Geoscience uses for calculus (and beyond)

I've heard many geology students rant about calculus. It's confusing. It's boring. It's useless, at least for geology. So why do they have to take it, then, other than as a form of ritual torture?

I admit that I don't make students use calculus in my classes. I don't want students to miss an intuitive understanding of a topic because they're bogged down in the math, and I don't want to set up pre-requisites that prevent students from graduating in a reasonable amount of time. So I take other approaches - analogies and demonstrations, mostly - to try to make sense of quantitative subject.

I don't do calculus directly in my research, either - at least, not in the sense of calculating derivatives and integrals. But the fieldwork that I do is based on theoretical work that, in many cases, would be impossible without calculus and other post-algebra math. I don't often have time to go into most of it in class (though I've been looking for excuses in Structure this fall, ever since my students made the mistake about ranting about calculus before lecture one day).

I'm putting this list together off the top of my head, and I know that it's horrendously incomplete. (Partly because I forget things, and partly because my research experience is limited to certain subdisciplines.) If you have examples to add, please do, and I will edit the post to include them. If you sign your post, I'll give you credit for your additions. (And if you've got any references to books or papers or online sources that explain more about the use of the technique, I would love to include them, as well.)

Geologic time

The radioactive age equation is probably the most obvious example of a geologic use of calculus. (It was even in my undergrad calc textbook.) The decay rate (dN/dt) is proportional to the amount of the parent isotope present. That makes a simple differential equation, and leads to an equation with natural logarithms. (It also makes radioactive dating challenging to explain to intro classes. I know I do a terrible job of it, and probably leave students figuring that if they're confused, it's probably wrong.)

Structural geology

Something as simple as the attitude of a layer is actually related to calculus. It's simple (well, with practice) to describe the orientation of a plane, but how many geologic layers are really, genuinely planar? And yet we measure the orientation of planes around the surface of a fold. But we aren't measuring the curved surface - we're measuring a plane that's tangent to the surface. It's actually kind of the derivative of the curved surface, except that it's in three dimensions, which makes it really complicated. (It also makes it a subject requiring 3rd semester calculus.) Lots more information about this (and other shapes) is in Dave Pollard and Ray Fletcher's structural geology textbook, in the chapter on differential geometry. (I found myself trying to explain the chapter by writing about an ant traveling along a wire. I suspect the book could be improved by somehow combining the math and some creative analogies.)

And then there's stress. And strain, actually. Both stress and strain are physical examples of tensors - functions that relate one vector to another vector. To work with tensors, you need to know matrix algebra (which I learned in linear algebra). Tensors are important for describing the properties of all kinds of things that vary depending on direction - the stress experienced by different fault planes, the changes in length of stretched fossils with different starting orientations, variations in compressibility with direction, and so forth. (They apply to subdisciplines other than structural geology, too - the weird ellipsoid called the "indicatrix" in optical mineralogy relates the speed of light to the orientation of a crystal, for instance. And if groundwater can flow through material with different permeabilities in different directions.)

Petrology and geochemistry

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.

Geophysics

Probably every subdiscipline within geophysics uses calculus, linear algebra, differential equations, and so forth.

I've played more with heat flow than with most other topics within geophysics. And heat flow is defined using ideas from calculus. The equation for conduction of heat in one dimension, for instance, relates the transfer of heat to the change in temperature with distance (dT/dx). Add more dimensions, or move the rocks around, and it gets even more complicated. Heat flow equations make it possible to predict how hot a rock can get beside a magma body, or can test whether a continental collision could really result in growth of a certain high-temperature mineral. I love it. It takes arm-waving and gives it substance and testability.

I'm sure there's more. (Groundwater hydrology, for instance, uses differential equations to describe the movement of water and the chemical reactions between the water and minerals.) I need help remember what some are, though.

Sedimentary geology

Brian points out that fluid dynamics is based in calculus. (Tell me more about the modeling of sedimentary systems, though - I mean, most modeling that I know of uses calculus at least, if not differential equations, because changes in time or in space (or both) are part of the model, and if changes are involved, calculus is useful.)