Some years ago I fell in love with Desmos as a visualization tool. I find it great for teaching for two reasons: it strikes a good balance between intuitiveness and power, and it's web-based so no installation required. Together this means that students actually use it. Here are a few of my favorite examples.
This is a classic example of a chaotic dyanimical system. It was discovered while trying to make a model of atmospheric conditions. It is deterministic (no random choice), yet it is effectively impossible to predict how many times it loops around the left node before it flips over the right, and vice versa. And two points that begin near each other can end up far apart. You can move the sliders to change the parameters \(\sigma,R,b\), which correspond to atmospheric properties, as well the starting point \((x_0,y_0,z_0)\).
Naturally these visualizations are a boon to teaching differential geometry. Here we have the isometric family that contains the helicoid (spirally shape) and catenoid (tubey shape). Notice that the although the red and blue curves are bending, their lengths are not changing: there is bending but not stretching. Thus the transformation is an isometry.
This one was a bit of an out-of-the-box experiment. It's very pretty, but I'm not sure how effective it was as a teaching aid. In Intro PDEs we solve the wave equation in 3D space. A typical example is a wave of light or sound. In particular, it is important to understand spherical waves, but it is hard to show both the overall shape and intensity at the same time. The part of the wave that moves outwards loses intensity, but the inward moving part actually concentrates. Move the time \(T\) slider to move the wave.
Not every visualization needs to be super fancy. Here we have the function \(f(x,y)=x^{3}-\frac{2}{3}y^{3}-2y^{2}-36x+6y\) and we want to find its critical points. As you move the blue dot, the value of \(f\) at this point is shown. The level set is also graphed. Can you find any local maxima and minima? What do you notice about the shape of the level set in these cases?
Because Desmos can integrate and differentiate, often you can just input the formula straight out of the lecture slides. Here we integrate out the Frenet equations in the plane to get a curve given its curvature function \(k(s)\). Try putting different functions and see what you get.
There is a lot going on here, because there are two different diagrams side-by-side. On the right, we have a pendulum. Press play on time \(T\) and it swings back and forth, eventually coming to rest hanging vertically down. The thin blue line shows the velocity.
On the left, we have a coordinate representation of the state of the system. The x-axis represents the angle, and the y-axis represents the velocity. If you move the red dot around, you can change the starting conditions of the pendulum. When the simulation is running, the purple dot shows the current state. The purple curve shows the evolution of the system. Finally, the light and dark yellow regions shows those states which have lower energy than the start and current state respectively. Since this pendulum has friction, it loses energy as time goes on. Therefore we know it is impossible to reach some states (for example, when it is impossible to flip over).