参数曲面

Curves get a second parameter

The sphere in spherical coordinates

Graphs, cylinders and surfaces of revolution

Choosing the parametrization

On the sphere parametrization $\mathbf{X}(\theta, \varphi)$, what does fixing $\varphi = \pi/3$ and varying $\theta$ trace?

For the graph parametrization $\mathbf{X}(x, y) = (x, y, f(x, y))$, what is $\mathbf{X}_x \times \mathbf{X}_y$?

The cylinder $\mathbf{X}(t, z) = (\cos t, \sin t, z)$ has which coordinate curve families?

A surface of revolution has profile $g(z) = \sqrt{4 - z^2}$ for $-2 \le z \le 2$. Give the radius of the latitude circle at height $0$, as a plain number.

The tangent vectors at a point are $\mathbf{X}_u = (1, 0, 1)$ and $\mathbf{X}_v = (0, 1, 0)$. Give $|\mathbf{X}_u \times \mathbf{X}_v|$, as a plain number.