Wandering the landscape

A contour map of Abshire Park (sledding hill!). Also bring along: meter sticks, cellphone with inclinometer, and calculator.

Questions and actions

Without the contour map, ascend the mountain. Partway up...

• The slope (derivative) plays a big role in Calculus. What is the slope at this point on the hillside?
• Or...a more precise way to put it is... What is the slope in a particular direction?
• Meter stick and inclinometer: Measure the angle. What trig function is "rise over run"? That is... slope!
• Now, if you turn around (by 180 degrees) what will the slope be?
• Try different directions to place your meter stick such that the slope is a maximum...This maximum slope is called the gradient. The gradient is a vector in the x-y plane. It points in the direction to move to go up as fast as possible. The magnitude of the gradient is the slope in the steepest direction.
• Turning around by 180 deg, you are looking in the direction of the fall line.
• What do you think the slope is if you turn 90 degrees away from the gradient?

Contour lines: Stick your right arm out. Rotate until your arm is pointing in the direction of the gradient or "straight uphill". Start walking. As you go, keep adjusting your direction to keep your arm pointing uphill.

• What do you notice?
• Going uphill? downhill? neither? **both**!!?
• Tracing out a contour line.
• Does the gradient point in the same direction everywhere on the hillside?