The Partial Derivative Machine

When students are first introduced to the common manipulations of thermodynamics they often have issues connecting the presented mathematics with the physical measurements the math represents. The Partial Derivatives Machine (PDM) allows students to investigate these relationships while gaining a deep understanding of the limitations placed on measurements in thermodynamics. The following sequences use the PDM to explore various derivatives and integrals through experimental measurements.

When beginning these sequences, it is often helpful to use the 'black box' to allow students to interact with the device first without seeing the internal mechanisms at work. This can prevent unproductive lines of work in which students attempt to predict the functional form of the variables rather than taking measurements. Some students may assume that the system obeys Hooke's law but encouraging careful measurements can lead students to find that the system is nonlinear and cannot be described by a known function.

We have included several sequences of activities which use the PDM. The first short sequence uses a modified one-dimensional version of the PDM to introduce students to taking data with the PDM and clarify some nuances prior to using the two-dimensional version. For instance, students often spend a non-trivial amount of time considering the size of “small changes”, $dx$, when taking data. The second sequence uses the two-dimensional PDM to experimentally measure partial derivatives and two-dimensional integration. The third sequence is a series of activities which can be done to verify through experiment some of the partial derivative relationships that often occur in thermodynamics.

In the Paradigms curriculum, the PDM is introduced in the Interlude, the one-week mathematics prep course prior to Energy & Entropy. Additional activities were created to address derivatives and integrals in one-dimensional contexts. This sequence can be easily incorporated into any physical contexts involving derivatives and integrals.

Activities: Introduction to the Partial Derivative Machine

  • Partial Derivative Machine Variables (Estimated time: 15 minutes): This small group activity introduces students to the PDM by asking them to determine how many measurable quantities exist within the system and how many of these quantities are simultaneously controllable. It is recommended to begin with this activity as it familiarizes students with the PDM and leads students to think about what is being held fixed when they measure a derivative.

Activities: 1-D Partial Derivative Machine

These activities use a modified PDM in order to measure one-dimensional derivatives and integrals from data. These activities have been used in Vector Fields and Symmetries & Idealizations as students begin to encounter physical applications of multivariable and vector calculus in electricity and magnetism.

  • Derivative Machine (Estimated time: x minutes): This small group activity has students find the relationship between $x$ and $F$ and then determine $\frac{dx}{dF}$. This encourages students to consider the derivative as an experimentally measurable ratio of small changes which may expand many students' working definitions of derivatives. This activity can conclude with a homework problem asking students to analyze their data.
  • Internal Energy of the Derivative Machine (Estimated time: x minutes): This small group activity introduces experimental measurements of integrals by determining the internal energy, $U$, of a nonlinear system at several locations. Students choose a point of zero internal energy and then add up, incrementally, the force at each small change in distance by numerical integration. This activity emphasizes that integration is “chopping and adding” infinitesimal pieces rather than only a mathematical operation on symbols.

Activities: Measuring Partial Derivatives and Potential Energy

In this sequence, students use the PDM to experimentally measure partial derivatives and two-dimensional integration in order to understand many of the mathematical manipulations frequently used in thermodynamics.

  • Isowidth and Isoforce Stretchability (Estimated time: x minutes): In this small group activity, students are challenged to measure a given partial derivative with the PDM. Through this, they observe that the fixed property affects the measurement of a partial derivative (e.g. $\left(\frac{\partial x}{\partial F_x}\right)_y$ and $\left(\frac{\partial x}{\partial F_x}\right)_{F_y}$ are different measurements). This activity can be followed by a whole class discussion of representations of derivatives, experimental measurement of derivatives, and the physical and mathematical consequences of the quantity held constant.
  • Easy and Hard Derivatives (Estimated time: 10 minutes): This small group activity asks students to write each partial derivative that can be formed from $x$, $y$, $F_x$, and $F_y$. Students then categorize each derivative they have written as either “hard” or “easy” to measure on the PDM. This prompts students to consider what makes a measurement difficult and the necessary manipulations in measuring “harder” partial derivatives.
  • Potential Energy of an Elastic System (Estimated time: 50 minutes): In this integrated laboratory activity, students use the PDM to determine the change in potential energy between two states of a nonlinear system. Through this process, students must review definitions of work and force in order to experimentally find the potential energy. The integration requires students to consider the “path”, how large their $dx$ is experimentally, and how their choice of independent variables affects the difficulty of performing the integration.

Activities: Partial Derivative Relations

In this sequence, students use the PDM to investigate mathematical relationships between partial derivatives which are commonly used in thermodynamics. This sequence does not have an established order but can be incorporated in a thermodynamics or math methods course while covering partial derivatives.

  • Upside Down Derivatives (Estimated time: 10 minutes): In this small group activity, students verify that an “upside-down” derivative is the same experiment as the reciprocal by measuring two “easy” derivatives. Then students attempt to determine a relationship between the two derivatives. This activity demonstrates to students that the result of a derivative being “flipped upside-down” is known, and using the physical system helps show that the math is true for physical systems.
  • Cyclic Chain Rule (Estimated time: 15 minutes): In this small group activity, students use the Partial Derivative Machine to experimentally verify the cyclic chain rule.
  • Deriving Change of Variables (Estimated time: 10 minutes): [FIXME: barely exists] This small group activity has students use variables from the Partial Derivative Machine to practice the mathematical technique of change of variables when given a partial derivative cannot be explicitly measured.

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