Examples from Lesson 3

For any of the following, click evaluate to render the image. It may take a few seconds to load. Once you see the image you may need to click again to be able to resize it by scrolling, or drag around to rotate the image. By right-clicking, you can reset the view to the first octant among other things.

Example 2

(a) Graph of \( \mathbf{r}(t) = \left< t^2-1,t \right> \)
(b) Graph of \( \mathbf{r}(t) = \left< 2 \cos t, 2\sin t, 1 \right> \)
(c) Graph of \( \mathbf{r}(t) = \left< \cos t, t, \sin t \right> \)

Example 4

Graphs of \(x^2 + y^2 = 1\) and \(z = x^2 - y^2\)

Curve of intersection of the above surfaces, given by the vector equation \(\mathbf{r}(t) = \left<\cos u, \sin u, \cos 2t \right>\).

Intersection of helix \(\mathbf{r}(t) = \left<\sin t, \cos t, t\right>\) with the sphere \(x^2+y^2+z^2 = 5\).