📐 Concept diagram

06-01 - Functions of Several Variables

Phase: 6 | Subject: 06-01 Prerequisites: 05-09-applications-of-integration.md (comfort with graphing in 2D), 04-03-the-derivative.md (function concepts), 02-10-vectors-basic.md (coordinate axes) Next subject: 06-02-limits-and-continuity-in-rn.md

Learning Objectives

By the end of this subject, you will be able to:

1. Define functions of two and three variables, identifying their domains and ranges
2. Sketch and interpret level curves (contour maps) of f(x,y)
3. Interpret level surfaces for functions of three variables f(x,y,z)
4. Visualize and describe graphs of multivariable functions
5. Identify common pitfalls: confusing domain, range, and graph dimensions

Core Content

From One Variable to Many

A function of one variable, y = f(x), maps real numbers to real numbers: its domain is a subset of ℝ, its range is a subset of ℝ, and its graph is a curve in ℝ².

A function of two variables, z = f(x,y), maps ordered pairs to real numbers: its domain is a subset of ℝ² (a region in the plane), its range is a subset of ℝ, and its graph is a surface in ℝ³.

A function of three variables, w = f(x,y,z), maps ordered triples to real numbers: its domain is a subset of ℝ³ (a region in space), its range is a subset of ℝ, and its graph lives in ℝ⁴ — impossible to fully visualize.

General form:

$f: ℝⁿ → ℝ
f(x₁, x₂, ..., xₙ) = some expression
$

We focus primarily on n = 2 and n = 3.

Domain and Range of f(x,y)

Domain: The set of all ordered pairs (x,y) for which the defining expression is defined. This is a region in the xy-plane.

Range: The set of all possible output values z = f(x,y).

Finding Domain: Examples

Example 1: f(x,y) = x² + y² - Domain: all of ℝ² (every (x,y) works). Range: [0, ∞).

Example 2: f(x,y) = √(1 - x² - y²) - Domain: 1 - x² - y² ≥ 0 → x² + y² ≤ 1, the unit disk (including boundary). - Range: [0, 1].

Example 3: f(x,y) = ln(x² - y) - Domain: x² - y > 0 → y < x², the region STRICTLY below the parabola y = x² (parabola not included — logarithm of 0 is undefined). - Range: (-∞, ∞) — as (x² - y) → 0⁺, ln → -∞; as (x² - y) → ∞, ln → ∞.

Example 4: f(x,y) = 1/(y - x²) - Domain: all (x,y) such that y ≠ x² (plane minus the parabola). Range: (-∞, 0) ∪ (0, ∞).

Common Domain Edge Cases

Misconception: The domain of f(x,y) is a region in the INPUT plane, not the output surface. Don't confuse the domain (xy-plane subset) with the range (z-values).

Graphs of Functions of Two Variables

The graph of z = f(x,y) is the set of points (x, y, f(x,y)) in ℝ³. It is a surface sitting above (or below) the xy-plane.

Common Surface Types

Plane: f(x,y) = ax + by + c - e.g., f(x,y) = 2x + 3y + 1 is a plane with normal vector ⟨-2, -3, 1⟩.

Paraboloid: f(x,y) = x² + y² - Circular paraboloid opening upward. Vertex at (0,0,0). - Cross-sections parallel to xy-plane are circles.

Hyperbolic paraboloid (saddle): f(x,y) = x² - y² - Saddle shape at origin. - Cross-sections: x = constant → downward parabola z = c² - y². - Cross-sections: y = constant → upward parabola z = x² - c².

Hemisphere (upper): f(x,y) = √(1 - x² - y²) - Graph is the upper half of the sphere x² + y² + z² = 1, z ≥ 0.

Exponential surface: f(x,y) = e^(-x² - y²) - Bell-shaped surface (Gaussian bump). Maximum at (0,0,1), decays to 0 as ||(x,y)|| → ∞.

