## Small Group Activity: Chain Rule Measurement

Vector Calculus I 2022
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.
What students learn Measure partial derivatives in $x$-, $y$-, and $r$-direction on a three-dimensional surface; Express a partial derivative in any arbitrary direction in terms of those in $x$-, $y$-directions; Verify the chain rule relation $\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}$ numerically by comparing their measured values.

##### Student Conversations

Some students may be confused by the (lack of) dimensions of slope

1. Measurement
1. Using the measurement tool, find the rate of change in the surface in the $x$-direction at the blue dot on your surface. Include units. $\frac{\partial f}{\partial x} = \hspace{2in}$
2. Using the measurement tool, find the rate of change in the surface in the $y$-direction at the blue dot on your surface. Include units. $\frac{\partial f}{\partial y} = \hspace{2in}$
3. Using the measurement tool, find the rate of change in the surface in the $s$-direction at the blue dot on your surface. Include units. $\frac{\partial f}{\partial s} = \hspace{2in}$
2. Computation
1. What are the rectangular coordinates of the blue dot (on the contour mat)? $(x,y) = \hspace{2in}$
2. What are the polar coordinates of the blue dot (on the contour mat)? $(s,\phi) = \hspace{2in}$
3. Use the chain rule to express $\frac{\partial f}{\partial s}$ in terms of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. $\frac{\partial f}{\partial s} = \hspace{2in}$
3. Comparison