Question

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y, z)=\ln \left(4-x^{2}-y^{2}-z^{2}\right)\)

Short answer

The largest set \(S\) is the interior of a sphere centered at the origin with radius 2: \(x^2 + y^2 + z^2 < 4\).

Step-by-step solution

Understanding Continuity of Logarithmic Functions

The function is continuous wherever its argument is positive. For a logarithm to be defined, the input inside the logarithm must be greater than zero, otherwise the logarithm is undefined.

Set Up the Inequality

To find where the function is continuous, set up the inequality for the argument of the logarithm. The argument is given by:\[4 - x^2 - y^2 - z^2 > 0\]

Solve the Inequality

Rearrange the inequality to define a valid region:\[x^2 + y^2 + z^2 < 4\]This inequality represents the interior of a sphere centered at the origin (0, 0, 0) with a radius of 2.

Describe the Set

The inequality \(x^2 + y^2 + z^2 < 4\) describes the set of all points (x, y, z) inside a sphere centered at the origin with radius 2. Thus, the largest set \(S\) is the open ball inside this sphere, excluding the surface where the radius is exactly 2.

Key concepts

Open Ball

An open ball in multivariable calculus is a set of points that are all within a certain distance from a given center point, but does not include the points exactly at that distance. Think of it as the inside of a sphere, without its surface.

In the context of our function, the open ball is defined by the inequality \(x^2 + y^2 + z^2 < 4\), which means all points \((x, y, z)\) that are less than 2 units away from the origin \(0, 0, 0\). This places all these points strictly within the sphere, excluding the points on the surface.

When we talk about "openness," we refer to the absence of boundary points. This mathematical distinction is crucial because functions like logarithms can encounter issues at boundaries, such as becoming undefined at zero. That's why an open ball naturally represents the safe zone for the continuity of our function.

Sphere

A sphere in three-dimensional space is the collection of all points that are at a fixed distance (the radius) from a given point (the center). However, unlike the open ball, a sphere includes its entire surface.

In our problem, the sphere has its center at the origin \((0,0,0)\) and a radius of 2. Mathematically, it is represented by the equation \(x^2 + y^2 + z^2 = 4\). This equation includes all points exactly 2 units away from the origin, defining the boundary surface.

When considering the continuity of functions, we need to be careful with surfaces like these, as they might introduce points where the function could become undefined or behave irregularly. This is why we prefer to work within the open ball when ensuring continuity of the logarithmic function given in our problem.

Inequalities in Multivariable Functions

Inequalities are crucial in defining regions where multivariable functions maintain specific properties, such as continuity.

In the given function \(f(x, y, z) = \ln(4 - x^2 - y^2 - z^2)\), we explore the inequality \(x^2 + y^2 + z^2 < 4\) to determine where the function is continuous. Here, the inequality indicates all points where the sum of squares of \(x, y,\) and \(z\) is less than 4.

This condition ensures that the argument of the logarithm remains positive, which is essential because logarithms are undefined for zero or negative values. Inequalities thus help carve out valid and continuous regions from function domains in multivariable calculus.

Logarithmic Functions

Logarithmic functions are mathematical operations that are the inverse of exponentiation. They can only take positive inputs, as trying to take the logarithm of zero or a negative number results in an undefined expression.

For the function \(f(x, y, z) = \ln(4 - x^2 - y^2 - z^2)\), the argument \(4 - x^2 - y^2 - z^2\) must be greater than zero to ensure the function is defined and continuous. This is because a logarithm requires its input to be positive.

Understanding this restriction helps us find specific regions in multivariable functions where they behave correctly. By applying these conditions, we ensure we only consider valid inputs that do not result in mathematical errors or undefined behavior.