How Can a Generic Vector Be Decomposed Using the Helmholtz Theorem?

[Legion81]

I have to show that a generic vector can be decomposed into an irrotational and solenoidal component:

V(r) = -Grad[phi(r)] + Curl[A(r)]

I'm not sure how to start. Do I need to take the curl or div of V and use a vector identity? Any help would be greatly appreciated!

 

[Legion81]

Helmholtz' Theorem starts with the two components in my original post and defines the divergence and curl as:

div[V] = s(r)
and
curl[V] = c(r), where div[c(r)] = 0

But I can't find anything about how we can define a generic vector as two components:

V = -grad[phi] + curl[A], where "phi" is the scalar potential and "A" is the vector potential. I need to do this before I can show that s(r) and c(r) uniquely specify the vector.

I hope that makes my problem a little more clear.

 

[gabbagabbahey]

I have to show that a generic vector can be decomposed into an irrotational and solenoidal component:

V(r) = -Grad[phi(r)] + Curl[A(r)]

I'm not sure how to start. Do I need to take the curl or div of V and use a vector identity?

Taking the divergence/curl of both sides of this equation seems like a good place to start. What do you get when you do that?

P.S. You may wish to use boldface font to denote vectors, to make things clear.

 

[Legion81]

I actually just found an easy way of showing it using projection operators. Thanks for the reply.

Consider this question solved.