Multivariable Functions: Domains, Level Sets & Early Limits

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Module Context

• Module 2: Calculus of Multivariable
  Chapter focus: Multivariable Functions and Partial Derivatives.
• Motivation: Functions with several independent variables dominate scientific applications (probability, statistics, fluid dynamics, electricity, etc.).
  Their calculus is "richer" because variables interact in multiple ways; integrals and derivatives proliferate.

Fundamental Definition

• Let $D$ be a set of $n$–tuples $(x<em>1,x</em>2,\dots,x<em>n)$. A real–valued function of several variables is a rule assigning a number $w=f(x</em>1,x<em>2,\dots,x</em>n)$
  to every tuple in $D$.
  • $D$ = domain (input set).
  • Range = all attainable $w$ values (output set).
  • $x<em>1,\dots ,x</em>n$ are independent / input variables.
  • $w$ is the dependent / output variable.

Naming Conventions

• 2-variable case → typically write $f(x,y)$ and imagine domain in the $xy$-plane.
• 3-variable case → write $f(x,y,z)$ and imagine domain in 3-space.
• In applications, letters often carry meaning
  • Example cylinder: $V=f(r,h) = \pi r^2 h$ (inputs $r,h$; output $V$).

Evaluating a Formula (Example 1)

Given $f(x,y,z)=\sqrt{x^2+y^2+z^2}$,
$f(3,0,4)=\sqrt{3^2+0^2+4^2}=\sqrt{25}=5.$

Domain Restrictions (Real-valued convention)

• Disallow inputs that would cause division by zero or complex results.
  Examples:
  • $f(x,y)=\sqrt{y-x^2}$ requires $y\ge x^2$.
  • $f(x,y)=1/(xy)$ requires $xy\ne 0$.

Mini-Catalog of Functions (Example 2 & 3)

| Function | Domain (informal) | Range |
|——|——|——|
| $w=\sqrt{y-x^2}$ | $y\ge x^2$ | $[0,\infty)$ |
| $w=\dfrac1{xy}$ | $xy\ne 0$ | $(-\infty,0)\cup(0,\infty)$ |
| $w=\sin(xy)$ | entire plane | $[-1,1]$ |
| $w=\dfrac1{\sqrt{x^2+y^2+z^2}}$ | $(x,y,z)\ne(0,0,0)$ | $(0,\infty)$ |
| $w=xy\ln z$ | half-space z>0 | $(-\infty,\infty)$ |

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Interior & Boundary in the Plane

• Interior point: $(x<em>0,y</em>0)$ is interior to a region $R$ if some disk centered there lies entirely in $R$ (Fig. 12.1a).
• Boundary point: every disk centered at $(x<em>0,y</em>0)$ intersects both $R$ and its complement (Fig. 12.1b).
  The point itself can be in or out of $R$.
• Interior = set of all interior points.
  Boundary = set of all boundary points.
• Open region: consists only of interior points.
  Closed region: contains its entire boundary.

Unit-Disk Examples (Fig. 12.2)

• Open disk {(x,y)\,|\,x^2+y^2<1} → every point interior; boundary excluded.
• Boundary circle ${(x,y)\,|\,x^2+y^2=1}$ → every point boundary; none interior.
• Closed disk ${(x,y)\,|\,x^2+y^2\le 1}$ → contains interior and boundary.

Bounded vs. Unbounded

• Bounded: lies inside some disk of finite radius.
• Unbounded: not bounded.
  • Bounded examples: line segments, triangles, rectangles, disks.
  • Unbounded examples: entire lines, axes, graphs over infinite intervals, half-planes, the plane itself.

Example: Domain of $f(x,y)=\sqrt{y-x^2}$ (Fig. 12.3)

• Domain = set ${(x,y)|y\ge x^2}$ including the parabola $y=x^2$.
• Classification: closed (contains boundary), unbounded (extends infinitely).

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(Visual content extends Page 2 ideas—definitions already captured.)

Key Takeaways

• Adding or removing some boundary points creates sets that are neither open nor closed.
• Graphical intuition: interior ↔ shaded region; boundary ↔ curve separating interior/exterior.

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Extension to 3-Space

• Replace disks with balls.
  • Closed ball = interior points + bounding sphere.
  • Open ball = interior without sphere.

Definitions in Space

• Interior / boundary points mirror 2-D case but use spheres (Fig. 12.4).
• Open region in space ↔ only interior points.
• Closed region ↔ contains full boundary.

Typical Sets (Example 6)

• Open: open balls; open half-space z>0; first octant without coordinate planes; entire space.
• Closed: lines; planes; closed balls; closed half-space $z\ge 0$; first octant with bounding planes.
• Neither: closed ball minus a spherical patch; solid cube missing a face/edge/corner.

Graphs & Level Curves (2 variables)

• Level curve: set ${(x,y)|f(x,y)=c}$.
• Graph (surface): set ${(x,y,f(x,y))}$ → written $z=f(x,y)$.

