Animations
Below is an example of an animation that I've used to help demonstrate the relationship between radii of circles of best fit (or turning radius of the animated buggy), the normal component of acceleration (of the buggy moving at unit speed), and curvature (or rate of turning of the tangent vector) of a plane curve.
This animation was again embedded within the course notes .pdf as well as embedded within the course Moodle page.
I made the animation below to help illustrate the cyclical nature of directional derivatives (as one varies the direction), and the relationship the directional derivative has with the gradient.
In class, for example, I can pause this video (say at a point where \(\mathbf{u}\) makes an acute angle with \(\nabla f\)) and point out how the surface is sloping up.
WebGL 3D objects
Below is a manipulatable object I embedded into an MA113 HW. The question prompt was as follows: Find the volume of the region bounded by the surface \(z=y-x^2\), \(y=2-x^2\), and \(z=0\). The 3D object below was embedded into the homework .pdf as well as provided as a WebGL 3D object on the course Moodle page as you see here:
This allowed students to have a clean illustration of the region in question, one that they could move around and view from multiple angles as they acclimate to the process of setting up bounds of integration.
Below is an illustration I made for MA478: Intro to Elliptic Curves. It's a visualization of the projective completion of the curve \(y^2 = (x+1)(x-\frac{1}{4})(x-\frac{1}{2})\):
A strange feature of elliptic curves is that they are tangent to the ``line at infinity.'' This can be a difficult concept for students; but in the illustration above, you can literally see it (rotate around so you can see the \(Y\) axis).
Manim Videos
Manim is a Python library used to make explanatory mathematics videos. In lieu of the new educational environment we find ourselves in, I've learned enough Python to start using this library. Below is a brief clip (no sound) of a manim demo I made to explain Riemann sums. Typically, I render these with a transparent background, so that I can talk ``eye-to-eye'' with the audience and gesture as the animation plays (visit Panopto to see examples).