Problem 18 - Thomas Calculus

Question

\begin{equation} \begin{array}{c}{\text { Volume of an ellipsoid Find the volume of the ellipsoid }} \\\ {\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1} \\ {\text { (Hint: Let } x=a u, y=b v, \text { and } z=c w . \text { Then find the volume of }} \\ {\text { an appropriate region in } u v w-\text { space.) }}\end{array} \end{equation}

Step-by-Step Solution

Answer

The volume of the ellipsoid is \( \frac{4}{3} \pi abc \).

Step 1: Substitute Variables

Consider the given ellipsoid equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). According to the hint, substitute \( x = a u \), \( y = b v \), and \( z = c w \). This gives the transformed equation \( \frac{(au)^2}{a^2} + \frac{(bv)^2}{b^2} + \frac{(cw)^2}{c^2} = 1 \). This simplifies to \( u^2 + v^2 + w^2 = 1 \), the equation of a unit sphere in \(uvw\)-space.

Step 2: Transform the Volume Element

Determine the volume element after substituting: \( dV = dx \, dy \, dz = |J| \, du \, dv \, dw \), where \( J \) is the Jacobian of the transformation. Compute \( J \) as the determinant of the transformation matrix: \( |J| = \begin{vmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{vmatrix} = abc \). Thus, \( dV = abc \, du \, dv \, dw \).

Step 3: Compute the Sphere's Volume

The region of integration is the unit sphere \( u^2 + v^2 + w^2 \leq 1 \) in \(uvw\)-space. The volume of this sphere is known to be \( \frac{4}{3} \pi \).

Step 4: Determine the Ellipsoid's Volume

Multiply the volume of the unit sphere by the Jacobian \(abc\) to find the volume of the ellipsoid. Therefore, the volume of the ellipsoid is \( \text{Volume} = abc \times \frac{4}{3} \pi = \frac{4}{3} \pi abc \).

Key Concepts

Transformation of Variables

In the study of finding the volume of an ellipsoid, transformation of variables is a crucial concept. This process helps us convert a complex geometric shape into a simpler one for easy integration and computation.
In the given exercise, we transform the ellipsoid equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]by substituting new variables:

• Let \( x = a u \),
• \( y = b v \),
• \( z = c w \).

This transformation allows us to rewrite the ellipsoid equation as a unit sphere in \(uvw\)-space: \[ u^2 + v^2 + w^2 = 1 \]This new sphere equation is much easier to handle mathematically. The transformed space now represents a unit sphere where the integration can be performed over a simple region.

Jacobian Determinant

The Jacobian determinant is a vital tool when transforming variables in multidimensional spaces. It helps calculate the factor by which volume elements are scaled during the transformation. Here's how it works: the transformation \( (x, y, z) \) to \( (u, v, w) \) involves taking the partial derivatives of the old variables with respect to the new variables.
For our ellipsoid example, the transformation matrix is:\[ J = \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} \]The Jacobian is the determinant of this matrix:\[ |J| = \text{det}(J) = abc \]This determinant, \(abc\), quantifies the scaling factor applied to the volume element during the transformation from ellipsoid space to the unit sphere space. We use this to adjust the volume calculation from the sphere to the ellipsoid by multiplying the sphere's volume with \(|J|\).

Unit Sphere Volume

Finding the volume of a unit sphere is a fundamental step in calculating the volume of an ellipsoid. A unit sphere is a sphere with a radius of 1.
The formula for volume \(V\) of a sphere with radius \(r\) is:\[ V = \frac{4}{3} \pi r^3 \]For the unit sphere with \(r = 1\), it simplifies to:\[ V = \frac{4}{3} \pi \times 1^3 = \frac{4}{3} \pi \]This known volume is crucial in our calculation process. After transforming the ellipsoid to the unit sphere, we compute this volume and then adjust it with the Jacobian determinant to find the ellipsoid's volume. The simplicity of the unit sphere equation and its uniform symmetry make these calculations straightforward.

Ellipsoid Equation

The ellipsoid equation serves as the starting point for our exploration into volume computation. An ellipsoid is a 3D shape characterized by having three principal semi-axes of different lengths. The general equation of an ellipsoid centered at the origin is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]Here, \(a, b,\) and \(c\) represent the semi-axis lengths along the x, y, and z axes respectively.
This quadratic surface equation describes a stretched sphere, or ellipsoid, depending on the values of \(a, b,\) and \(c\). By transforming this equation into the unit sphere form through variable changes, we can streamline our calculations and effectively find the ellipsoid's volume. Ultimately, by understanding this foundational equation, we set the stage for transforming, scaling, and integrating within a more manageable geometric framework.