Functions of several variables
Why this matters
A scalar output may depend jointly on location and time; domain geometry matters before evaluating or plotting.
Learning objectives
- Specify domain/codomain
- compute level sets
- distinguish a slice from the whole surface.
Concept and explanation
f:D⊆R²→R assigns one scalar to each ordered pair (x,y); a level set f=c collects all domain points giving c. For f=x²+y², circles are level sets; holding y fixed produces one-dimensional slices, not a complete description. Domain restrictions such as sqrt(4−x²−y²) require x²+y²≤4.
Worked examples
Worked example 1
For f=x²+y² on R², f(1,2)=5; f=5 is x²+y²=5, circle radius √5. Slice y=2 gives f(x,2)=x²+4 and intersects level 5 at x=±1; two points do not represent the whole circle.
Worked examples
Worked example 1
g(x,y)=sqrt(9−x²−y²) has closed-disk domain x²+y²≤9 and nonnegative range [0,3]. At (0,0), g=3; at (3,0), g=0; (4,0) is not real-valued. Level g=2 yields x²+y²=5, inside domain.
Guided practice
- For f=2x+y, set f=4, solve y=4−2x and name a line, then check (1,2).
- For ln(1−x²−y²), write strict disk x²+y²<1 before evaluating (0,0).
Independent practice
- Find level set x²+4y²=4 (ellipse).
- Give domain of 1/(x−y) (exclude x=y).
- Compare slices f(x,0),f(0,y) for xy and explain why both zero miss f(1,1)=1.
- Interpret units for T(x,y) °C with x,y meters.
Common mistake and counterexample
Claiming two coordinate slices identify xy as zero throughout is refuted at (1,1); ask for an off-axis test and level set.
Knowledge check and mastery evidence
M1 correct disk domain of g; M2 level set f=5; M3 distinguish slice and surface; M4 exclude (4,0). 4/4, M1 essential.
Related laboratory
3D Function Visualization: graph on a bounded domain and list level-set coordinates in text; plots are not proof of global domain behavior. Next partial derivatives.
Prerequisites
Functions, coordinate geometry and domains.
Key terms
- independent variable
- dependent variable
- domain
- range
- surface
- level curve
- level surface
Notation and definitions
z=f(x,y); D⊆R²; f:D→R; f(x,y)=c.
A · Interactive foundation practice
Give the real domain of g(x,y)=√(9−x²−y²).
B · Application
Apply the current calculus concept to motion, energy, measurement or another meaningful model. Define variables, domain and units.
C · Reasoning and error detection
Diagnose a topic-specific error, explain why it fails, correct it and verify the correction independently.
D · Challenge / transfer
Connect symbolic, graphical and numerical representations in a less-structured problem; state assumptions and limitations.
Check and summary
Connect the symbolic, graphical, numerical and verbal representations; state units, one likely error and an independent validation method.
Next class: Partial and directional derivatives