Problem 69 - Calculus Early Transcendentals

Question

Find an equation for the family of level surfaces corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=\sqrt{x^{2}+2 z^{2}}.$$

Step-by-Step Solution

Answer

Answer: The level surfaces of f(x, y, z) = sqrt(x^2 + 2z^2) represent a family of ellipsoids with y as a free parameter. The ellipsoids are centered at the origin, with their major axis along the z-axis and minor axis along the x-axis.

Step 1: Set f(x, y, z) equal to a constant k

We are given the function f(x, y, z) = \sqrt{x^2 + 2z^2}. We need to set this function equal to a constant k, which results in the equation: \[ \sqrt{x^2 + 2z^2} = k. \]

Step 2: Square both sides of the equation

Next, we'll square both sides of the equation to get rid of the square root and simplify the equation: \[ (x^2 + 2z^2) = k^2. \]

Step 3: Describe the level surfaces

We'll now analyze the equation obtained in Step 2 to describe the level surfaces. Notice that the equation is in the form: \[ \frac{x^2}{k^2}+\frac{2z^2}{k^2}=1. \] This equation represents a family of ellipsoids with y as a free parameter (i.e., the y-value can be any real number), and the semi-major and semi-minor axes are determined by the value of k. The ellipsoids are centered at the origin, with their major axis along the z-axis and minor axis along the x-axis. As k increases, the ellipsoids expand, and when k = 0, the level surface degenerates to a single point at the origin.

Key Concepts

Ellipsoids

An ellipsoid is a three-dimensional, closed surface that forms a smooth, elongated shape. It is like an elongated sphere. In mathematical terms, an ellipsoid can be seen as a set of points that satisfy a specific equation. Generally, the equation of an ellipsoid takes the form:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \)

Here, \(a\), \(b\), and \(c\) are constants that represent the distances from the center of the ellipsoid to its surface along the x, y, and z axes, respectively. These are known as the semi-axes.

For our exercise, we have an ellipsoid defined by the equation:

• \( \frac{x^2}{k^2} + \frac{2z^2}{k^2} = 1 \)

This means the semi-axis along x is \(k\) and along z is \(\frac{k}{\sqrt{2}}\). There is no restriction along the y-axis, implying an infinite extent in this direction, forming cylindrical ellipsoid surfaces.

As \(k\) changes, different "slices" or level surfaces of the ellipsoid are created, showcasing its scalability based on the constant used.

Surfaces in Three Dimensions

In multivariable calculus, surfaces in three dimensions play a crucial role in understanding spatial relationships and geometry. They represent sets of points that fulfill a particular condition or equation. Surfaces can have simple forms like planes or more complex shapes like ellipsoids or hyperboloids.

In our problem, the level surfaces form what can be thought of as 'sheets' stacked in three dimensions. These sheets do not rely on the y-coordinate due to its absence in our defining function equation: \( \sqrt{x^2 + 2z^2} = k \).

Hence, any value of \(y\) forms part of the solution, translating into a surface that extends into infinite space along the y-axis. This is a defining characteristic of these level surfaces, where they rise and curve around based on the values set by \(x\) and \(z\). This results in cylindrical sheets with ellipsoidal cross-sections, oriented along the y-axis.

Multivariable Calculus

Multivariable calculus extends calculus into the arena of several variables. It deals with functions of two or more variables and involves differentiation and integration within higher-dimensional spaces.

A key component in multivariable calculus is understanding how functions behave based on different variables, which is where level surfaces come into play. Level surfaces help represent functions in three-dimensional spaces for a constant value, essentially taking snapshots of function behavior at a particular 'height'.

By analyzing the level surfaces for \(f(x, y, z)=\sqrt{x^2 + 2z^2}\), we see how changes in \(x\) and \(z\) result in different possible 'slices' of the function, while \(y\) provides flexibility, not altering the defined surface layers. These insights are crucial for visualizing and solving complex scenarios in multivariable calculus, aiding in the comprehension of broader geometrical and physical contexts.