Functions of Several Variables: Comprehensive Study Guide

This guide covers the transition from single-variable calculus to multivariable calculus, focusing on how we define, visualize, and differentiate functions that depend on multiple independent inputs.

Learning Objectives

By the end of this study session, you should be able to:

• Recognize functions of two or more variables and identify their domain and range.
• Sketch the graph of a function of two variables using traces.
• Identify and sketch level curves for functions of two variables and level surfaces for functions of three variables.
• Calculate limits and verify continuity for multivariable functions.
• Compute first and higher-order partial derivatives.

Key Terms & Glossary

• Function of Two Variables: A rule that assigns to each ordered pair $(x, y)$ in a set $D$ a unique real number $f(x, y)$.
  • Example: $f(x, y) = x^2 + y^2$ (The paraboloid surface).
• Domain: The set of all possible input pairs $(x, y)$ for which the function is defined.
  • Example: For $f(x, y) = \ln(x+y)$, the domain is $x + y > 0$.
• Level Curve: The set of points $(x, y)$ where $f(x, y) = c$ for some constant $c$. Also known as an isotherm (for temperature) or isobar (for pressure).
  • Example: For $f(x, y) = x^2 + y^2$, the level curve for $c=4$ is the circle $x^2 + y^2 = 4$.
• Trace: The intersection of a surface with a plane (usually a coordinate plane like $x=k$ or $y=k$).
• Partial Derivative: The derivative of a multivariable function with respect to one variable while holding all other variables constant.

The "Big Idea"

In single-variable calculus, we look at curves on a 2D plane. In multivariable calculus, we move to 3D space and beyond. A function $f(x, y)$ represents a surface in 3D. The core challenge is that "change" is no longer just left-to-right; it can happen in infinite directions. Partial derivatives allow us to simplify this by looking at change in just the $x$ or $y$ directions specifically.

Formula / Concept Box

Hierarchical Outline

1. Introduction to Multivariable Functions
   • Definition: Mapping $\mathbb{R}^n \to \mathbb{R}$.
   • Domain Restrictions: Watch for square roots of negatives and denominators of zero.
2. Visualizing Functions
   • Surfaces: The 3D graph $z = f(x, y)$.
   • Level Curves: "Flattening" the 3D surface into 2D contour lines.
   • Level Surfaces: For $f(x, y, z) = w$, the level set is a 3D surface.
3. Limits and Continuity
   • Path Dependence: If Path A and Path B yield different limits, the limit Does Not Exist (DNE).
   • Continuity: Requires the limit to exist and equal the function value: $\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)$.
4. Partial Differentiation
   • First-Order: $f_x$ and $f_y$.
   • Higher-Order: $f_{xx}, f_{yy}, f_{xy}, f_{yx}$.

Visual Anchors

Concept Map: Relationship of Variables

Level Curves of a Cone

Definition-Example Pairs

• Term: Domain of a Multivariable Function

  • Definition: The set of points in the plane where the function formula is mathematically valid.
  • Example: For $g(x,y) = \sqrt{9 - x^2 - y^2}$, the domain is the disk $x^2 + y^2 \leq 9$. In the real world, this could represent the footprint of a hemispherical dome.

• Term: Level Surface

  • Definition: For a function of three variables $f(x, y, z)$, the set of points where the output is a constant $k$.
  • Example: In a room, if $T(x,y,z)$ is the temperature, a level surface $T=72^\circ$ is a 3D shape (like a curtain) where every point is exactly 72 degrees.

Worked Examples

Example 1: Finding the Domain

Problem: Find the domain of $f(x, y) = \frac{\sqrt{y-x^2}}{x^2 + (y-1)^2}$.

Solution:

1. Numerator restriction: The term inside the square root must be non-negative: $y - x^2 \geq 0 \implies y \geq x^2$.
2. Denominator restriction: The denominator cannot be zero. $x^2 + (y-1)^2 = 0$ only at the point $(0, 1)$.
3. Result: The domain is the set of all points on or above the parabola $y = x^2$, excluding the point $(0, 1)$.

Example 2: Calculating Partial Derivatives

Problem: Let $f(x, y) = x^2e^{3y} + \sin(xy)$. Find $f_x$ and $f_y$.

Solution:

• To find $f_x$: Treat $y$ as a constant. $\frac{\partial}{\partial x}(x^2e^{3y}) + \frac{\partial}{\partial x}(\sin(xy)) = 2xe^{3y} + y\cos(xy)$
• To find $f_y$: Treat $x$ as a constant. $\frac{\partial}{\partial y}(x^2e^{3y}) + \frac{\partial}{\partial y}(\sin(xy)) = 3x^2e^{3y} + x\cos(xy)$

Checkpoint Questions

1. How do you prove that a limit of a function of two variables does not exist at a point?
2. What is the geometric interpretation of a level curve $f(x, y) = k$ in relation to the surface $z = f(x, y)$?
3. If $f(x, y, z) = x^2 + y^2 + z^2$, what shape do the level surfaces form?
4. Under what condition is the mixed partial derivative $f_{xy}$ equal to $f_{yx}$?

[!TIP] When looking for the domain, always look for three "red flags": zeros in denominators, negatives under even roots, and non-positive values in logarithms.