Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)

(b) Let S be the surface parametrized by $\mathit{r}(u,z)=x(u,z)\mathit{i}+y(u,v)\mathit{j}+z(u,v)\mathit{k}\mathbf{.}$

Give two different unit normal vectors to S at the point $r(u_0,v_0).$

(c) A smooth surface that can be smoothly parametrized as

We are given statements

(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)

(b) Let S be the surface parametrized by $\mathit{r}(u,z)=x(u,z)\mathit{i}+y(u,v)\mathit{j}+z(u,v)\mathit{k}\mathbf{.}$

Give two different unit normal vectors to S at the point$r(u_0,v_0).$

(c) A smooth surface that can be smoothly parametrized as

One surface is the portion of the paraboloid $z=x^2+2y^2$, another surface is the portion of the saddle$z=2y^2-x^2$, both determined by the region R.

The two different unit normal vectors to S at the point $r\left(u_0,v_0\right)$, if S is a surface parameterized by $\mathit{r}(u,v)=x(u,v)\mathit{i}+y(u,v)\mathit{j}+z(u,v)\mathit{k}is,$

$\pm \frac{\left(y_u{z}_v-z_u{y}_v\right)i+\left(z_u{x}_v-x_u{z}_v\right)j+\left(x_u{y}_v-y_u{x}_v\right)k}{\sqrt{x_u^2{y}_v^2+y_u^2{x}_v^2+y_u^2{z}_v^2+z_u^2{y}_v^2+z_u^2{x}_v^2+x_u^2{z}_v^2-2\left(x_u{x}_v{y}_u{y}_v+y_u{y}_v{z}_u{z}_v+x_u{x}_v{z}_u{z}_v\right)}}$.

An example for a smooth surface that can be smoothly parameterized as is a generalized cylinder S defined by extending the curve $y=x^{2}fromz=0toz=1$which is parameterized by as .