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.

 

[paranormal]

Yes, you are correct. To show that a generic vector can be decomposed into an irrotational and solenoidal component, you can use the Helmholtz decomposition theorem, which states that any vector field can be decomposed into a sum of an irrotational (curl-free) component and a solenoidal (divergence-free) component. To start, you can use the vector identity:

V = -Grad[phi] + Curl[A]

where V is the generic vector, phi is a scalar potential, and A is a vector potential.

Next, you can take the curl of both sides of the equation:

Curl[V] = Curl[-Grad[phi]] + Curl[Curl[A]]

Using the vector identity Curl[Curl[A]] = -Grad[Div[A]] + Laplacian[A], we can rewrite the equation as:

Curl[V] = -Grad[phi] - Grad[Div[A]] + Laplacian[A]

Now, we know that the curl of a gradient is always zero, so we can simplify the equation to:

Curl[V] = -Grad[Div[A]] + Laplacian[A]

Since we want to show that V can be decomposed into an irrotational (curl-free) and solenoidal (divergence-free) component, we can set the right-hand side of the equation to zero:

-Grad[Div[A]] + Laplacian[A] = 0

This is known as the Helmholtz equation. By solving this equation, we can find the scalar and vector potentials, phi and A, which will allow us to decompose V into its irrotational and solenoidal components.

I hope this helps to get you started on showing the vector decomposition using the Helmholtz theorem. Remember to always check your work and make sure that your final solution satisfies all the necessary conditions.