Dreamy Draw Recreation Area

Dreamy Draw Recreation Area

Week 11

The Fall 2014 S-STEM Field Trip was to the Dreamy Draw Recreation Area which is the gateway into diverse desert landscapes that offer everything from fantastic views to secluded valleys. We went to the Peak Summit Trail to enjoy the area -- the area boasts dozens of miles of trails to enjoy the glory of the Sonoran Desert in relative solitude.

While I'm looking at those mountains, I tried to calculate the directional derivatives and gradients by using mathematics theories.

First, how to find ‘The directional derivative”:

If the function f(x,y) be the height of a mountain range at each point x=(x,y) and I stand at some point x=a, the slope of the ground in front of me will depend on the direction I am facing. It might slope steeply up in one direction, be relatively flat in another direction, and slope steeply down in yet another direction. The partial derivatives of f will give the slope ∂f∂x in the positive x direction and the slope ∂f∂y in the positive y direction. We can generalize the partial derivatives to calculate the slope in any direction. The result is called the directional derivative.

 

Second, how to find “The gradient”:

There is one direction where the directional derivative Duf(a) is the largest. This is the “uphill” direction. Let's call this direction of maximal slope m. Both the direction m and the maximal directional derivative Dmf(a) are captured by something called the gradient of f and denoted by ∇f(a). The gradient is a vector that points in the direction of m and whose magnitude is Dmf(a). In math, we can write this as

 ∇f(a)∥∇f(a)∥=m and ∥∇f(a)∥=Dmf(a).