Representing Gravity as a Vector Field

[schaefera]

In my book, it says that gravity can be thought of as a force in the form of this vector:

F= (-GMm)/(x²+y²+z²)*u

where u is a unit vector in the direction from the point to the origin. How would this be represented as a vector field (this is not a homework problem, just me wondering...)?

Is u, the unit vector, able to be split up into u= {(x)i + (y)j + (z)k}/(sqrt( x²+y²+z²), then you can sub in for that and get a vector field of the form

F=(-xGMm)/(((sqrt( x²+y²+z²)³) i + ... and so on?

Because then you can find the divergence of this vector field, but you can't find the divergence of that first equation I listed above because it's not explicitly a vector field...

 

[nasu]

You can take the divergence in the first case too. Just use the operator in spherical coordinates.

 

[HallsofIvy]

Yes,
$$F= -\frac{GmM}{x^2+ y^2+ z^2}\vec{u}$$
and
[tex]u= \frac{1}{\sqrt{x^2+ y^2+ z^2}}(x\vec{i}+ y\vec{j}+ z\vec{k})[/itex]
so that
$$F= -\frac{GmM}{r^3}(x\vec{i}+ y\vec{j}+ z\vec{k})$$
or
$$F= -\frac{GmM}{r^3}\vec{r}$$