Parametric Surfaces.sagews

Parametric Surfaces

Let's get some practice setting up parametric representations of surfaces.

Recall that the Sage command for plotting a parametric surface defined with coordinate functions $x(u, v)$, $y(u, v)$, and $z(u, v)$, where $a \leq u \leq b$, and $c \leq v \leq d$ looks like this: parametric_plot3d((x(u, v), y(u, v), z(u, v)), (u, a, b), (v, c, d)).

Remember that it is easy to parametrize a graph:

var("u, v")
parametric_plot3d((u, v, u^2 - v^2), (u, -2, 2), (v, -2, 2))

(u, v)

3D rendering not yet implemented

But sometimes you want a different domain. For example, in section 15.3, we considered the integral [ \int_0^1 \int_0^y x^2 + 3 y^2 , dx , dy ] To get this, we need to parametrize the $x$ and $y$ variables over the region between $x = 0$ to $x = y$, with $y$ between $0$ and $1$. One way to do this would be to think of it this way: [$$\begin{array}{rcl} x & =& u v \\ y & =& v \end{array}$$ ] where $u$ and $v$ both run from $0$ to $1$. (Then note that the $x$ coordinate will run from zero to whatever the $y$ coordinate is.) So my parametric plot looks like this:

var("u, v")
reg1top = parametric_plot3d((u*v, v, (u*v)^2 + 3*v^2), (u, 0, 1), (v, 0, 1))
reg1top

(u, v)

3D rendering not yet implemented

(I am storing the plot in a variable reg1top so I can use it again later.)

If I want, I can even draw in the sides. (We can combine plots by adding them.)

reg1back = parametric_plot3d((0, u, v*3*u^2), (u, 0, 1), (v, 0, 1))
reg1left = parametric_plot3d((u, 1, v*(u^2 + 3)), (u, 0, 1), (v, 0, 1))
reg1front = parametric_plot3d((u, u, v*(u^2 + 3*u^2)), (u, 0, 1), (v, 0, 1))
reg1top + reg1left + reg1back+reg1front

3D rendering not yet implemented

Practice

Try generating the following:

1. The top half of a hemisphere of radius 3.

2. The surface $z = x + 2 y$ over the region where $0 \leq y \leq x^2$, $0 \leq x \leq 1$. Then see if you can add the sides to make a picture of the region under consideration when computing [ \int_0^1 \int_0^{x^2} x + 2 y , dy , dx ] from section 15.3.

3. The region trapped between $z = x^2 + 3 y^2$ and $z = 8 - x^2 - y^2$, which we found the volume of in section 15.10.

4. On Test 3, problem 1(c) asked you to find the region being integrated over for [ \int_{x=0}^3 \int_{y=0}^{9-x^2} \int_{z = 0}^{3-x} f(x, y, z) , dz , dy , dx ] (It was choice (G) of the options presented, and is shown below.)

︠21e3f986-d7d1-40af-9133-57bf1eb029fc︠
︠7acd1648-f46d-4aa5-9b07-a813791b3ac6︠