Example 7 – Paraboloid

Given $f(x,y)=100-x^2-y^2$:

• Domain: entire $xy$-plane.
• Range: $(-\infty,100]$.
• Graph: paraboloid opening downward (Fig. 12.5).
• Level curves $f=c$ → circles $x^2+y^2=100-c$.
  • $c=0$ → radius $10$.
  • $c=51$ → radius $7$.
  • $c=75$ → radius $5$.
• Level curve vs. Contour line:
  • Level curve lies in the domain (plane).
  • Contour line is the intersection of the surface with plane $z=c$ (Fig. 12.6).
  Context (e.g., topographic maps) often calls both "contours".

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Special Note on $f=100$

• The set $f(x,y)=100$ yields only the single point $(0,0)$ (radius $0$) yet still qualifies as a level curve.

From 2-D to 3-D: Level Surfaces

• For a function $f(x,y,z)$ a level surface is ${(x,y,z)|f(x,y,z)=c}$.

Example 8 – Radial Distance

$f(x,y,z)=\sqrt{x^2+y^2+z^2}$:

• Each level surface $\sqrt{x^2+y^2+z^2}=c$ (with c>0) is a sphere of radius $c$ centered at the origin (Fig. 12.8).
• The "surface" $f=0$ collapses to the single point at the origin.
• Graph of $f$ would require 4-D space (points $(x,y,z,f)$); here we focus only on level surfaces in 3-D domain.
• Direction matters: function increases when moving away from origin, decreases when moving toward it.

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Computer-Generated Graphing (Example 9)

Function describing subsurface temperature fraction:
$w=\cos\bigl(1.7\times10^{-2}t-0.2x\bigr)\,e^{-0.2x},$
where
• $t$ = time in days,
• $x$ = depth in feet.

Observations from Fig. 12.9:

1. Amplitude decay with depth (factor $e^{-0.2x}$).
   • At $x=15\text{ ft}$ → variation ≈ 5 % of surface.
   • At $x=30\text{ ft}$ → variation < 0.25 %.
2. Phase shift: at 15 ft the cycle lags about half a year; deepest layers experience "seasonal inversion" (warmest in surface winter).

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Exercise Themes (12.1)

1. Domain & Range analysis (problems 1–12).
   • Tasks: determine domain, range, level curves, boundary, openness/closedness, boundedness.
2. Identifying Surfaces via Level Curves (13–18).
3. Graph + Level Curves Production (19–28).
4. Level Surfaces Sketching (29–36).
5. Find specific Level Curve/Surface passing a point (37–44).
   • Illustrates solving $f(x,y)=k$ or $f(x,y,z)=k$ for $k$ using given point.
6. Theory Problems (45–48)
   • Max/min of $xyz$ on lines.
   • Concorde sonic-boom carpet width $w=4(Th/d)^{1/2}$ (Fig. 12.10).
   • Abstract definition of graphs in higher dimensions.

Concorde Example (Problem 47)

• Width formula: $w=4\bigl(\dfrac{Th}{d}\bigr)^{1/2}$.
  For $T=290\,\text{K},\,h=16.8\,\text{km},\,d=5\,\text{K/km}$ compute $w$ to decide safe flight path south of Nantucket.
  (Exercise sets numerical work for the student.)

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CAS (Computer Algebra System) Explorations

• Explicit-surface tasks (49–52):
  • Plot $z=f(x,y)$ over rectangles.
  • Produce level curves and specific level curve through a point.
• Implicit-surface tasks (53–56): plot level surfaces given equations.
• Parametrized surfaces (57–60): provide $x=f(u,v),\,y=g(u,v),\,z=h(u,v)$ descriptions for helicoids, cones, tori, etc., to visualize along with level curves in the $xy$-plane.

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Introduction to Limits & Continuity (Section 12.2)

Informal Idea

• $f(x,y)$ approaches limit $L$ as $(x,y)\to(x<em>0,y</em>0)$ if values of $f$ get arbitrarily close to $L$ whenever $(x,y)$ is sufficiently close to $(x<em>0,y</em>0)$.
• Multivariable nuance: approach can occur from any direction in the plane (or space).

Formal \varepsilon$–\delta Definition

We write
\lim{(x,y)\to(x0,y0)}f(x,y)=L if: For every \varepsilon>0$, there exists$\delta>0 such that 0

Equivalently (by rectangular neighborhood):
If 0<|x-x0|

Boundary Points

• Definition applies whether (x0,y0)$is interior <strong>or</strong> boundary of the domain; we merely insist$(x,y) stay within the domain in taking the limit.

Synthesized Connections & Implications

1. Geometry of Domains (open/closed, bounded/unbounded) parallels interval properties from single-variable calculus but requires spatial reasoning with disks/balls.
2. Graphs vs. Level Sets:
   • Graph of f$$ lives in one extra dimension.
   • Level curves/surfaces keep dimensionality same as domain, providing alternative visualization.
3. Directional Dependence foreshadows concepts like partial derivatives, gradient vectors, and directional derivatives (topics ahead in the chapter).
4. Real-world relevance:
   • Cylinder volume, temperature diffusion, sonic-boom footprints illustrate why multivariable functions matter.
   • Exponential damping in Example 9 is a physical manifestation of heat equation solutions.
5. Ethical / Practical Consideration: Concorde problem links mathematical prediction to minimizing noise pollution over inhabited areas.
6. Computational Tools: CAS and 3-D graphing greatly aid intuition—emphasized via numerous exploration exercises.