Class videos

These are a couple of videos that show a bit of how my teaching looks like. Both correspond to the class of Math 230 (Vector Analysis and Calculus). The full material of this class can be found here.

Integrating with Spherical Coordinates

When we switch our coordinates to spherical coordinates, we change the coordinates x, y, and z to:

x = ρcos(θ)sin(φ)

y = ρsin(θ)sin(φ)

z = ρcos(φ)

But what about the differential of volume, dxdydz? Why is it not dρdθdφ?

Let the watermelon explain!

15-4(third)_Math230 (online-video-cutter.com).mp4

Using a watermelon to demonstrate how the Jacobian of the differential of volume in spherical coordinates depends on ρ and φ. Class notes here.

Defining curvature

Say you are taking a break from studying. You decide to go for a ride on your bike, thinking you left math at your office. Suddenly, you realize your path is given by a vector-valuated function

r(t)= <x(t), y(t), z(t)>,

where t is time.

How can we measure how much your path "bends"? This is given by the curvature, and the answer is hidden in your hands.

13-4(first)_Math230 (online-video-cutter.com).mp4

Curvature with bikes. Class notes here.