Problem 4 - Thomas Calculus

Question

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Spherical cap The cap cut from the sphere \(x^{2}+y^{2}+z^{2}=9\) by the cone \(z=\sqrt{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The spherical cap is parametrized by \((3 \sin \phi \cos \theta, 3 \sin \phi \sin \theta, 3 \cos \phi)\) for \(0 \leq \phi \leq \frac{\pi}{4}\), \(0 \leq \theta < 2\pi\).

Step 1: Understanding the problem

We are asked to find a parametrization for a spherical cap. The surface is defined by the intersection of a sphere and a cone. The sphere is given by the equation \(x^2 + y^2 + z^2 = 9\), and the cone is given by \(z = \sqrt{x^2 + y^2}\). This means our spherical cap is the region of the sphere above this cone.

Step 2: Switch to spherical coordinates

Spherical coordinates are ideal for a sphere, defined by \( (\rho, \theta, \phi) \) where \(\rho\) is the radial distance, \(\theta\) is the azimuthal angle, and \(\phi\) is the polar angle. For the sphere \(x^2 + y^2 + z^2 = 9\), we have \(\rho = 3\). In spherical coordinates, the equations are: \(x = \rho \sin \phi \cos \theta\), \(y = \rho \sin \phi \sin \theta\), \(z = \rho \cos \phi\). Substitute \(\rho = 3\).

Step 3: Implement the constraints

The spherical cap is also above the cone \(z = \sqrt{x^2 + y^2}\) which in spherical coordinates becomes \(\cos \phi = \sin \phi\). Solving this gives \(\phi = \frac{\pi}{4}\). Therefore, the cap is defined for \(\phi \leq \frac{\pi}{4}\).

Step 4: Parametrization of the surface

Using the constraints, our parametrization of the spherical cap is given as:\[\begin{align*}x &= 3 \sin \phi \cos \theta, \y &= 3 \sin \phi \sin \theta, \z &= 3 \cos \phi\end{align*}\]with \(0 \leq \phi \leq \frac{\pi}{4}\) and \(0 \leq \theta < 2\pi\).

Step 5: Final parametrized coordinates

After fixing the radius \(\rho = 3\), and the constraints from the sphere and the cone, the parametrization of the spherical cap is:\[(3 \sin \phi \cos \theta, 3 \sin \phi \sin \theta, 3 \cos \phi)\]where \(0 \leq \phi \leq \frac{\pi}{4}\) and \(0 \leq \theta < 2\pi\).

Key Concepts

Spherical Coordinates

Spherical coordinates are a powerful tool when dealing with three-dimensional shapes like spheres. They simplify the equations by using three parameters: the radial distance \( \rho \), the azimuthal angle \( \theta \), and the polar angle \( \phi \).
When describing a sphere, \( \rho \) represents the constant radius. In our scenario, the sphere is defined by the equation \( x^2 + y^2 + z^2 = 9 \), where the radius is 3. Thus, \( \rho = 3 \).
The azimuthal angle \( \theta \) is measured in the \( xy \)-plane from the x-axis, sweeping from 0 to \( 2\pi \). Meanwhile, the polar angle \( \phi \) measures the angle from the positive z-axis, ranging from 0 to \( \pi \).
By using these parameters, the conversion from spherical to Cartesian coordinates is:

• \( x = \rho \sin \phi \cos \theta \)
• \( y = \rho \sin \phi \sin \theta \)
• \( z = \rho \cos \phi \)

Using spherical coordinates can greatly ease the process of parametrizing surfaces, such as the spherical cap in this problem.

Sphere and Cone Intersection

When a sphere and a cone intersect, the result is often a unique shape like a spherical cap. In this problem, our sphere is described by the equation \(x^2 + y^2 + z^2 = 9\), and our cone is described by \(z = \sqrt{x^2 + y^2}\).
Where they intersect determines the boundary of the spherical cap. To find this boundary, we translate the cone's equation into spherical coordinates: since \(z = \rho \cos \phi\) and \(\sqrt{x^2 + y^2} = \rho \sin \phi\), we equate these under the cone condition which gives us \(\cos \phi = \sin \phi\).
Solving \(\cos \phi = \sin \phi\) provides \(\phi = \frac{\pi}{4}\), identifying the critical angle where the cone cuts the sphere.
This means our spherical cap forms for all \(\phi\) values up to \(\frac{\pi}{4}\). This constraint helps define our spherical cap's shape, ensuring accurate parametrization.

Trigonometric Parametrization

Trigonometric parametrization utilizes trigonometric functions to model the coordinates of points on a given surface, in this case, a spherical cap. We apply trigonometric relationships within the spherical coordinate system, which involves the conversion formulas:

• \( x = 3 \sin \phi \cos \theta \)
• \( y = 3 \sin \phi \sin \theta \)
• \( z = 3 \cos \phi \)

To parametrize the spherical cap, we already determined the range of \( \phi \) from 0 to \( \frac{\pi}{4} \) due to the cone-sphere intersection. The variable \( \theta \), as the azimuthal angle, covers one full rotation, hence ranges from 0 to \( 2\pi \).
These trigonometric relations help in expressing each point on the surface of the spherical cap as a function of \( \phi \) and \( \theta \). The result is a complete parametrization of the surface which is both elegant and functional. This type of expression is crucial for analyzing and visualizing complex three-dimensional shapes like the spherical cap formed in our scenario.