# Vectors or vector fields? (Notation, physics example)

This is from physics, however I need help with the math.

In the wikipedia-article of Newton's law of universal gravitation two forms are stated. One as vector form and one as a field.

I can't see any difference, aren't both vector fields?

Vector form: $$\mathbf{F}_{12}=-G\frac{m_1m_2}{\lvert \mathbf{r}_{12} \rvert^2} \underbrace{ \frac{\mathbf{r}_2-\mathbf{r}_1}{{\lvert \mathbf{r}_2-\mathbf{r}_1 \rvert}}}_\text{\mathbf{\hat r}_{12}}$$ where $\mathbf{F}_{12}$ is the force applied on object 2 due to object 1.

And gravitational field: $$\mathbf{g}(\mathbf{r}_{12})=-G\frac{m_1}{\lvert\mathbf{r}_{12} \rvert^2}\mathbf{\hat r}_{12}$$

What is the difference?

Isn't $\mathbf{F}_{12}$ just a abbreviation for $\mathbf{F}_{12}(\mathbf{r}_{12})$, i.e. a vector field?

• the wiki value seems satisfactory en.wikipedia.org/wiki/…, Note that $F(r)=mg(r)$. $g$ and $F$ are not the same at all, Not that the unites are different. If you are familiar with the electro static force, The gravitational field is an analog for the electric field. – sha Oct 9 '16 at 13:56

## 1 Answer

The difference is this: imagine you are a point mass feeling the gravitational force of other masses we can take two perspective to what is happening,either we can assume there is an instantaneous force mediated between you and other particles that makes you accelerate in some direction or you can imagine a gravitational field that already exists at each point in space as a consequence of the existence of other particles that tells you where to accelerate. Either way the result is the same.

Instantaneous things make physicists feel bad because of special relativity but a field concept is more palatable and is a more general concept that survives special relativity intact. But as long as we do classical mechanics we do not care.