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Relationship Between Gravitational Potential and Gravitational Field Strength.

Equipotential Lines drawn to indicate a constant Graviataional Potential
From the equation V_g=-Gm₁/r , we can see that a constant Grviational potential will occur at a constant radius from the centre of mass of a point mass. In a 2-D drawing this shell could be represented by a circle.
From the equation V_g=gh, for a uniform field, the constant Gravitational potential will occur in horizontal lines perpendicular to the direction of the force field.
Since gravitational potential is defined as the gravitational potential energy per unit mass you can see why a mass in a circular orbit can continue indefinitely without gaining or loosing energy. (the gravitational potential and hence the gravitational potential energy of a mass placed in the circle would have a constant potential energy)
Equipotential lines can often be best understood by comparison to contour lines on a map. wikipedia link contour lines Contour lines are lines drawn at a contant height (altitude). Lines with a higher altitude would have a higher gravitational potential (eg. in a uniform force field). A 10.0 kg mass, for example, would have the same gravitational potential energy anywhere on the contour line.
Try this assignment Drawing Equipotential Lines , Solution

Gradient is a measure of change. I would like to say that it is the rate of change but that would be wrong since the word rate implies change per uint time. A gradient on a topigraphical map is the change per uint distance. When the equipotential lines are very close together on the map that would mean that there is a very steep slope there. Technically the magnitude of the gradient is the slope. The direction of the gradient is perpendicular to the contour lines. The gradient lines point down the steepest path down the hill. (Skiiers call this the fall line.)
Some help in describing this try tis link http://www.falstad.com/mathphysics.html run the 2-D Vector Fields Applet.
- display none
- floor is equipotential (switch on and off flat view) to show equipotential lines
- floor is streamlines to show potential gradient lines

Draw lines perpendicular to the equipotential lines. Every intersection of an equipotential line and a gradient line should be a right angle. As the gradient is greater the density of gradient lines (the distance between gradient lines gets smaller.
Draw the gradient lines on the assignment. (a gradient has a direction and a magnitude so can generate a vector from a scalar field)

Gravitational Field Strength	Gravitational Potential
Defines the force (magnitude and direction) per unit mass. (this refers to the force that would occur if a mass was located at the specific point in the field)	Defines the gravitational potential energy per unit mass. (this refers to the gravitational potential energy that would occur if a mass was placed at the specific point in the field)
It is a vector	It is a scalar.

DV=DEg/m

DV=Dmgh/m

DV=Dgh

g=DV/Dh

or the gravitational field strength is the gradient of the gravitational potential. For the general case calculus is needed g=dV/dh, or you can find the slope of the curve on a plot of V vs r.
In the example below the slope should be close to 9.8 at the smallest radius. Data extracted from your example above. Forgive the quality of the plot.

Questions: Looking at your example, where is the lowest gradient and where is the highest? How does this correspond to the gravitational field strength?
Estimate the radius of the earth on the above graph, then calculate the slope of the line at that point. Does the magnitude of the slope equal 9.8 Nkg^-1?