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Among the central ideas of calculus are straightness and nonstraightness (curvature) of continuous sets of points (curves, surfaces, etc), and the idea of zooming in to straighten them out in order to quantify rates of change or directions of change, and measuring nonstraightness (curvature) by how the linearized zoom changes along the set. First derivatives are loosely correlated with how things change in a zoomed in view, while second derivatives are loosely correlated with how those zoomed in quantities themselves change as we move around, which measures failure to be straight, which means measuring how things curve or bend, which is called curvature.
To extend what we do in the calculus of a single variable to higher dimensions (more variables) and to have the mathematical tools to study curvature in that context, we need to revisit linearity of one variable versus another (straight lines in the plane) to generalize it to multilinear functions (called tensors! which generalize vectors and matrices). Baby steps in this direction were already taken in multivariable calculus, but we need to ramp up the sophisitication a bit to have a serious mathematical environment to fully describe curved spaces of any dimension and their symmetries and differential properties. The first part of this course (Part 1: Algebra of bob's book) develops this foundation.
We learned about quantifying curvature of curves in the plane or in space in multivariable calculus, so we have a starting point to think about these questions. No measure of curvature was discussed for surfaces in space then, but since surfaces can be thought of as made up of curves, we can extend our notion of curvature of curves to curvature of surfaces. We need to distinguish two kinds of curvature of a surface: intrinsic and extrinsic. The intrinsic curvature refers to the intrinsic geometry of points within the surface itself without regard for points off the surface as will become clear with our plane, cylinder, cone and sphere examples, while extrinsic curvature refers to how the surface bends (or not) in the enveloping space. Calculus is a tool for quantifying how mathematical objects change, so we need to wait till Part 2: Calculus for investigating curvature and related properties.
A few things we take for granted in calculus have to be rethought. The existence of global parallelism of vectors in the plane and in flat space has to be given up when we study curved spaces. In calculus this allows us to think of a vector without regard to where it is located, but as we will see immediately with the cone example, even in an (locally) intrinsically flat surface like a cone, the location of the initial point of the vector matters, so we need to start thinking of vectors as not only having length and direction but also a location, and in order to compare vectors at different points, we need to transport them along some path between the two points. If we try to keep the vector parallel to itself in some sense, this is called parallel transport, and on curved surfaces this depends on the path. (I will try to motivate this in the first class.)
This is analogous to one of the final topics in multivariable calculus: line integrals of vector fields along different paths between two points. If the vector field is the gradient of a function (a so called "conservative vector field," which is the gradient of a "potential function"), its line integral does not depend on the path and is simply given by the difference in values of that function. Path dependence or not of parallel transport is the sign of curvature or flatness, and can be used to quantify or measure the curvature. When the curl of the vector field is zero, it is conservative. In a similar way something called the metric (local Pythagorean theorem for nearby points in a space) acts as a potential for curvature but through second derivatives instead of first. When the curvature defined in this way is zero, the geometry is locally flat and parallel transport is path independent (locally).
When we move on to Part 2: Calculus after Part 1: Algebra, we need to develop the idea of tangent spaces at each point of our spaces that we study, in the same way that we have a different tangent vector attached to each point along a curve: the tangent vector is naturally tied to its location on the curve itself in terms of its interpretation, similar to a vector field in the plane or in space where each value of the vector field is tied to its location. This is the first step to understanding how to describe more complicated spaces.
The second thing we take for granted but need to rethink is the dot product of vectors in calculus. This plays two distinct roles which we do not distinguish: first in representing a linear function of a number of variables in terms of its coefficients, and second in determining lengths of and angles between vectors. In words these sound very different and indeed they are, but the dot product notation identifies them hiding their differences. We need to separate them back out in order to be able to understand multilinearity (tensors!) and metric geometry (lengths and angles) as two independent ideas whose interaction together leads to the study of curved geometry, like that on the surface of a sphere, known for millenia from celestial navigation. Chapter 0 attempts to begin the process of separating these two concepts. Much of differential geometry does not need metric geometry at all, so it is important to be able to develop tools without relying on it.
To support these activities regarding curvature and parallel transport, besides the paper cone exercise on our daily log [bad pdf], I will first show how to plot and zoom in on graphs in 2 and 3 dimensions by right clicking mouse activities in Maple, showing the importance of the grid to the zoom and the analogous linearization of nonlinear sets that underlies calculus. No Maple knowledge is necessary before this class, though it is helpful if you have used it a bit in a previous calculus class.
Then on the white board I will relate cones to tangent cones to the sphere (same Maple worksheet) to transfer our parallel transport idea from the cone to the sphere, then to a parabola of revolution which will later underlie our calculation of the geodetic precession effect for gyros freely falling in orbit around the Earth.
This should more than cover our first brief 50 minute encounter, and if time is insufficient, we will finish next time.