Topic 8 / 17intermediate

Functions of Several Variables

Determine domains, sketch surfaces $z = f(x,y)$, and visualize them with traces, level curves, and contour maps.

A function $f (x, y)$ assigns a real number to each pair $(x, y)$ in its domain. The graph $z = f (x, y)$ is a surface in 3D. We understand surfaces by **traces** (slices at fixed values) and **contour maps** (level curves).

Domain of $f(x, y)$

Find restrictions: denominators

\neq = 0

, square root expressions

\geq 0

, log arguments

> 0

, etc. Express as a region in the

x y

-plane.

Surfaces and traces

Set one variable to a constant and sketch the resulting cross-section in the other two: -

z = c

trace: solve

f (x, y) = c

— produces a **level curve** (contour). -

x = c

trace: substitute and solve in the

y z

-plane. -

y = c

trace: substitute and solve in the

x z

-plane. Combining these traces builds intuition for the full surface.

Level curves and contour maps

The level curve

f (x, y) = c

is the set of

(x, y)

where

f

takes value

c

. A contour map plots several level curves at evenly spaced

c

-values; **closely spaced contours indicate steep terrain**, widely spaced contours indicate flat terrain.

Strategy tips

Use level curves to spot circular symmetry: $f (x, y) = x^{2} + y^{2}$ has concentric circles.
Saddle points show as crossed contours.

Level surfaces (3 variables)

For

f (x, y, z)

, the **level surface**

f (x, y, z) = c

is a 2D surface in 3D. Common examples:

x^{2} + y^{2} + z^{2} = c

(sphere shells),

z - x^{2} - y^{2} = c

(paraboloid shells).

Recognizing common surfaces

Quick visual cues: -

z = a x + b y + c

→ plane. -

z = x^{2} + y^{2}

→ upward paraboloid (level curves are circles). -

z = x^{2} - y^{2}

→ saddle (hyperbolic level curves). -

z = x^{2} + y^{2}

→ upper half cone.

Worked examples

Example 1

Find the domain of

f (x, y) = ln (4 - x^{2} - y^{2}) + 1 - x^{2}

1
Log argument positive: $4 - x^{2} - y^{2} > 0 \Rightarrow x^{2} + y^{2} < 4$ (open disk of radius 2).
2
Square root nonnegative: $1 - x^{2} \geq 0 \Rightarrow - 1 \leq x \leq 1$ .
3
Intersect.
${(x, y) : x^{2} + y^{2} < 4 and - 1 \leq x \leq 1}$

Answer

Strip

- 1 \leq x \leq 1

inside the open disk of radius 2.

Example 2

Sketch level curves of

f (x, y) = x^{2} - y^{2}

for

c = - 2, - 1, 0, 1, 2

1
$c = 0$ : $x^{2} = y^{2} \Rightarrow y = \pm x$ — two lines.
2
$c > 0$ : $x^{2} - y^{2} = c$ — hyperbola with branches opening left/right.
3
$c < 0$ : $y^{2} - x^{2} = - c > 0$ — hyperbola with branches opening up/down.
4
Together: a saddle pattern with crossed asymptotes through the origin.

Answer

Hyperbolas; sign of

c

flips orientation. Asymptote lines

y = \pm x

pass through saddle.

Example 3

Match the function to its graph: (i)

z = sin x^{2} + y^{2}

, (ii)

z = e^{- (x^{2} + y^{2})}

1
(i) Concentric ripples — sine of distance from origin, periodic.
2
(ii) Bell-shaped peak at origin, decays to 0 outward.

Answer

(i) ripple pattern; (ii) Gaussian bell.

Interactive visualizations

Loading visualization…

Saddle:

z = x^{2} - y^{2}

. Note how it goes up along

x

and down along

y

Loading visualization…

Contour map of

f (x, y) = x^{2} + y^{2}

. Concentric circles; closer rings indicate steeper slope.

Formulas in this topic

Level curve

f (x, y) = c

Set of points where f equals c.

Level surface

f (x, y, z) = c

Set in 3D where 3-variable f equals c.

Circular paraboloid

z = x^{2} + y^{2}

Upward; level curves are circles.

Saddle

z = x^{2} - y^{2}

Hyperbolic level curves.

Cone (upper half)

z = x^{2} + y^{2}

Apex at origin, opens upward.

Domain of $f(x, y)$

Surfaces and traces

Level curves and contour maps

Level surfaces (3 variables)

Recognizing common surfaces

Worked examples

Interactive visualizations

Formulas in this topic

Related topics