Question

Let \(D\) be the region bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a > 0, b > 0\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v, z=c w\). Find the volume of \(D\)

Short answer

Answer: The volume of the region D bounded by the given ellipsoid is \(V(D) = \frac{4}{3}\pi abc\).

Step-by-step solution

State the transformation and the ellipsoid

Given transformation is T: \((u, v, w) \rightarrow (x, y, z)\), where \(x = au, y = bv, z = cw\). The equation of the given ellipsoid in x, y, and z coordinates is: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$

Transform the ellipsoid equation

Replace x, y, z in the ellipsoid equation with their corresponding expressions from the given transformation: $$\frac{(au)^2}{a^2} + \frac{(bv)^2}{b^2} + \frac{(cw)^2}{c^2} = 1$$ Simplify the equation by canceling the constants: $$u^2 + v^2 + w^2 = 1$$ Now our ellipsoid equation is in terms of u, v, and w.

Compute the Jacobian determinant

Now we need to compute the Jacobian determinant of the transformation: $$J(u, v, w) = \left|\begin{matrix} \frac{dx}{du} & \frac{dx}{dv} & \frac{dx}{dw} \\ \frac{dy}{du} & \frac{dy}{dv} & \frac{dy}{dw} \\ \frac{dz}{du} & \frac{dz}{dv} & \frac{dz}{dw} \end{matrix}\right|$$ Differentiate x, y, and z with respect to u, v, and w to obtain: $$J(u, v, w) = \left|\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{matrix}\right|$$

Calculate the absolute value of the Jacobian determinant

Now we compute the absolute value of the Jacobian determinant. Since it is a diagonal matrix, the determinant is simply the product of its diagonal elements: $$|J(u, v, w)| = |abc|$$

Determine the transformed region and find the volume

The region D in terms of u, v, and w corresponds to a unit sphere: $$u^2 + v^2 + w^2 \le 1$$ To find the volume of D, we need to multiply the volume of the sphere in u, v, w coordinates by the absolute value of the Jacobian determinant: $$V(D) = V_{u,v,w}(u^2 + v^2 + w^2 \le 1) \times |abc|$$ The volume of a unit sphere with u, v, w coordinates is: $$V_{u,v,w} = \frac{4}{3}\pi$$ Now we can calculate the volume of D as follows: $$V(D) = \frac{4}{3}\pi \times |abc|$$ So, the volume of the region D bounded by the given ellipsoid is: $$V(D) = \frac{4}{3}\pi abc$$

Key concepts

Ellipsoid Volume

When we talk about the volume of an ellipsoid, we're referring to the three-dimensional space inside this elongated sphere-like shape. The formula for calculating the volume of an ellipsoid is derived from the transformation of coordinate variables, which simplifies the complex ellipsoid into a unit sphere in a different coordinate system.
For an ellipsoid defined by the formula \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), where \( a, b, \) and \( c \) are lengths of the semi-principal axes, the volume is given by:

• Replacing the ellipsoid coordinates \( (x, y, z) \) with transformed coordinates \( (au, bv, cw) \).
• Transforming the ellipsoid equation to a unit sphere \( u^2 + v^2 + w^2 = 1 \).
• The volume of this unit sphere is \( \frac{4}{3} \pi \).
• Multiplying it by the absolute value of the product of the semi-principal axes \( abc \).

Thus, the volume of the ellipsoid is \( \frac{4}{3} \pi abc \), where the terms \( a, b, \) and \( c \) stretch the unit sphere into the ellipsoid's distinct shape.

Jacobians

Jacobians play a crucial role in transforming coordinates and understanding their properties, especially in calculus when dealing with multiple dimensions. They aid in understanding how transformations affect areas, volumes, and integrals. In our context, the Jacobian matrix is created when mapping from one set of coordinates to another.
When considering the transformation \( x = au, y = bv, z = cw \), the Jacobian matrix becomes:

• A three by three matrix with derivatives \( \frac{dx}{du}, \frac{dy}{dv}, \frac{dz}{dw} \).
• The specific matrix looks like \( \begin{matrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{matrix} \).
• The determinant of this diagonal matrix is \( abc \), calculated by multiplying the diagonal elements.

The absolute value of this Jacobian determinant \( |abc| \) relates to how volumes change under the transformation, turning the original elliptical volume into a sphere and understanding how stretched or compressed it is.

Coordinate Transformations

Coordinate transformations are alterations in a coordinate system that facilitate solving complex equations by changing how we see or define a shape in space using simpler terms. This mechanism works by using a mathematical function that maps old coordinates to new ones, preserving the geometric properties.
In the ellipsoid problem, we transition from coordinates \( (x, y, z) \) to \( (u, v, w) \) with definitions like \( x = au, y = bv, z = cw \):

• Such transformations simplify the ellipsoid equation to a unit sphere, \( u^2 + v^2 + w^2 = 1 \), making integration more straightforward.
• Coordinate transformations help visualize and solve for volumes by converting complex shapes into unit spheres.
• They also aid in applying physical scenarios where different coordinate systems align better with the problem's conditions.

This concept is invaluable in calculus for evaluating integrals over regions that change easily into more manageable forms, such as spheres, and in understanding spatial relations in geometry and physics.