# Any vector contribute to both the divergence role（component） and curl role（component）

Acording to Wikipedia, Helmholtz Decompose theorem states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field

I want to show you a simple example in two dimension cooerdinates system as below figure.

Fig 1 Calculus equation for the divergence and the curl

The divergence of vector F at a point o is calculated by using blue vectors, or V1, U2,V3 and U4 in the figure above , and the curl of vector F is calculated by using red vectors, or U1,V2,U3 and V4.

All of vectors which are decomposed from vector F for calculating the divergence and curl have been used.

So, you can say this is Helmholtz Decomposition.

Certainly, you can think about an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field with any vector field.

But if vector F has some component that contribute to both vector field, you can’t say that the vector field decomposed into an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field as showing bellow.

Fig 2 If F(r) has some component that contribute to both

If vector F is smooth, U4 bears distinct relationship to U3 in Fig 1.

Fig 3 the conditions for smooth vector field

U4 that is constitutive part of divergence is given by a equation include U3 that is constitutive part of curl.

So, you can say that Any vector field has some component which contributes to both roles(components) of divergence and curl.

Therefore, Helmholtz Decomposition is wrong.