## Small Group Activity: Chain Rule

Vector Calculus I 2022
This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.
What students learn Uses the chain rule to relate partial derivatives in different coordinate systems.

Suppose that $x=s\,\cos\phi$ and $y=s\,\sin\phi$.

1. Let $h(x,y)=xy$. Determine $\frac{\partial h}{\partial s}$ (with $\phi$ held constant) and $\frac{\partial h}{\partial\phi}$ (with $s$ held constant).
2. Find at least one more way to compute these derivatives.
Make sure you get the same answer!
3. Determine $\frac{\partial s}{\partial x}$ and $\frac{\partial \phi}{\partial x}$ (with $y$ held constant).
4. Find at least one more way to compute these derivatives.
Make sure you get the same answer!

Copyright 2014 by The Raising Calculus Group

If surfaces are available, Chain Rule Measurement is a possible alternative to this activity.

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