Question

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

Short answer

Parametrize the cone as \((r\cos\theta, r\sin\theta, r+1)\) and the plane as \((r\cos\theta, r\sin\theta, 0)\).

Step-by-step solution

Identify the Cone Equation

We are given the cone equation in the form \(x^2 + y^2 = (z-1)^2 \). This cone has its vertex at the point \((0,0,1)\) and opens downwards.

Identify the Plane

The plane equation is \(z = 0\), which is the XY-plane. This plane intersects the cone and forms one of the bounding surfaces of the domain \(D\).

Parametrize the Cone Surface

To parametrize the cone, we express the coordinates \((x,y,z)\) in terms of two parameters. One common parametrization is to use cylindrical coordinates: let \(r\) be the radius and \(\theta\) be the angle. Hence, \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = r + 1\). For the portion of the cone above the plane \(z=0\), \(r\) ranges from \(0\) to \(1\) as \(z = 0\) makes \(r = 1\). \(\theta\) ranges from \(0\) to \(2\pi\).

Parametrize the Plane Surface

The plane \(z=0\) can be parametrized simply by \((x,y,0)\). As the plane intersects the cone, \(x^2 + y^2 = 1\) when plugged into the cone equation for \(z=0\). Using polar coordinates, we can parametrize this circular region as \(x = r\cos\theta, y = r\sin\theta, z = 0\) with \(r\) from \(0\) to \(1\) and \(\theta\) from \(0\) to \(2\pi\).

Conclude the Parametrization

Now that we have parametrized both the cone and the plane, the domain \(D\) is completely described. The cone parametrization describes the outside surface from the vertex down to where it meets the plane, and the plane parametrization covers the circular section where the cone intersects \(z=0\).

Key concepts

Cone Parametrization

In mathematics, when asked to parametrize a cone, we are basically trying to express the cone's surface using parameters that simplify both visualization and calculations. Cones, typically, have rotational symmetry, making them perfect candidates for cylindrical coordinates. This process consists of deriving the equations of a cone in terms of parameters like the radius from the central axis and the angle around that axis.

In our context, the core equation for the cone is given by the equation: \[x^2 + y^2 = (z-1)^2\]This tells us that all points \(x, y, z\) lie on a conical surface with its vertex at point \((0,0,1)\) and opening downwards. In cylindrical coordinates, which utilize radial distance and angles, we can set:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(z = r + 1\)
• \(r\) varies from \(0\) to \(1\)
• \(\theta\) ranges from \(0\) to \(2\pi\)

This parametrization sweeps around the cone from the vertex down, perfectly encapsulating all points on the cone's upper half.

Plane Parametrization

When it comes to parametrizing a plane, it's often much simpler compared to other surfaces such as cones or spheres. For a plane, we just need two parameters to describe any point on the plane. In this scenario, the task was to parametrize the plane \(z = 0\), which cuts through space and acts as the base of our domain.

Since the plane is horizontal, our parametrization can be derived directly by setting:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(z = 0\)
• \(r\) varies from \(0\) to \(1\)
• \(\theta\) ranges from \(0\) to \(2\pi\)

Here, the parameters \(r\) and \(\theta\) effectively trace out a circle in the \(XY\)-plane, making it straightforward to represent all possible locations on this plane within the bounds created by its intersection with the cone. It results in a circular region centered at the origin in the \(XY\)-plane.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional extension of polar coordinates. This system is crucial for parametrizing symmetric shapes like cylinders, cones, and more because it aligns well with their geometry.

This coordinate system translates between Cartesian coordinates \(x, y, z\) and cylindrical counterparts \(r, \theta, z\):

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(z = z\)

The parameter \(r\) represents the radial distance from the \(z\)-axis, \(\theta\) is the angular position around the \(z\)-axis, and \(z\) is the vertical height. Thus, cylindrical coordinates are particularly beneficial since they naturally fit the geometry of surfaces like cones, where symmetry around a central axis is involved.

Using cylindrical variables, one can easily express and manipulate equations of complex three-dimensional objects into more manageable forms, hence their popularity in problems dealing with rotational or cylindrical symmetry.

Polar Coordinates

Polar coordinates, although typically two-dimensional, are crucial in setting up specific three-dimensional parametrizations, especially for planes and circles in space. They provide a convenient alternative to Cartesian coordinates when dealing with circular authorities or when rotational symmetry simplifies the problem.

In polar coordinates, a point in a plane is represented as:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

Here, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle from a fixed direction (usually the positive \(x\)-axis). The beauty of polar coordinates lies in their ability to conveniently describe circular paths, which then sets the basis for complex parametrizations in three dimensions.

When extending polar coordinates into three dimensions, as seen with the plane parametrization, they help build cylindrical descriptions by accommodating the height or depth as an extra parametric layer over the 2D concept, making them indispensable in understanding circular or rotationally symmetric regions.