91Ó°ÊÓ

Understanding the domain of a function is crucial in multivariable calculus. The domain tells us all the possible input values (or points) for which the function is defined. Consider the function \( f(x, y) = \ln(x^2 + y^2) \). For this function, the expression inside the logarithm \(x^2 + y^2\) must be greater than zero because the natural logarithm is only defined for positive numbers.
This results in the domain being all points \((x, y)\) in the plane except for the origin \((0, 0)\), because at the origin, \(x^2 + y^2 = 0\), making the logarithm undefined.
Therefore, the domain can be written as: all real-numbered points except \((0, 0)\), which you might denote as \(\{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 > 0\}\).

The range of a function is the set of all possible output values it can produce. For \(f(x, y) = \ln(x^2 + y^2)\), understanding the range involves exploring what values \(x^2 + y^2\) can take.
Since \(x^2 + y^2 > 0\) spans all positive numbers, \(\ln(x^2 + y^2)\) can generate any real number. When \(x^2 + y^2\) approaches larger positive numbers, the logarithm produces larger positive outputs. Conversely, when \(x^2 + y^2\) gets closer to zero (but remains positive), the logarithm heads towards negative infinity.
Thus, the range of this function is all real numbers, written as \((-\infty, \infty)\).

Level curves offer a fascinating glimpse into the structure of multivariable functions. They are curves that represent points where the function has constant values. For our function, level curves are defined by setting \( \ln(x^2 + y^2) = c \) for some constant \(c\).
Solving this gives \( x^2 + y^2 = e^c \), which represents circles with radius \( \sqrt{e^c} \) centered at the origin. Each value of \(c\) corresponds to a different circle, forming concentric rings around the origin.

• When \(c\) is large, the circles increase in size.
• When \(c\) approaches negative values, circles become smaller but never vanish.

The concept of open and closed sets helps us classify domains of functions further. In multivariable calculus, a set is open if it does not include its boundary points, while a closed set includes these points.
For the function \(f(x, y) = \ln(x^2 + y^2)\), its domain \(x^2 + y^2 > 0\) is open because it excludes its boundary point, which occurs at \((0, 0)\) where \(x^2 + y^2 = 0\).
An intuitive way to think about an open set is imagining a dotted circle where touching the edge is not allowed. So, for our function, despite approaching close to the point \((0, 0)\), it is never included in the domain.

Bounded and unbounded sets introduce another layer of understanding to the domain. A set is bounded if it is contained within some finite region of space, while it is unbounded if it extends infinitely in at least one direction.
For the function \(f(x, y) = \ln(x^2 + y^2)\), the domain \(x^2 + y^2 > 0\) spans all around the plane except for the origin. This means the set is unbounded since it stretches out infinitely away from \((0, 0)\) in any direction.
This characteristic allows the function to take a wide range of inputs without stopping or being contained, portraying the unbounded nature of multivariable domains.