gravitational potential

The potential is the integration over space of the gravitational field. Vice versa, the gravity field, is the spatial derivative of the potential, i.e. \[\mathbf{g}=-\nabla U\]. Note that the gradient of a scalar field \[U\], \[\nabla U\], is a vector that determines the rate and direction of change in \[U\].

We may easily see this in a more general way by expressing \[d\mathbf{s}\] (the incremental distance along the line joining two point masses) into a set of coordinates:

\begin{align*} dU&=-\mathbf{g}\cdot d\mathbf{s}\\ &=-\left( g_{x}\hat{\mathbf{i}}+g_{y}\hat{\mathbf{j}}+g_{z}\hat{\mathbf{k}} \right)\cdot \left( dx\hat{\mathbf{i}}+dy\hat{\mathbf{j}}+dz\hat{\mathbf{k}} \right)\\ &=-\left( g_{x}\,dx+g_{y}\,dy+g_{z}\,dz \right) \end{align*}

By definition, the total derivative of \[U\] is given by \[dU=\frac{\partial U}{\partial x}dx+\frac{\partial U}{\partial y}dy+\frac{\partial U}{\partial z}dz\]. Therefore, \[\frac{\partial U}{\partial x}dx+\frac{\partial U}{\partial y}dy+\frac{\partial U}{\partial z}dz=-\left( g_{x}\,dx+g_{y}\,dy+g_{z}\,dz \right)\], which implies that \[\mathbf{g}=- \left( \frac{\partial U}{\partial x},\frac{\partial U}{\partial y},\frac{\partial U}{\partial z}\right)=-\nabla U \].