Visualizing: Cross-Sections

Fix one variable to see a cross-section:

For f(x,y) = x² + y²: - Fix y = 0: z = x² (parabola opening up) - Fix y = 1: z = x² + 1 (parabola, shifted up by 1) - Fix y = 2: z = x² + 4 (parabola, shifted up by 4)

These vertical cross-sections show how the surface builds.

Level Curves (Contour Maps)

A level curve of z = f(x,y) at level c is the set of points (x,y) such that f(x,y) = c. It's a curve in the xy-plane along which the function has constant value c.

Interpretation: Level curves are the projection of the intersection of the surface z = f(x,y) with the horizontal plane z = c onto the xy-plane.

⚠️ CRITICAL FOUNDATION: Level curves (2D) and level surfaces (3D) are the primary visualization tools for multivariable functions. Everything from gradients to Lagrange multipliers builds on understanding what constant-value sets look like.

Examples

f(x,y) = x² + y²: - Level curves: x² + y² = c → circles of radius √c (for c ≥ 0). - c = 0: point at origin. - c = 1: circle radius 1. - c = 4: circle radius 2. - Spacing: evenly spaced z-levels produce widening gaps in the contour map because the surface gets steeper (farther from origin, contour lines get closer together).

f(x,y) = y - x²: - Level curves: y - x² = c → y = x² + c (parabolas shifted up/down). - c = 0: y = x². - c = 1: y = x² + 1. - c = -1: y = x² - 1.

f(x,y) = xy: - Level curves: xy = c → y = c/x (hyperbolas). - c > 0: hyperbolas in quadrants I and III. - c < 0: hyperbolas in quadrants II and IV. - c = 0: x-axis and y-axis.

Reading Contour Maps

• Closely spaced contours: steep surface (rapid change).
• Widely spaced contours: gentle surface (slow change).
• Closed loops: peaks (concentric circles shrinking to a point) or pits (often marked with hachures or indicated by values).
• Parallel lines: constant slope (plane).

Misconception: Level curves are in the xy-plane, NOT on the surface. They are the "footprint" of constant-height slices.

Level Surfaces (Functions of Three Variables)

For w = f(x,y,z), the graph lives in ℝ⁴. Instead, we use level surfaces: the set f(x,y,z) = c.

Example: f(x,y,z) = x² + y² + z² - Level surfaces are spheres: x² + y² + z² = c (c ≥ 0). - c = 0: point at origin. - c = 1: unit sphere. - c = 4: sphere radius 2.

Example: f(x,y,z) = x² + y² - z - Level surface for c = 0: x² + y² = z (paraboloid). - Level surface for c = 1: x² + y² = z + 1 (paraboloid shifted down).

Example: f(x,y,z) = 2x + 3y - z - Level surfaces are planes: 2x + 3y - z = c (parallel planes).

Physical interpretation: Level surfaces are common in physics: - Temperature T(x,y,z) = c → isothermal surfaces. - Pressure P(x,y,z) = c → isobaric surfaces. - Potential V(x,y,z) = c → equipotential surfaces.

Comparing Dimensions

Key Terms

• 06 01 Functions Of Several Variables
• Common Domain Edge Cases
• Common Surface Types
• Comparing Dimensions
• Correct: B)
• Correct: C)
• Domain and Range of f(x,y)
• Example 1: Domain and Range
• Example 2: Level Curves
• Example 3: Level Surfaces
• Examples
• Expression

Worked Examples

Example 1: Domain and Range

Problem: Find the domain and range of f(x,y) = √(4 - x² - y²).

Solution:

Step 1: For the square root, we need 4 - x² - y² ≥ 0.

Step 2: x² + y² ≤ 4. This is the closed disk of radius 2 centered at the origin.

Step 3: Domain: {(x,y) : x² + y² ≤ 4}.

Step 4: Range: When (x,y) = (0,0), f = √4 = 2 (maximum). When x² + y² = 4, f = 0 (minimum). So range = [0, 2].

