Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe geometrically the level surfaces for the functions defined in Problems 33-38. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} ; k>0 $$

Step-by-Step Solution

Answer

The level surfaces are spheres centered at the origin with radius \(\sqrt{k}\>0\).

Step 1: Understand Level Surfaces

The level surface for the function \(f(x, y, z) = x^2 + y^2 + z^2\) is defined by setting \(f(x, y, z) = k\), where \(k\) is a constant. This means solving the equation \(x^2 + y^2 + z^2 = k\).

Step 2: Identify the Geometric Shape

The equation \(x^2 + y^2 + z^2 = k\) represents the equation of a sphere centered at the origin \((0,0,0)\) in three-dimensional space, with a radius of \(\sqrt{k}\).

Step 3: Define the Conditions for \(k\)

Since \(k > 0\), the radius is a positive real number. This ensures the sphere is defined in space and not degenerate (a point or non-existent).

Step 4: Visualize the Level Surfaces

For any given positive \(k\), the level surface is a sphere with a specific radius. For different values of \(k\), the size of the sphere changes, but the center remains at the origin.

Key Concepts

Sphere

A sphere is a perfectly symmetrical three-dimensional geometric shape. It's defined by all the points in space that are equidistant from a central point, known as the center. In mathematical terms, the radius of the sphere is the distance from this center point to any point on the surface of the sphere. The formula for a sphere centered at the origin in 3D space is expressed as \(x^2 + y^2 + z^2 = r^2\) where \(r\) is the radius. In this equation, the three variables \(x, y,\) and \(z\) represent the coordinates of any point on the surface of the sphere. This ensures the uniformity and balance of the sphere since each point on the surface maintains the same distance from the origin. When considering the function \(f(x, y, z) = x^2 + y^2 + z^2\) and setting it equal to a constant \(k\), we are essentially describing a family of concentric spheres with radii \(\sqrt{k}\). As \(k\) increases, the sphere's radius increases, creating a larger sphere, but always centered at the origin.

3D Geometry

Three-dimensional (3D) geometry involves the study of shapes in a space with three dimensions: length, width, and height. It allows us to analyze and understand the positioning and properties of objects not just on a flat plane, but in a real-world context. Key concepts in 3D geometry include:

• Coordinates: Points are located using three coordinates \((x, y, z)\) which represent their positions along the length, width, and height dimensions.
• Planes: Flat two-dimensional surfaces that extend infinitely in the 3D space, described by equations involving the variables \(x, y,\) and \(z\).
• Shapes: Includes spheres, cubes, cylinders, etc., each defined by specific mathematical equations.
• Distance: Measured using the Pythagorean theorem extended into three dimensions: \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\).
• Volume: The amount of space enclosed by a 3D shape, for a sphere, it is \(\frac{4}{3}\pi r^3\).

Understanding these concepts is essential to solving problems in 3D geometry, such as finding the level surfaces of a function like \(x^2 + y^2 + z^2 = k\), which represent spheres with radii \(\sqrt{k}\).

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of more than one variable, such as \(f(x, y, z)\). It explores how these functions change and behave in response to changes in all their variables, providing valuable insights into complex systems.In the context of functions like \(f(x, y, z) = x^2 + y^2 + z^2\), multivariable calculus is used to:

• Analyze level surfaces, which are curves, surfaces, or hypersurfaces where the function attains a constant value. These provide a visual representation of the problem.
• Explore gradients, which point in the direction of the greatest rate of increase of the function, giving insights into how the function changes.
• Integrate over regions within 3D space to find volumes or solve physics-based problems.

The level surfaces given by the equation \(x^2 + y^2 + z^2 = k\) exemplify key ideas in multivariable calculus, such as how changes in \(k\) result in varying radii of the sphere, highlighting the relationship between equations and 3D space.