Dreamy Draw Recreation Area
Week 10
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 looking
to 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 u 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).
