Section 9.4 Visualizing the Geometry of the Gradient
The activity below is designed to help you understand the geometry of topographic maps and the gradient vector.
Activity 9.1.
Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates \((x,y)\) measured in miles. Your global positioning system says your present location is at one of the following points (pick one):
Your guidebook tells you that the height \(h\) of the hill in feet above sea level is given by
where \(a=5000\hbox{ ft}\text{,}\) \(b=30\,{\hbox{ft}\over\hbox{ mi}^2}\text{,}\) and \(c=10\,{\hbox{ft}\over\hbox{ mi}^2}\text{.}\)
Where is the top of the hill located?
How high is the hill?
Draw a topographic map of the hill. Your map should have at least 3 level curves; label your location on the map.
What is your height?
Starting at your present location, in what map direction (2-d unit vector) do you need to go in order to climb the hill as steeply as possible? Draw this vector on your topographic map.
How steep is the hill if you start at your present location and go in this compass direction? Draw a picture which shows the slope of the hill at your present location.
In what direction in space (3-d vector) would you actually be moving if you started at your present location and walked in the map direction you found above? To simplify the computation, your answer does not need to be a unit vector.
Stand up and imagine yourself standing at your chosen point on the hill — you will need to decide where the top of the hill is located. Now point in the direction of the gradient at your location.
Figuring out the location of the top of the hill, and its height, should have been straightforward. But drawing level curves — the topo map for the hill, shown in Figure 9.2 — is not so easy. Don't skip this step! Feel free to use a graphics program to generate these drawings, but it is important to develop the ability to translate between equations and topo maps.
A key feature that you should have realized while answering these questions is that the gradient lives in the topo map, not on the hill. Although we think of the gradient as pointing “uphill”, the gradient of a function of two variables is a vector in the \(xy\)-plane. In particular, when answering the last question, your hand should have been horizontal!
In order to find the 3-dimensional vector direction of travel, you will need to combine a horizontal vector in the direction of the gradient with a vertical vector scaled so that the ratio is the steepness of the hill at the given point — which is given by the magnitude of the gradient. It is probably easiest, but not necessary, to choose the horizontal part of this vector to be a unit vector. Make sure you get the units right!
A possible extension would be to answer the same questions for the hill given by