Problem 13 - Calculus: Early Transcendentals

Question

Show that for all values of \(\theta\) and \(\phi\), the point \((a \sin \phi \cos \theta, a \sin \phi \sin \theta, a \cos \phi)\) lies on the sphere given by \(x^{2}+y^{2}+z^{2}=a^{2}\).

Step-by-Step Solution

Answer

The point always lies on the sphere since the coordinates satisfy the sphere equation for all values of $\theta$ and $\phi$.

Step 1: Identify the coordinates

We start by identifying the coordinates given in the problem \((a \sin \phi \cos \theta, a \sin \phi \sin \theta, a \cos \phi)\). These coordinates are denoted as \(x\), \(y\), and \(z\), respectively.

Step 2: Substitute coordinates into the sphere equation

Substitute \(x = a \sin \phi \cos \theta\), \(y = a \sin \phi \sin \theta\), and \(z = a \cos \phi\) into the equation \(x^{2} + y^{2} + z^{2} = a^{2}\).

Step 3: Simplify expression for $x^2$

Calculate \(x^2 = (a \sin \phi \cos \theta)^2 = a^2 \sin^2 \phi \cos^2 \theta\).

Step 4: Simplify expression for $y^2$

Calculate \(y^2 = (a \sin \phi \sin \theta)^2 = a^2 \sin^2 \phi \sin^2 \theta\).

Step 5: Simplify expression for $z^2$

Calculate \(z^2 = (a \cos \phi)^2 = a^2 \cos^2 \phi\).

Step 6: Add the expressions for $x^2$, $y^2$, and $z^2$

Sum \(x^2\), \(y^2\), and \(z^2\): \(x^2 + y^2 + z^2 = a^2 \sin^2 \phi \cos^2 \theta + a^2 \sin^2 \phi \sin^2 \theta + a^2 \cos^2 \phi\).

Step 7: Simplify the expression using trigonometric identities

Factor \(a^2\) from the expression: \(a^2(\sin^2 \phi \cos^2 \theta + \sin^2 \phi \sin^2 \theta + \cos^2 \phi)\). Recognize that \(\cos^2 \theta + \sin^2 \theta = 1\) leading to \(a^2(\sin^2 \phi + \cos^2 \phi)\). Since \(\sin^2 \phi + \cos^2 \phi = 1\), the expression simplifies to \(a^2\). This matches the sphere equation \(x^2 + y^2 + z^2 = a^2\).

Key Concepts

Sphere Equation

In the realm of 3D geometry, a sphere is a perfectly symmetrical object in which all points on its surface are equidistant from its center, at a distance termed the radius. The mathematical notation for a sphere centered at the origin with radius \(a\) is \(x^2 + y^2 + z^2 = a^2\). This equation describes the set of points that lie on the surface of the sphere.
In the given exercise, we're working to prove that certain points defined in spherical coordinates lie on this sphere. These points are expressed as \((a \sin \phi \cos\theta, a \sin \phi \sin\theta, a \cos \phi)\). Here, the parameters \(\phi\) and \(\theta\) represent angles in spherical coordinates, mapping the point somewhere onto the surface of the sphere.
The key understanding here is recognizing these expressions can transform into our spherical equation, thereby demonstrating the points given indeed lie on the sphere. By substituting the spherical coordinates directly into the sphere equation and after simplifications, we arrive exactly to \(a^2\), confirming that the conditions are met.

Trigonometric Identities

Trigonometric identities play a fundamental role in simplifying expressions, particularly those involving angles. Among the most crucial are \(\sin^2\theta + \cos^2\theta = 1\), a universally utilized identity derived from the Pythagorean Theorem. This identity is pivotal in reducing complex trigonometric expressions to simpler forms.
In our sphere problem, this identity simplifies the combination of terms resulting from spherical coordinate substitution. We recognize that after substituting, expressions like \(\sin^2 \phi \cos^2 \theta + \sin^2 \phi \sin^2 \theta\) emerge. By applying \(\cos^2\theta + \sin^2\theta = 1\), it condenses to \(\sin^2 \phi\), allowing further simplification when paired with another identity, \(\sin^2 \phi + \cos^2 \phi = 1\), to completely simplify to 1.
Mastery of these identities allows us to verify our given points reside on the sphere, facilitating smooth execution of tedious algebraic manipulations.

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another. In this exercise, we're focusing on transforming spherical coordinates to Cartesian coordinates. This transformation is key when proving that a point lies on a given geometric object, such as a sphere.
Spherical coordinates are described by three quantities: the radial distance, and two angles, typically denoted by \(\rho\), \(\theta\), and \(\phi\). The transformation equations for a point \((x, y, z)\) in Cartesian coordinates based on spherical coordinates are:

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

To validate a point's position on a sphere, these expressions are substituted into the sphere's equation. The initial complexity is peeled away to reveal a simple equality, demonstrating the alignment of the spherical coordinate system with the Cartesian system.
Successful transformations like these make analyzing 3D geometric problems intuitive, as they allow the use of familiar equations in differing coordinate systems, providing flexibility and convenience in problem-solving.