Example 2: Level Curves

Problem: Describe and sketch level curves of f(x,y) = x² + 4y² at c = 0, 1, 2, 4.

Solution:

Step 1: Equation x² + 4y² = c.

Step 2: Divide by c: x²/c + 4y²/c = 1 → x²/c + y²/(c/4) = 1.

Step 3: These are ellipses centered at origin: - c = 0: just the point (0,0). - c = 1: ellipse with semi-major axis 1 in x-direction, semi-minor axis 1/2 in y-direction. - c = 2: ellipse with semi-major axis √2, semi-minor axis √(2)/2 ≈ 0.707. - c = 4: ellipse with semi-major axis 2, semi-minor axis 1.

Step 4: As c increases, the ellipses grow. The surface is an elliptical paraboloid.

Example 3: Level Surfaces

Problem: Describe the level surfaces of f(x,y,z) = x² + y² - z² at c = -1, 0, 1.

Solution:

Step 1: x² + y² - z² = c.

Step 2: For c = 0: x² + y² = z². This is a double cone (both nappes). In cylindrical coordinates: r² = z² → r = |z|.

Step 3: For c = 1: x² + y² - z² = 1. This is a hyperboloid of one sheet. Cross-section in xy-plane: circle radius 1. As |z| → ∞, radius grows.

Step 4: For c = -1: x² + y² - z² = -1 → z² - (x² + y²) = 1. This is a hyperboloid of two sheets. Only exists for |z| ≥ 1.

Quiz

Q1: What does the concept of Common Domain Edge Cases primarily refer to in this subject?

A) The definition and application of Common Domain Edge Cases B) A historical anecdote about Common Domain Edge Cases C) A computational error related to Common Domain Edge Cases D) A visual representation of Common Domain Edge Cases

Correct: A)

• If you chose A: Common Domain Edge Cases is defined as: the definition and application of common domain edge cases. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Common Domain Edge Cases is defined as: the definition and application of common domain edge cases. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Common Domain Edge Cases is defined as: the definition and application of common domain edge cases. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Common Domain Edge Cases is defined as: the definition and application of common domain edge cases. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Common Surface Types?

A) It is used only in advanced research contexts B) It is used to common surface types in mathematical analysis C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: B)

• If you chose A: This is incorrect. Common Surface Types serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Common Surface Types serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Common Surface Types serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Common Surface Types serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Comparing Dimensions is TRUE?

A) Comparing Dimensions is a fundamental concept covered in this subject B) Comparing Dimensions is mentioned only as a historical footnote C) Comparing Dimensions is not related to this subject D) Comparing Dimensions is an advanced topic beyond this subject's scope

Correct: A)

• If you chose A: Comparing Dimensions is a fundamental concept covered in this subject. This subject covers Comparing Dimensions as part of its core content. Correct!
• If you chose B: This is incorrect. Comparing Dimensions is a fundamental concept covered in this subject. This subject covers Comparing Dimensions as part of its core content.
• If you chose C: This is incorrect. Comparing Dimensions is a fundamental concept covered in this subject. This subject covers Comparing Dimensions as part of its core content.
• If you chose D: This is incorrect. Comparing Dimensions is a fundamental concept covered in this subject. This subject covers Comparing Dimensions as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) ** {(x,y) : x² + y² < 9} (open disk radius 3, boun C) A different result from a common mistake D) The inverse of the correct answer

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is ** {(x,y) : x² + y² < 9} (open disk radius 3, boun. The other options represent common errors.
• If you chose B: The worked examples show that the result is ** {(x,y) : x² + y² < 9} (open disk radius 3, boun. The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is ** {(x,y) : x² + y² < 9} (open disk radius 3, boun. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is ** {(x,y) : x² + y² < 9} (open disk radius 3, boun. The other options represent common errors.

Q5: How are Comparing Dimensions and Expression related?

