A level set of f is the set S={(x, y) | f(x, y)=k}, where k...

A level set of $f$ is the set $S=\{(x, y) | f(x, y)=k\},$ where $k$ is in the range of $f .$ Let $$f(x, y)=\left\{\begin{array}{ll}
0 & \text { if } x^{2}+y^{2} < 1 \\
x^{2}+y^{2}-1 & \text { if } x^{2}+y^{2} \geq 1
\end{array}\right.$$
Sketch the level set of $f$ for $k=0$ and 3

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Level Sets

A level set is the collection of points in the domain of a function where the function takes on a constant value. This concept is widely used to visualize contours (in two dimensions) or isosurfaces (in higher dimensions) of scalar fields.

Piecewise Functions

Piecewise functions are defined by different expressions over separate parts of their domain. Understanding how the function is partitioned in its definition is key to analyzing its behavior in different regions.

Implicit Equations

Many level sets are described by implicit equations, such as those defining circles. Interpreting and sketching these implicit equations, like x² + y² = constant, is essential for visualizing the level sets.

Domain Partitioning

When a function is defined differently on different regions—such as inside and outside a circle—it’s important to identify which part of the domain a specific level set comes from, ensuring correct interpretation of the boundaries and interior.

Circular Geometry

Since the function involves expressions like x² + y², circular geometry naturally arises. Recognizing the role of circular boundaries and understanding properties of circles is fundamental when sketching level sets that involve circular regions.

Transcript

Please give Ace some feedback