Problem 33 - Thomas Calculus

Question

a. Parametrization of an ellipsoid The parametrization \(x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leq \theta \leq 2 \pi\) gives the ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1 .\) Using the angles \(\theta\) and \(\phi\) in spherical coordinates, show that $$ \mathbf{r}(\theta, \phi)=(a \cos \theta \cos \phi) \mathbf{i}+(b \sin \theta \cos \phi) \mathbf{j}+(c \sin \phi) \mathbf{k} $$ is a parametrization of the ellipsoid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\) \(\left(z^{2} / c^{2}\right)=1\) b. Write an integral for the surface area of the ellipsoid, but do not evaluate the integral.

Step-by-Step Solution

Answer

\(\mathbf{r}(\theta, \phi)\) is a correct parametrization for the ellipsoid, and the surface area integral is set up using partial derivatives of \(\mathbf{r}\).

Step 1: Review Parametrization of Ellipse

The given parametric equations for an ellipse are \(x = a \cos \theta\), \(y = b \sin \theta\), which satisfy the equation \(\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1\). This equation represents an ellipse in a plane.

Step 2: Understanding Parametrization of Ellipsoid

We extend the ellipse into three dimensions to form an ellipsoid. The ellipsoid is represented by the equation \(\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) + \left(\frac{z^2}{c^2}\right) = 1\). Our goal is to show that \(\mathbf{r}(\theta, \phi) = (a \cos \theta \cos \phi) \mathbf{i} + (b \sin \theta \cos \phi) \mathbf{j} + (c \sin \phi) \mathbf{k}\) is a valid parametrization for this ellipsoid.

Step 3: Relating Parametric Equations to Spherical Coordinates

In spherical coordinates, \(\theta\) is the azimuthal angle, and \(\phi\) is the polar angle. The parametric form incorporates these angles: \(x = a \cos \theta \cos \phi\), \(y = b \sin \theta \cos \phi\), and \(z = c \sin \phi\). When squared and divided by their respective semi-axis square values, these parametric equations satisfy the ellipsoid equation.

Step 4: Verification of the Parametric Form

Verify the parametric form for all three axes: \((a \cos \theta \cos \phi)/a = \cos \theta \cos \phi\), \((b \sin \theta \cos \phi)/b = \sin \theta \cos \phi\), and \((c \sin \phi)/c = \sin \phi\). Substitute these into the ellipsoid equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) where each term will reduce to form:\((\cos^2 \theta \cos^2 \phi) + (\sin^2 \theta \cos^2 \phi) + (\sin^2 \phi) = 1\). This is always true because \(\cos^2 \phi + \sin^2 \phi = 1\).

Step 5: Setting Up Surface Area Integral

The surface area \(S\) of the ellipsoid can be expressed as \( S = \int_{0}^{2\pi} \int_{0}^{\pi} \left| \frac{\partial \mathbf{r}}{\partial \theta} \times \frac{\partial \mathbf{r}}{\partial \phi} \right| \, d\phi \, d\theta \), where \(\frac{\partial \mathbf{r}}{\partial \theta}\) and \(\frac{\partial \mathbf{r}}{\partial \phi}\) represent partial derivatives of the parameterization vector \(\mathbf{r}\) with respect to \(\theta\) and \(\phi\) respectively. This integral provides the surface area, but does not need evaluation here.

Key Concepts

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space. In this system, a point is defined by three variables:

• \( r \) - the radial distance,
• \( \theta \) - the azimuthal angle (usually measured from the positive x-axis in the xy-plane),
• \( \phi \) - the polar angle (measured from the positive z-axis).

This system is particularly useful for problems involving symmetry about a point, such as spheres and ellipsoids.
When parametrizing an ellipsoid using spherical coordinates, the angles \( \theta \) and \( \phi \) allow us to navigate across the surface of the ellipsoid. Parametrizing with these angles maps the three-dimensional shape based on the rotational and positional symmetries.

Parametric Equations

Parametric equations define a geometric shape in terms of parameters, rather than fixed variables. For instance, a parametric equation for an ellipse can be written as:

• \( x = a \cos \theta \)
• \( y = b \sin \theta \)

Here, \(\theta\) acts as a parameter that traces out the curve.
In the case of an ellipsoid, these equations are extended into three dimensions:

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

These parametric equations not only describe the spatial coordinates but also inherently account for the rotational aspect through \( \theta \) and positional height through \( \phi \). This ensures any point on the ellipsoid's surface is accessible and uniquely identified by these parameters.

Surface Area Integral

The surface area integral is a powerful tool for calculating the area of a complex surface, like an ellipsoid. To find the surface area, you express the integral in terms of the parameters that cover the surface.
The formula for the surface area \( S \) of an ellipsoid using parametric equations is:\[S = \int_{0}^{2\pi} \int_{0}^{\pi} \left| \frac{\partial \mathbf{r}}{\partial \theta} \times \frac{\partial \mathbf{r}}{\partial \phi} \right| \, d\phi \, d\theta\]Here, the surface is divided into small patches using the parameter space \(\theta\) and \(\phi\), and each patch's area is calculated using the cross product of the partial derivatives. This cross product gives the normal vector's magnitude to the elemental patch on the surface, ensuring proper accounting for the warp and stretch across the surface.

Partial Derivatives

Partial derivatives are a method of taking derivatives with respect to one variable while holding other variables constant. In the context of parameterized surfaces, they are essential for finding the tangential vectors on the surface.
For an ellipsoid parametrized by \( \mathbf{r}(\theta, \phi) = (a \cos \theta \cos \phi) \mathbf{i} + (b \sin \theta \cos \phi) \mathbf{j} + (c \sin \phi) \mathbf{k} \), the partial derivatives \( \frac{\partial \mathbf{r}}{\partial \theta} \) and \( \frac{\partial \mathbf{r}}{\partial \phi} \) express how changes in the parameters \(\theta\) and \(\phi\) affect the surface coordinates:

• These derivatives form vectors that lie on the tangent plane at each point on the ellipsoid, which are crucial for forming the cross product used in the surface area integral.
• Partial derivatives tell us the rate of change in each parametric direction, giving us geometric insight into the surface's structure and intricacies.