What’s calc III got to do with it?

Calc III teaches you about the differentiability of multivariable functions. In this talk, we’ll look at the origins of differential geometry, where the notion of differentiability is extended to more general settings. We’ll see how certain familiar shapes, such as spheres, can themselves be thought of as differentiable or, as geometers like to say, “smooth”. If time permits, we’ll compute the differential of some standard maps between topological spaces, and interpret the Jacobian’s meaning in these contexts. This talk is a good fit for anyone curious to see how basic calc III principles apply to geometry and topology.