# Decompostion of scalar field

1. Apr 12, 2014

### Jhenrique

If a vector field can be decomposed how a sum of a conservative + solenoidal + harmonic field...

so, BTW, a scalar field can be decomposed in anothers scalar fields too???

2. Apr 13, 2014

### chogg

What types of scalar fields are there?

3. Apr 14, 2014

### Geometry_dude

$$\phi (q) = \frac{\phi(q) + \lvert \phi(q)\rvert}{2} +\frac{\phi(q) - \lvert \phi(q)\rvert}{2}$$
for a point $q \in Q$ on a smooth manifold and a section $\phi \in C^{\infty}(Q, Q \times \mathbb R)$ of the trivial vector bundle?
This is a decomposition into positive and negative parts.
If the operation $q \to - q$ makes sense, then you can also take
$$\phi (q) = \frac{\phi(q) + \phi(-q)}{2} +\frac{\phi(q) - \phi(-q)}{2}$$
using the same trick. This is a decomposition into symmetric and anti-symmetric parts.
EDIT: You might be interested in this: http://en.wikipedia.org/wiki/Hodge_decomposition
Note that a "scalar field" is a $0$-form since $\bigwedge^0 T^*Q \simeq Q \times \mathbb R$.

Last edited: Apr 14, 2014