Question

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$f(w, x, y, z)=\sqrt{1-w^{2}-x^{2}-y^{2}-z^{2}}$$

Short answer

$$ Answer: The domain of the function is all points (w, x, y, z) inside and on the boundary of a 4-dimensional sphere of radius 1 centered at the origin.

Step-by-step solution

Identify the condition for the function to be defined.

For the function to be defined, the expression inside the square root must be greater than or equal to zero. This means that: $$1-w^{2}-x^{2}-y^{2}-z^{2} \geq 0$$

Rewrite the inequality to find the boundaries.

We can rewrite the inequality as: $$w^{2}+x^{2}+y^{2}+z^{2} \leq 1$$

Recognize the equation form and calculate the domain.

The left-hand side of the inequality represents the equation of a 4-dimensional sphere. The inequality implies that all points inside and on the boundary of the sphere are part of the domain. The domain can be described as all points inside a 4-dimensional sphere of radius 1 centered at the origin.

Key concepts

Square Root Function

The square root function is a classic mathematical function, often seen as \( f(x) = \sqrt{x} \). However, in more advanced contexts, you might encounter multi-variable forms like \( f(w, x, y, z) = \sqrt{1-w^2-x^2-y^2-z^2} \). The defining feature of the square root function is that it only accepts non-negative inputs. This means that the expression within the square root, known as the radicand, cannot be negative. This constraint is crucial for determining the domain of the function.

In our example, the condition \( 1-w^2-x^2-y^2-z^2 \geq 0 \) ensures the radicand is valid. This constraint helps identify which values \((w, x, y, z)\) can be plugged into the function without resulting in an undefined situation.

Understanding these constraints helps us explore and understand advanced mathematical structures like spheres in higher dimensions.

4-Dimensional Sphere

A 4-dimensional sphere, also known as a 3-sphere, is an extension of the concept of circles and spheres into higher dimensions. While we are familiar with 2-dimensional circles and 3-dimensional spheres, the notion of a 4-dimensional sphere might seem abstract.

The equation \( w^2 + x^2 + y^2 + z^2 \leq 1 \) describes the set of all points \((w, x, y, z)\) that lie inside or on the surface of a sphere with radius 1 centered at the origin in four-dimensional space.

This means that any combination of \(w, x, y, z\) values, when squared and summed, should not exceed 1. Such a constraint effectively defines a boundary in this high-dimensional space, much like how the equation \( x^2 + y^2 \leq r^2 \) defines a 2-dimensional circle.

• Visualizing a 4-dimensional sphere is challenging, but think of it as the set of points that reach out equally in all four directions (w, x, y, z) up to the distance of 1.
• In practical terms, it represents a unique mathematical construct useful in various theoretical and applied contexts.

Inequalities in Calculus

Inequalities in calculus often involve expressions where one side is restricted to being greater than, less than, or equal to the other. This is crucial for understanding the behavior of functions, especially when dealing with roots and variables in multiple dimensions.

In our square root function, the inequality \( w^2 + x^2 + y^2 + z^2 \leq 1 \) is foundational. It tells us that the domain of the function is constrained within the 4-dimensional sphere. Those constraints reveal the possible input values which lead to valid outputs.

When dealing with inequalities in calculus, it is essential to:

• Analyze the inequality to determine which values satisfy it.
• Understand that they describe regions, boundaries, or conditions where a function behaves in a particular way.
• Use these insights to explore more complex, multi-variable functions.

Such understanding is crucial for solving real-world problems using calculus, as it allows us to correctly model and interpret various phenomena.