Problem 36 - Calculus 3

Question

Find the domain of the following functions. $$ f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}} $$

Step-by-Step Solution

Answer

The domain is the sphere \( x^2 + y^2 + z^2 \leq 49 \).

Step 1: Understanding the Function

The function is given by \( f(x, y, z) = \sqrt{49-x^{2}-y^{2}-z^{2}} \). For a square root to be defined in the real numbers, the expression inside the square root must be non-negative.

Step 2: Setting the Condition

To find where the function is defined, set the inside of the square root as greater than or equal to zero: \[ 49 - x^{2} - y^{2} - z^{2} \geq 0 \] This can be rearranged to: \[ x^{2} + y^{2} + z^{2} \leq 49 \]

Step 3: Interpreting the Condition

The inequality \( x^{2} + y^{2} + z^{2} \leq 49 \) represents the set of all points \((x, y, z)\) that fall within or on the surface of a sphere centered at the origin with radius 7, since \(7^2 = 49\).

Step 4: Conclusion of the Domain

The domain for the function \( f(x, y, z) \) is all triples \((x, y, z)\) that satisfy the inequality \( x^{2} + y^{2} + z^{2} \leq 49 \), including points inside this sphere and on its surface.

Key Concepts

Multivariable Calculus

When exploring functions with more than one variable, we enter the realm of multivariable calculus. This area of mathematics extends concepts from single-variable calculus to higher dimensions.

• Functions of several variables: In contrast to single-variable functions which depend on one input, multivariable functions are defined over coordinate points like \((x, y, z)\), using multiple inputs to produce a single output.
• Domain of a function: The domain refers to all possible input values for the function that produce real, defined results. For the function \(f(x, y, z) = \sqrt{49-x^{2}-y^{2}-z^{2}}\), the domain is determined by ensuring the expression within the square root is non-negative.

This understanding lets us explore more dynamic systems and solve complex problems involving multiple interdependent parameters.

Spherical Coordinates

Spherical coordinates offer a way to describe points in three-dimensional space using a different system than Cartesian coordinates. Instead of \((x, y, z)\), we use \((\rho, \theta, \phi)\) where:

• \(\rho\) is the radial distance from the origin to the point.
• \(\theta\) is the angle formed with the positive x-axis in the xy-plane.
• \(\phi\) is the angle formed with the positive z-axis.

These coordinates are particularly useful for problems involving spheres or radial symmetry. For example, when considering the inequality \(x^2 + y^2 + z^2 \leq 49\), which forms a sphere of radius 7 centered at the origin, spherical coordinates simplify the analysis and understanding by making the symmetry of the sphere explicit.

Inequalities in 3D

Inequalities in three-dimensional space can define regions that functions such as \(f(x, y, z)\) are limited to. In this context:

• The inequality \(x^2 + y^2 + z^2 \leq 49\) defines all points within or on the boundary of a sphere with radius 7.
• Visualizing the domain helps understand the behavior and limitations of functions.
• Similarly, other forms of inequalities can describe various geometric surfaces or volumes, such as cylinders, planes, or ellipsoids.

Understanding these geometric constraints is crucial for accurately defining the domain and other properties in multivariable calculus contexts. Inequalities give insight into the spatial characteristics and boundaries of mathematical and real-world systems.