Multivariate Functions – Lecture 09

Review: Functions of one variable

• Form: $y = f(x)$
  • One independent variable $x$, one dependent variable $y$.
• Graph lives in Cartesian $\mathbb R^2$.
  • Horizontal axis: $x$.
  • Vertical axis: $y$.
• Domain = set of permissible $x$–values (interval(s) on the $x$–axis).
• Range = set of attained $y$–values (interval(s) on the $y$–axis).
• Vertical–Line-Test (VLT): every vertical line intersects the graph at most once –necessary and sufficient condition for "single-valuedness" of $f$.

Functions of two variables

• General form: $z = f(x,y)$.
  • Two independent variables $(x,y)$, one dependent variable $z$.
• Graph lives in $\mathbb R^3$.
  • Axes follow the Right-Hand Rule:
  • Fingers along $+x$, curl toward $+y$, thumb gives $+z$.
• Domain = subset of the $xy$–plane (a 2-D region). Often written as a Cartesian product
  $(x,y) \in [a,b]\times[c,d]$.
• Range = intervals on the $z$–axis.
• VLT still applies: a vertical line (parallel to the $z$–axis) can meet the surface at most once.

"Usual Suspects" when finding domains

• Square root: $\sqrt{u}\;\Rightarrow\;u\ge 0$.
• Natural logarithm: \ln(u)\;\Rightarrow\;u>0.
• Denominator: $\dfrac{1}{u}\;\Rightarrow\;u\ne 0$.

Worked domain examples (14.1.1 & 14.1.2)

(a) $z = \sqrt{4 - x^2 - y^2}$

• Condition: $4 - x^2 - y^2 \ge 0 \;\Rightarrow\;x^2 + y^2 \le 4$.
• Domain: closed disk of radius $2$, centre at origin. Boundary circle $x^2+y^2=4$ is included (solid line).
  • Sketch: shade the interior; label $R = 2$.

(b) $z = \ln\bigl(4 - x + y^2\bigr)$

• Condition: 4 - x + y^2 > 0 \;\Rightarrow\;y^2 - x + 4 > 0.
• Domain: open region to the right of the curved boundary $y^2 - x + 4 = 0$ (dotted line excluded).
• Quick test: point $(0,0)$ satisfies 4>0, so it belongs to the domain.
• Boundary represents a sideways-opening parabola.

(c) Mixed "criss-cross, gaps & holes" example (14.1.2)

• Function (implied by conditions): something like $z = \dfrac{ \sqrt{x-y^2}\,}{y-1}$.
• Combined restrictions
  • Square root: $x-y^2 \ge 0 \;\Rightarrow\;x \ge y^2$ (region to the right of the parabola $x=y^2$, boundary mostly included).
  • Denominator: $y-1 \ne 0 \;\Rightarrow\;y \ne 1$ (horizontal line $y=1$ removed; shown as an open, dashed line).
• Resulting domain: half-parabolic region $x \ge y^2$ with a "gap" (hole) along $y=1$. Interior contains points such as $(4,0)$; the point on the parabola at $y=1$ is still excluded.

Identifying conic boundaries (visual tips)

• $x^2 + y^2 = r^2$ Circle.
• $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Ellipse.
• $x^2 = 4py$ or $y^2 = 4px$ Parabola.
• $x^2 - y^2 = 1$ (or scaled) Hyperbola.
  • Recognising the boundary curve helps decide which side of it a domain occupies.

Level sets & contour lines (14.1)

• Level set: collection of all domain points where $f(x,y)=k$ for fixed constant $k$.
  • For $z=f(x,y)$ these appear as contour lines in the $xy$-plane.
• Key facts
  • Each contour lives inside the domain.
  • Contours belonging to different $k$ values must not cross if the original surface is a function (otherwise one $(x,y)$ would yield two different $z$-values).

Example 14.1.3

Function: $f(x,y)=\sqrt{x^2+y^2}=r$ in polar coordinates.

• Level values: $k=0,1,2$.
• Equations: $x^2+y^2=k^2$ circles of radius $k$ centred at origin.
• Domain is all $(x,y)$ (no restrictions: inside/outside all circles acceptable).

Example 14.1.4

Function: $f(x,y)=e^{y/x}$.

• Domain: $x\ne 0$ (division by zero forbidden).
• Requested levels: $k=1,e,e^2$.
• Solve $e^{y/x}=k\;\Rightarrow\;y/x=\ln k \;\Rightarrow\;y=(\ln k)x$.
  • $k=1$: $\ln 1=0\Rightarrow y=0$ (the $x$-axis, but x \ne 0).
  • $k=e$: $y=x$.
  • $k=e^2$: $y=2x$.
• Each contour is a straight line through the origin; the line $x=0$ is missing from the domain, so each level set has an infinitesimal gap at the origin.

Example 14.1.5

Function: $f(x,y)=\sqrt{x+y+4}$.

• Domain condition: $x+y+4\ge 0 \Rightarrow y\ge -x-4$ (half-plane above this line; boundary included).
• Levels: $k=0,1,2,3$.
  • Square and rearrange: $k^2=x+y+4\Rightarrow y=-x+(k^2-4)$.
  • Lines have slope $m=-1$, vertical spacing $\Delta b= (k<em>2^2-k</em>1^2)$.
• Qualitative 3-D shape
  • Along any ray perpendicular to the domain boundary, $z$ increases like a square-root: steep near the boundary, shallower farther away.

Interpreting contour spacing

• Draw contours at equal vertical intervals ($\Delta z$ constant).
  • Close contour lines large change in $z$ over small planar distance steep surface.
  • Widely spaced lines small gradient shallow slope.

Functions of three variables

• Notation: $w = T(x,y,z)$ (e.g.temperature inside a solid).
  • Domain: a 3-D region in $\mathbb R^3$ ("blob" of material, room, etc.).
• Level surfaces ("isotherms" when $T$ is temperature): sets where $T(x,y,z)=k$.
  • Examples: $T=35^{\circ}\mathrm C, 50^{\circ}\mathrm C, 65^{\circ}\mathrm C, 80^{\circ}\mathrm C$.
  • Each level surface is a 2-D sheet inside the 3-D domain.
• As with contours, densely packed level surfaces indicate regions of large spatial temperature gradient.

Quick comparison table

• 1-var: curve in $\mathbb R^2$, VLT, domain $x$-intervals.
• 2-var: surface in $\mathbb R^3$, VLT along $z$-axis, domain planar regions, contour lines.
• 3-var: hypersurface in $\mathbb R^4$ (visualised by level surfaces in $\mathbb R^3$).

Ethical & practical remarks (implicit)

• Accurate domain identification prevents using formulas where they are undefined ( avoid dividing by zero, complex logs, etc.).
• Contour/level-set reading underpins practical tools: topographic maps, weather isotherms, medical CT/isodose plots.
• Understanding steepness via contour density links directly to gradient vectors and optimisation (topics that follow).