I'm a mathematics professor who is seeking to find interesting, application-driven ways of teaching freshmen college students differential/integral calculus. I am in no way versed in crystallography or much of earth science beyond the usual education that the average non-expert gets, but I did enjoy and do well in a single-semester college geology course, and rocks and minerals have always fascinated me. I am wondering if the crystallographers and earth scientists here might know off-hand applications of tangent lines, derivatives, optimization, integration, etc., in crystallography. I've been browsing, and it certainly seems that vectors and linear algebra and group theory are central in this field, but it has been difficult to find uses for undergraduate calculus skills in the field. Please help?
This site may give you some ideas. In particular, look at the Petrology and Geochemistry section:
Metamorphic phase diagrams show which minerals are stable under various temperatures and pressures. The slopes of the boundaries between the areas where different minerals are stable depend on the properties of the minerals (like the entropy and the molar volume).
And... well, mineralogy, petrology, and geochemistry are full of ideas taken from chemical thermodynamics. And thermodynamics is full of multivariable calculus. (How does changing the pressure change the amount of calcium in garnet vs the amount of calcium in plagioclase? The question is a partial derivative.) From my own favorite, metamorphic petrology, there are all kinds of techniques for using metamorphic minerals to tell stories of heating and burial and cooling and exhumation. And none of the techniques would be possible without calculus. (Or, in many cases, differential equations.) I'm sure igneous rocks and ore deposits have similar uses of thermodynamics.
Here's an example of a metamorphic P-T Diagram with the source link with more examples: