MATH 42 Differential Geometry I
As the Greek roots of the word suggest, geometry has its origins in our natural and practical need to measure the world around us. Over the millennia, geometry has evolved to encompass the general study of the measurable properties of "space." Consequently, any problem that involves a notion of "space" and "measurement" has the potential to be "geometrized"; i.e., turned into a statement about geometry. A prime example of this is the geometrization of gravity in Einstein's general theory of relativity. With applications across STEM, this course will serve as an introduction to differential geometry, which applies ideas from linear algebra and multivariable calculus to the exploration of geometry. Topics might include: vector fields & integral curves; covariant differentiation, parallel transport & geodesics; (Gaussian) curvature and Gauss’ Theorema Egregium; the Gauss-Bonnet Theorem; Isometries & Riemannian metrics.