Newtonian Gravity: Describing as Vector Field

[captain]

how would you describe Newtonian gravity as a vector field?

 

[arildno]

Well, as a simple case, suppose you've got a mass positioned at location $$\vec{x}_{0}=(x_{0},y_{0},z_{0})$$ with mass $m_{0}$

Then, for any spatial point $$(x,y,z)=\vec{x}\neq\vec{x}_{0}$$
that mass generates at that point a force per unit mass:
$$\vec{f}(x,y,z)=-\frac{Gm_{0}}{||\vec{x}-\vec{x}_{0}||^{3}}(\vec{x}-\vec{x}_{0})$$

The force $\vec{F}$ acting upon an object of mass M situated at (x,y,z) is then found by multiplying f with M.

 

[jtbell]

Another form you sometimes see assumes that the mass is at the origin, and uses spherical coordinates:

$$\vec F (r, \theta, \phi) = - \frac{G m_0}{r^2} \hat r$$

where $\hat r$ is the unit vector in the outward radial direction at that particular point.