Question

A domain \(D\) in space is given. Parametrize each of the bounding surfaces of \(D\). \(D\) is the domain bounded by \(y=1-z^{2}, y=1-x^{2}, x=0\), \(y=0\) and \(z=0\).

Short answer

Parametrize each surface using its related constraints: (x,z) for \(y=1-z^2\), (x,y) for \(y=1-x^2\), and (x,y), (x,z), (y,z) for planes.

Step-by-step solution

Understanding the problem statement

We are asked to parametrize the surfaces bounding a domain in 3D space. The given bounding surfaces are the parabolas: \(y = 1-z^2\) and \(y = 1-x^2\), along with the planes: \(x = 0\), \(y = 0\), and \(z = 0\). We need to parametrize each surface individually.

Parametrizing the parabolic surface: \(y = 1 - z^2\)

In this case, the parameters are \(x\) and \(z\). Since \(y = 1 - z^2\), the parametric representation can be: \(\mathbf{r}(x, z) = (x, 1 - z^2, z)\). Here, \(x\) varies within the bounds defined by the other surfaces: \(0 \leq x \leq 1\), and \(z\) varies where \(y \geq 0\), or \(0 \leq z \leq 1\).

Parametrizing the parabolic surface: \(y = 1 - x^2\)

Here, the parameters are \(x\) and \(y\). Since \(y = 1 - x^2\), the parametric representation is: \(\mathbf{r}(x, y) = (x, y, \sqrt{1-y}\) or \(\mathbf{r}(x, y) = (x, y, -\sqrt{1-y})\) for each \(y\). Parameters are bounded by \(0 \leq x \leq 1\), and \(0 \leq y \leq 1\).

Parametrizing the plane: \(x = 0\)

For the plane \(x = 0\), we use \(y\) and \(z\) as parameters. The parametrization is \(\mathbf{r}(y, z) = (0, y, z)\), where the boundaries are determined by \(0 \leq y \leq 1\) and \(0 \leq z \leq 1\).

Parametrizing the plane: \(y = 0\)

For the plane \(y = 0\), use \(x\) and \(z\) as parameters. The parametrization is \(\mathbf{r}(x, z) = (x, 0, z)\). The boundaries are defined by \(0 \leq x \leq 1\) and \(0 \leq z \leq 1\).

Parametrizing the plane: \(z = 0\)

For the plane \(z = 0\), parameters are \(x\) and \(y\). The parametrization is \(\mathbf{r}(x, y) = (x, y, 0)\) with \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\).

Key concepts

Understanding 3D Space

In mathematics, 3D space, also known as three-dimensional space, refers to a geometric setting where three parameters determine a point's position. This can be imagined like a room, where we need three numbers—representing length, width, and height—to locate a point in space. These three numbers usually correspond to coordinates on the x, y, and z axes. In contexts like the one described in the exercise, understanding 3D space is crucial because the task involves visualizing surfaces and their boundaries within this volume. By using coordinates

• x for the horizontal position,
• y for the vertical position,
• and z for depth or third dimension,

we can describe any point in this space. Visualizing the gelation of all these points creates surfaces and volumes that are important in geometry and real-world applications.

Bounding Surfaces

When we talk about bounding surfaces, we're looking at the edges and limits that define a given volume in 3D space. In our exercise, we have a domain bounded by several mathematical surfaces. Each specific surface, like a parabola or a plane, acts as a boundary, marking where the domain starts or ends. The key is understanding each type of surface:

• A plane is a flat, two-dimensional surface extending infinitely. In the exercise, planes are identified by equations like \(x=0\), \(y=0\), and \(z=0\), indicating they run perpendicular to one of the coordinate axes.
• Parabolas are curved shapes defined by quadratic equations, and in 3D they create parabolic surfaces. The parabolic surfaces are set by equations like \(y = 1-z^2\) or \(y = 1-x^2\).

These bounds together create a specific, enclosed volume — a domain within 3D space. Understanding these definitions helps us outline and define the extent of areas we are interested in, often through parametrization.

Parametric Representation

Parametric representation is a method of defining a geometry using parameters, often simplifying the equations needed to describe these shapes. For bounding surfaces, this involves expressing coordinates (x, y, z) as a function of one or more parameters. For example, to parametrize the parabolic surface \(y = 1 - z^2\), we set

• \(x\) as one parameter,
• and \(z\) as another.

Thus, the position vector \(\mathbf{r}(x, z) = (x, 1 - z^2, z)\) succinctly represents this curved space. Similarly:

• \(\mathbf{r}(x, y) = (x, y, \sqrt{1-y})\) or \(\mathbf{r}(x, y) = (x, y, -\sqrt{1-y})\) for \(y = 1 - x^2\).
• \(\mathbf{r}(y, z) = (0, y, z)\) for plane \(x=0\).

Each parametrization is tailored to the surface it describes, marking the volume's outer limits in a clean and efficient manner.

Planes and Parabolas

Planes and parabolas are common in the study of parametric surfaces due to their well-defined properties. Planes are straight flat surfaces characterized by an equation like \(x = 0\), which essentially means the surface's orientation is fixed along the other axes. This is very useful in defining boundaries since they create simple limits in one direction. Also, these can be easily expressed in parameters, such as \(\mathbf{r}(y, z) = (0, y, z)\), maintaining simplicity in their mathematical expressions. On the other hand, parabolas are more complex due to their curvature. Typical in equations like \(y = 1 - x^2\) or \(y = 1 - z^2\), parabolas represent surfaces that curve outward or inward in three-dimensional space. Parametric equations for these surfaces require handling their quadratic nature, often involving taking square roots to properly describe their shape. But this complexity allows more versatility, showing how curved boundaries can also exist alongside straight ones.