Can Scalar Fields Be Decomposed Similar to Vector Fields?

[bsaucer]

If a vector field can be decomposed into a curl field and a gradient field, is there a similar decomposition for scalar fields, say into a divergence field plus some other scalar field?

 

[Delta2]

Well, perhaps you are not stating your question in a very precise manner to help us understand what exactly you mean, but I believe scalar field decomposition is easy. Suppose we have scalar field $\phi$ and $\vec{A}$ is any vector field, then we can write the equation

$$\phi=\nabla\cdot \vec{A}+\rho$$ where $\rho$ is another scalar field. Since in this equation both $\phi$ and $\nabla\cdot\vec{A}$ are scalar their difference $\rho=\phi-\nabla\cdot \vec{A}$ is well defined and is a scalar too. So in this way a scalar field can be decomposed in infinitely many ways, each way for each random vector field A we choose.

On the other hand if we first chose the scalar field $\rho$ randomly and then we seek for a vector field $\vec{A}$ that satisfies the equation

$$\nabla\cdot\vec{A}=\phi-\rho$$ then this vector field A is not uniquely defined since knowing only the divergence of a vector field does not uniquely determine the field (we also must know its curl and know some other conditions as well to uniquely determine it).