A) Comparing Dimensions and Expression are closely related concepts B) Comparing Dimensions and Expression are completely unrelated topics C) Comparing Dimensions is the inverse of Expression D) Comparing Dimensions is a special case of Expression

Correct: A)

• If you chose A: Both Comparing Dimensions and Expression are covered in this subject as interconnected topics. Correct!
• If you chose B: This is incorrect. Both Comparing Dimensions and Expression are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Comparing Dimensions and Expression are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Comparing Dimensions and Expression are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with From One Variable To Many?

A) From One Variable To Many has no common misconceptions B) A common mistake is confusing From One Variable To Many with a similar concept C) From One Variable To Many is always computed the same way in all contexts D) The main error with From One Variable To Many is using it when it is not needed

Correct: B)

• If you chose A: This is incorrect. Students often confuse From One Variable To Many with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse From One Variable To Many with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse From One Variable To Many with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse From One Variable To Many with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Domain And Range Of F(X,Y)?

A) Apply Domain And Range Of F(X,Y) to solve problems in this subject's domain B) Use Domain And Range Of F(X,Y) only in pure mathematics contexts C) Avoid Domain And Range Of F(X,Y) unless explicitly instructed D) Domain And Range Of F(X,Y) is not practically useful

Correct: A)

• If you chose A: Domain And Range Of F(X,Y) is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose B: This is incorrect. Domain And Range Of F(X,Y) is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Domain And Range Of F(X,Y) is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Domain And Range Of F(X,Y) is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Find the domain of f(x,y) = ln(9 - x² - y²).

2. Find the domain and range of f(x,y) = √(x² + y² - 1).

3. Describe the level curves of f(x,y) = x - y.

4. Describe the graph of f(x,y) = 5 - x² - y².

5. Find and describe level surfaces of f(x,y,z) = x + 2y + 3z at c = 0, 6.

6. For f(x,y) = 1/(y - x²), state the domain and describe what happens near the boundary.

7. Sketch the level curves of f(x,y) = y/(x² + 1) at c = 0, 1, -1.

Summary

Key takeaways:

• f(x,y) maps ℝ² → ℝ: domain is a region in the plane, range is an interval of real numbers
• Graphs of z = f(x,y) are surfaces in ℝ³ — visualize with cross-sections
• Level curves f(x,y) = c are 2D contours in the xy-plane showing constant-value paths
• Level surfaces f(x,y,z) = c are the primary visualization tool for 3-variable functions
• Domain restrictions follow the same algebraic rules as single-variable functions (no division by zero, no even root of negative, no log of non-positive)

Pitfalls

• Confusing domain with range. The domain of f(x,y) is a subset of ℝ² (a region in the xy-plane), while the range is a subset of ℝ. Describing the domain as an interval of z-values, or the range as a region in the plane, is a fundamental conceptual error.
• Applying single-variable domain reasoning incorrectly. For f(x,y) = √(x − y), the constraint x − y ≥ 0 describes a half-plane y ≤ x, not just an interval on the x-axis. Both variables contribute to the domain, and the result is typically a 2D region with a boundary curve.
• Misinterpreting level curves as lying on the surface. Level curves f(x,y) = c live in the xy-plane (the domain), not on the 3D graph z = f(x,y). Thinking of them as curves drawn on the surface is a common visualization mistake — they are the "footprint" of horizontal slices.
• Forgetting that the graph dimension is n + 1. The graph of f(x,y) lives in ℝ³ (2 inputs + 1 output = 3 dimensions), and the graph of f(x,y,z) lives in ℝ⁴ (which is impossible to fully visualize — hence the need for level surfaces instead of graphs).
• Misidentifying fundamental surface types. f(x,y) = x² + y² is a paraboloid, f(x,y) = √(x² + y²) is a cone, and f(x,y) = √(1 − x² − y²) is a hemisphere. Confusing these shapes or mispredicting their cross-sections makes it much harder to connect algebraic forms to geometric intuition.

Next Steps

Next up: 06-02-limits-and-continuity-in-rn.md