Multivariate Functions and Level Surfaces

Definition and Examples

Multivariate functions are functions that depend on two or more variables. Their graphs and level surfaces provide geometric insight into their behavior.

• Level Surface: For a function , a level surface is the set of points where for some constant .

• Example 1: - For , the level surface is the origin. - For , the level surface is a sphere of radius .

• Example 2: - For , the surface is a hyperboloid of one sheet. - For , the surface is a cone. - For , the surface is a hyperboloid of two sheets.

Level Curves in Two Variables

Level curves are the set of points where for a fixed . Their shapes depend on the function's form.

• Example: Which functions have circles centered at the origin as level curves?

Conclusion: Only yields circles as level curves.

Limits and Continuity of Functions of Two Variables

Definition of Limit

The limit of a function as approaches is if, for every , there exists such that:

• whenever is in the domain of and .

Limit Laws for Functions of Two Variables

Standard limit laws extend to multivariate functions:

• Sum:

• Difference:

• Constant Multiple:

• Product:

• Quotient: , provided

• Power:

• Root: , assuming if is even

Note: Polynomial functions are continuous everywhere.

Examples of Limits

• Example 1:

• Example 2:

• Example 3:

• Example 4: - This uses the single-variable result .

Exercise Example

• Compute:

• Solution: Factor numerator and denominator:

Nonexistence of Limits: The Two-Path Test

Procedure

If a function approaches different values along two distinct paths as , then the limit does not exist.

• Reason: If the limit existed, all paths would yield the same value.

Example

• Compute:

• Path 1 (y-axis): as

• Path 2 (x-axis): as

• Conclusion: Since the limits along these paths differ, the limit does not exist.

Exercise

• Compute:

• Answer: The limit does not exist. - Even if horizontal and vertical paths yield the same limit, other paths (e.g., ) may yield different values.

Summary Table: Limit Laws for Multivariate Functions

Additional info: These notes cover foundational concepts in multivariate calculus, including level surfaces, limit definitions, and techniques for determining the existence of limits. The two-path test is a critical tool for analyzing limits in higher dimensions.