
Cheers,
Norm