Problem 9 - Calculus with Concepts in Calculus

Question

Find the domain of the function. \(f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}\)

Step-by-Step Solution

Answer

The domain is all points \((x, y, z)\) such that \(x^2 + y^2 + z^2 \leq 1\).

Step 1: Understanding the Function

The function given is a multivariable function defined as \( f(x, y, z) = \sqrt{1 - x^2 - y^2 - z^2} \). The expression under the square root, \(1-x^2-y^2-z^2\), must be non-negative for the function to be defined in the real numbers.

Step 2: Set Condition for Non-Negativity

To ensure the expression under the square root is non-negative, we must have: \[ 1 - x^2 - y^2 - z^2 \geq 0 \] Rearranging gives: \[ x^2 + y^2 + z^2 \leq 1 \].

Step 3: Interpret the Condition

The condition \(x^2 + y^2 + z^2 \leq 1\) represents the set of all points \((x, y, z)\) that lie on or within the surface of a sphere of radius 1 centered at the origin in three-dimensional space. This is also known as a closed ball or a disk when in 3D.

Step 4: Define the Domain

The domain of the function \( f(x, y, z) \) is all points \((x, y, z)\) such that they satisfy the condition \(x^2 + y^2 + z^2 \leq 1\). This can be expressed as: \[ \{(x, y, z) \mid x^2 + y^2 + z^2 \leq 1\} \].

Key Concepts

Domain of a Function

The domain of a function in mathematics represents all the possible input values that the function can accept without leading to any undefined behavior. For a multivariable function like the one given, \(f(x, y, z) = \sqrt{1-x^2-y^2-z^2}\), finding its domain requires analyzing the expression for any restrictions on the variables \(x\), \(y\), and \(z\).
In this function, the expression \(1 - x^2 - y^2 - z^2\) is under the square root. Hence, it needs to be non-negative, as the square root of a negative number is not defined within the realm of real numbers. This means:

• \(1 - x^2 - y^2 - z^2 \geq 0\)
• Rewriting this inequality, we have \(x^2 + y^2 + z^2 \leq 1\)

The domain, therefore, includes all points \((x, y, z)\) that satisfy this condition. Simply put, these are the points where the function remains real and valid.
Understanding the domain helps determine where the function can be applied, ensuring valid computations. This is particularly useful in multivariable calculus and in visualizing functions within a defined space.

Inequalities in Three Dimensions

Inequalities in three-dimensional coordinates describe a variety of geometric shapes and regions in space. When dealing with inequalities involving three variables like \(x\), \(y\), and \(z\), such as \(x^2 + y^2 + z^2 \leq 1\) from our exercise, it defines a specific region in \(\mathbb{R}^3\).

• Here, \(x^2 + y^2 + z^2 \leq 1\) describes all the points inside or on the boundary of a sphere.
• The inequality describes a bounded region because of the less than or equal condition \(\leq\), encompassing everything from the center of the sphere up to its surface.

These kinds of inequalities are essential to understanding domains, constraints, and valid regions for multivariable functions. They often help clarify whether certain points belong to a solution set or not.
In the context of calculus and physics, these inequalities help delineate regions where functions are applicable or where physical phenomena can occur.

Sphere in Three-Dimensional Space

In three-dimensional space, a sphere represents a set of all points that are at the same distance from a central point, known as the radius. When defining a sphere using an inequality, as in the exercise, we are considering the area enclosed within the sphere.

• The standard equation for a sphere centered at the origin with radius \(r\) is \(x^2 + y^2 + z^2 = r^2\).
• In our scenario, the restriction \(x^2 + y^2 + z^2 \leq 1\) describes a solid sphere or a ball with a radius of 1, centered at the origin.

The notion of a sphere, and consequently its mathematical representation, plays a crucial role in fields such as multivariable calculus, physics, and engineering. They are not only of geometric importance but also arise frequently in problems involving distances, potential fields, and when assessing symmetrical properties of regions in space.