Problem 5 - Thomas Calculus

Question

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Spherical cap The cap cut from the sphere } x^{2}+y^{2}+z^{2}=9} \\ {\text { by the cone } z=\sqrt{x^{2}+y^{2}}}\end{array} $$

Step-by-Step Solution

Answer

A parametrization is \( x = 3 \sin \varphi \cos \theta \), \( y = 3 \sin \varphi \sin \theta \), \( z = 3 \cos \varphi \) for \( \varphi \in [0, \frac{\pi}{4}] \), \( \theta \in [0, 2\pi] \).

Step 1: Understand the Surfaces

We have a sphere described by the equation \( x^2 + y^2 + z^2 = 9 \) and a cone described by the equation \( z = \sqrt{x^2 + y^2} \). The task is to find the parameterization of the region where these two surfaces intersect, specifically the spherical cap.

Step 2: Convert to Spherical Coordinates

The sphere can be expressed using spherical coordinates. The conversion is as follows: \( x = \rho \sin \varphi \cos \theta \), \( y = \rho \sin \varphi \sin \theta \), \( z = \rho \cos \varphi \). For the sphere, \( \rho = 3 \).

Step 3: Parametrize the Sphere Surface

Using \( \rho = 3 \), we parameterize as: \( x = 3 \sin \varphi \cos \theta \), \( y = 3 \sin \varphi \sin \theta \), \( z = 3 \cos \varphi \). This describes the sphere with radius 3.

Step 4: Condition from the Cone

From the cone \( z = \sqrt{x^2 + y^2} \), we know that in spherical coordinates this is \( z = \rho \cos \varphi = \rho \sin \varphi \). With \( \rho = 3 \), it simplifies to \( 3 \cos \varphi = 3 \sin \varphi \) or \( \cos \varphi = \sin \varphi \).

Step 5: Simplify the Condition

The condition \( \cos \varphi = \sin \varphi \) leads to \( \tan \varphi = 1 \), which means \( \varphi = \frac{\pi}{4} \). Hence, the spherical cap is described by \( \varphi \) ranging from 0 to \( \frac{\pi}{4} \).

Step 6: Final Parametrization

The resulting parameterization for the spherical cap is: \( x = 3 \sin \varphi \cos \theta \), \( y = 3 \sin \varphi \sin \theta \), \( z = 3 \cos \varphi \) with \( \varphi \in [0, \frac{\pi}{4}] \) and \( \theta \in [0, 2\pi] \).

Key Concepts

Spherical Coordinates

Spherical coordinates provide a way to describe points in a 3-dimensional space using three numbers: the radius \( \rho \), polar angle \( \varphi \), and azimuthal angle \( \theta \). This system is particularly useful for shapes that exhibit symmetry around some point, such as spheres or cones. Unlike Cartesian coordinates which use \((x, y, z)\), spherical coordinates can sometimes make the equations and computations simpler.

• The radius \( \rho \) measures the distance from the origin to the point.
• The polar angle \( \varphi \) is the angle from the positive z-axis to the point.
• The azimuthal angle \( \theta \) represents the angle in the x-y plane from the positive x-axis.

To convert from Cartesian to spherical coordinates, the formulas are:

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

Choosing the right coordinate system helps reduce complexity and provides clearer insights into the geometry of surfaces.

Spherical Cap

A spherical cap is a portion of a sphere that is "cut off" by a plane. In this exercise, the cap is part of the sphere given by the equation \( x^2 + y^2 + z^2 = 9 \). This represents a sphere with a radius of 3. The spherical cap is created where this sphere intersects with the cone described by \( z = \sqrt{x^2 + y^2} \).Understanding spherical caps involves visualizing how the cone slices through the sphere. This intersection resembles a cap, akin to a cup or lid that sits on top of the sphere. For this problem:

• The sphere has a fixed size, defined by \( \rho = 3 \).
• The cap's boundary is determined by the cone, leading to a specific range for \( \varphi \), turned out to be \( [0, \frac{\pi}{4}] \).
• Spherical coordinates help simplify the understanding of the limits and conditions used in parametrization.

Studying the geometry of spherical caps helps to learn not just about spheres but also about how different surfaces can interact, which has many practical applications.

Sphere and Cone Intersection

The intersection between a sphere and a cone is an intriguing geometric scenario. This exercise involves determining where the given sphere and the cone intersect, described by their respective equations.The sphere given by \( x^2 + y^2 + z^2 = 9 \) is a 3-dimensional boundary centered at the origin. On the other hand, the cone \( z = \sqrt{x^2 + y^2} \) is symmetric around the z-axis forming a circular cross-section along it.Visualization and analysis of this interaction are critical:

• The intersection forms a circle on the sphere's surface.
• In this scenario, the derivation shows that where \( \cos \varphi = \sin \varphi \), leading to \( \varphi = \frac{\pi}{4} \), provides the boundary of the spherical cap.
• Being familiar with such intersections is key for solving geometrical problems where multiple types of surfaces coexist.

Understanding this intersection gives insight not only into geometry but also in practical applications involving physical objects.

Trigonometric Identities

Trigonometric identities are fundamental tools in dealing with equations involving angles, especially in the context of spherical coordinates.During the problem's resolution, trigonometry plays a role when transitioning from Cartesian to spherical coordinates and understanding conditions set by the cone. A key identity used is \( \cos \varphi = \sin \varphi \), which simplifies to \( \tan \varphi = 1 \).Some important aspects of this include:

• The equation \( \tan \varphi = 1 \) suggests \( \varphi = \frac{\pi}{4} \), which is crucial for determining the limits of the spherical cap.
• Trigonometric identities help solve geometric problems by simplifying conditions or setting bounds for angle variables.
• Knowing identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) becomes essential in analysis and solving of equations.

Mastering these identities can significantly ease the process of understanding geometric surface-related problems and facilitate clearer mathematical reasoning.