More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”.
Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space. They also introduce the names and images for all of the quadratic surfaces (so we’ve left the linear world in this chapter). This is a foreshadowing of the process of the course (start with the linear and then move to the nonlinear).
01.02.00 (0:35) “Curves and surfaces: Intro”
01.02.01 (5:10) “Implicit and parametric curves and surfaces”. Two ways to define curves. Implicitly or parametrically. Implicit: “the solutions to an equation yields a curve in the plane”. Parametric: “specifying coordinates as a function of a parameter”. We’ll want to move between these representations. Surfaces in 3d can also be expressed implicitly or parametrically (requires 2 parameters).
Examples: parameterization for a surface expressed implicitly (he shows the conversion where you set x = s, y = t, and then express z); go from a parameterization to an implicit equation (the example has a square root so he suggests caution).
01.02.02 (0:29) “Some examples please…?”. it’s important to learn the quadratic surfaces.
01.02.03 (5:33) “Examples of quadratic surfaces”. start with the sphere, then modify to get the ellipsoids, change a sign to get a hyperboloid, with two negatives signs get a 2-sheeted hyperboloid, then an elliptic paraboloid, hyperbolic paraboloids, cones (degenerate hyperbola), cylinders.
Then the narrator provides a reassurance: this isn’t about memorizing these or drawing pictures of these; it’s just worth recognizing them. Then the narrator mentions these won’t come back for a while and names surface integrals as an application. These are in there as a motivation to build up geometry and algebra skills.
01.02 (0:26) “The big picture”. Previous chapter was lines and planes. This one was curves and surfaces. Progression from linear to nonlinear is what will happen in the course.