Notes on “Calculus Blue” Volume 1, Chapter 2

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.