Activity 11.7.4.
This activity considers directional derivatives and gradients for a function that measures elevation. Suppose you are hiking in a foggy park and can only see a few feet in front of you. There is nothing blocking you from walking in any particular direction, but because of the fog you cannot see where the highest point on the mountain is. You want to try to find the top of the mountain, but you don’t have a map, trail, or line of sight to other landmarks. You do have a compass, which works in the fog. This allows you to identify the cardinal directions.
In order to use calculational tools from multivariable calculus, you overlay an \(xy\)-coordinate system on the park. The \(x\)-coordinate measures east/west location, with eastward movement associated with increasing \(x\)-values. The \(y\)-coordinate measures north/south location, with northward movement associated with increasing \(y\)-values. The elevation in meters above sea level of the park’s terrain at the location \((x,y)\) is given by a function \(h(x,y)\text{.}\)
(a)
Let \(P_1\) be your current location in the foggy park. You use your compass to find the east and north directions. At \(P_1\text{,}\) you find that the ground rises 1 meter per 50 meters traveled to the east and the ground rises 2.5 meters per 50 meters traveled to the north.
Use this information to find \(\nabla h (P_1)\text{.}\)
(b)
Use your answer to the previous part to say what direction is “uphill” at \(P_1\) and state the rate of elevation increase in this direction.
(c)
You decide to walk uphill from your location \(P_1\) in order to try to find the top of the mountain. After walking in the same direction for a while, you are at a point \(P_2\) and notice that you are no longer walking in the steepest direction. You again locate east and north and measure the steepness of the mountain in these directions. You find that the ground rises 1.5 meters per 75 meters traveled to the east and the ground goes down 0.5 meters per 100 meters traveled to the north.
Use this new information to calculate \(\nabla h (P_2)\text{,}\) find the uphill direction, and find how steep the mountain is in the uphill direction at \(P_2\text{.}\)
(d)
Suppose you continue this method of walking in the uphill direction for a while, finding the new uphill direction, and walking in the new uphill direction. Do you think you must eventually find the top of the mountain? How you will know that you have reached the top of the mountain? Remember that you can’t see very far in front of you. Write a few sentences to explain your reasoning.
