Problem 9 - Advanced Engineering Mathematics

Question

Describe the level surfaces but do not graph. $$ F(x, y, z)=x^{2}+3 y^{2}+6 z^{2} $$

Step-by-Step Solution

Answer

The level surfaces are ellipsoids centered at the origin with axes scaled by coefficients 1, 3, and 6.

Step 1: Understand the Function

The given function is \( F(x, y, z) = x^2 + 3y^2 + 6z^2 \). This function consists of quadratic terms for each variable with different coefficients. It represents a quadratic function in three dimensions.

Step 2: Identify the General Form of Level Surfaces

A level surface of a function \( F(x, y, z) \) is defined for a constant \( c \) such that \( F(x, y, z) = c \). Therefore, the level surface for this function is given by \( x^2 + 3y^2 + 6z^2 = c \).

Step 3: Recognize the Type of Surface

The equation \( x^2 + 3y^2 + 6z^2 = c \) represents an ellipsoid when \( c > 0 \). This is because it is a sum of squares equal to a positive constant, which fits the standard form of an ellipsoid \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \) upon rearranging.

Step 4: Analyze the Coefficients

The coefficients 1, 3, and 6 correspond to the directions of the \( x \), \( y \), and \( z \) axes, respectively. These coefficients imply that the axes of the ellipsoid will scale differently, resulting in different semi-axis lengths: \( \sqrt{c/1} \), \( \sqrt{c/3} \), and \( \sqrt{c/6} \) for the \( x \), \( y \), and \( z \) axes.

Step 5: Summarize the Level Surfaces

Each level surface \( x^2 + 3y^2 + 6z^2 = c \) is an ellipsoid centered at the origin with axes aligned along the coordinate axes differently scaled by the coefficients in the equation.

Key Concepts

Quadratic Function

Quadratic functions are fundamental tools in mathematics and are quite versatile. They can model a variety of real-world phenomena. A quadratic function in three dimensions takes the form \( F(x, y, z) = ax^2 + by^2 + cz^2 \), where \( a, b, \) and \( c \) are coefficients. These coefficients determine the shape and orientation of the function in space.

• Degree: Quadratic functions are degree 2 polynomials. This means their highest exponent on any variable is 2.
• Symmetry: Such functions often exhibit symmetry about the axes, which helps in understanding their graphs.
• Shape: In a three-dimensional setup, quadratic functions can form shapes like ellipsoids, hyperboloids, or paraboloids, depending on the coefficients of the terms.

Understanding quadratic functions is crucial for advancing in higher-dimensional mathematics, as they form the basis for many equations in physics and engineering.

Ellipsoid

An ellipsoid is a three-dimensional shape that is the generalization of an ellipse. If you were to slice an ellipsoid in a plane, the slice would be an ellipse. The equation of an ellipsoid can be written in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). Here, \( a, b, \) and \( c \) are the lengths of the semi-axes.

• Axes: The semi-axes dimensions determine how elongated or flattened the ellipsoid appears.
• Orientation: Conventionally, the axes are aligned with the coordinate axes unless rotated or translated in space.
• Level Surface: The level surfaces of the function \( F(x, y, z) = x^2 + 3y^2 + 6z^2 \) describe ellipsoids for different values of \( c \). For these surfaces, the semi-axis lengths vary based on the coefficients.

In our exercise, for a given constant \( c > 0 \), the equation \( x^2 + 3y^2 + 6z^2 = c \) translates to an ellipsoid form, showing how algebraic forms connect to geometric concepts.

Three-Dimensional Geometry

When we talk about three-dimensional geometry, we refer to the study of points, lines, surfaces, and shapes in a space with three dimensions: length, width, and height. Understanding this concept requires familiarity with coordinates and spatial reasoning.

• Coordinate System: The Cartesian coordinate system is commonly used, where any point in space is represented by a set of three numbers \( (x, y, z) \).
• Shapes: Common three-dimensional shapes include spheres, cubes, cylinders, and ellipsoids. Each shape is defined by specific geometric properties.
• Transformations: In 3D, transformations such as translations, rotations, and scaling affect how shapes like ellipsoids appear and are analyzed.

Three-dimensional geometry facilitates our understanding of the physical world, allowing us to represent objects mathematically and analyze their properties. It serves as the foundation for fields like physics, engineering, and computer graphics.