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Jeff Crabill – Linn Benton Community College January 2011

I begin this activity with a question to the class. If the top of a hill is in that corner of the room, point in the direction of the gradient vector from where you are sitting. Invariably, most students point up toward the corner of the room, meaning they are pointing uphill. I tell them to keep that in mind as they progress through this activity.

Part 1 is usually not a problem for students as they are comfortable with the gradient vector from a Multivariable calculus course, but they tend to write a 2-D vector and move on to part 2.

Students approached part 2 in several different ways. Some went to difference quotients using level curves as best they could. Others went to the master formula and *discovered* that the magnitude of the gradient was the answer to the question when they realized that $ d \vec{r} $ vector is parallel to the gradient vector. Many thought like this and wound up in the right place:

\begin{eqnarray} df & = & \nabla f \cdot d \vec{r} \\ \frac{df}{ds} & = & \nabla f \cdot \frac{d \vec{r}}{ds} \\ & = & \nabla f \cdot \frac{\nabla f}{| \nabla f | } \\ & = & \frac{\nabla f \cdot \nabla f}{| \nabla f | } \end{eqnarray}

i.e. the magnitude of the gradient. Many students remembered at that moment that the magnitude of the gradient was the rate of change they were looking for.

There is payoff in part 3 when students take their magnitude from part 2 and use it for the $ \hat{k} $ vector. At this point, several students pointed out to me that they should have been pointing their arms parallel to the floor earlier in the class period.