4.1 Functions of Several Variables

← Chapter 4 §4.1 Functions of Several Variables 多元函数 · Chapter 4 · Differentiation of Functions of Several Variables

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Access for free at openstax.org · CC BY-NC-SA 4.0

The formula has	So we need	Example
a square root	inside $\ge 0$	Ex 4.1b: $9-x^2-y^2\ge 0$ , a closed disk
a fraction	bottom $\ne 0$	Ex 4.6b: $x^2-y^2\ne 0$ , so $y\ne\pm x$
a square root in the bottom	inside $>0$	Ex 4.6a: $x^2+y^2+z^2<9$ , an open ball
a logarithm	inside $>0$	Exercise 7: $y^2-x>0$ , so $x<y^2$
only polynomials, cube roots	nothing	Exercise 37: all of space

Objective	Takeaways; say them first, then click
4.1.1 Recognize a function of two variables and identify its domain and range.	one $z$ per point; domain = where the formula works; range = the heights reached
4.1.2 Sketch a graph of a function of two variables.	graph = surface over the domain; fix one variable to find known curves
4.1.3 Sketch several traces or level curves of a function of two variables.	$f(x,y)=c$ : level curves, bunched = steep; $z=f(a,y)$ , $z=f(x,b)$ : traces
4.1.4 Recognize a function of three or more variables and identify its level surfaces.	$f(x,y,z)$ : a value at each point of space; level surface $f(x,y,z)=c$

Mistake	Result	Fix
Squaring $\sqrt{9-x^2-y^2}=-2$	a "level curve" $x^2+y^2=5$	a square root is never negative: levels need $c\ge 0$ (p308)
$\le$ with a root in the bottom	the sphere $x^2+y^2+z^2=9$ put in the domain of Ex 4.6a	the bottom cannot be 0: strict $<$
$x^2\ne y^2$ read as $y\ne x$	the line $y=-x$ left in	$y\ne\pm x$
Range read off the domain	$g=\sqrt{9-x^2-y^2}$ given range $[-3,3]$	range is outputs: heights 0 to 3
$4(x^2-2x+1)$ balanced by $+1$	$4(x-1)^2+(y+2)^2=13$ in Ex 4.4	add $4\cdot 1$ : the right side is 16
A trace drawn in the $xy$ -plane	the curve $z=f(a,y)$ lost	it lives in the plane $x=a$ , with axes $y$ and $z$
Level surface named without the sign of $c$	$4x^2+9y^2-z^2=-1$ called one sheet	$c<0$ : two sheets; $c=0$ : a cone

#	task	answer
4.1	domain and range of $\sqrt{36-9x^2-9y^2}$	$x^2+y^2\le 4$ , a disk of radius 2; range $[0,6]$
4.2	level curve of $x^2+y^2-6x+2y$ at $c=15$	$(x-3)^2+(y+1)^2=25$ : circle, centre $(3,-1)$ , radius 5
4.3	trace of $-x^2-y^2+2x+4y-1$ at $y=3$	$z=3-(x-1)^2$ : a parabola opening down in the plane $y=3$
4.4	domain of $(3t-6)\sqrt{y-4x^2+4}$	$\{(x,y,t): y\ge 4x^2-4\}$
4.5	level surface of $x^2+y^2+z^2-2x+4y-6z$ at $c=2$	$(x-1)^2+(y+2)^2+(z-3)^2=16$ : sphere, centre $(1,-2,3)$ , radius 4

#	task	answer	key step
1	$W=4x^2+y^2$ : $W(2,-1)$ , $W(-3,6)$	17, 72	$16+1$ ; $36+36$
3	$V=\pi x^2y$ : $V(2,5)$	$20\pi\approx 62.83$	a cylinder of radius 2, height 5
7	domain of $4\ln(y^2-x)$	$x<y^2$	domain lab, preset Exercise 7
9	domain of $y^2-x^2$	all of $\mathbb R^2$	a polynomial
11	range of $\sqrt{16-4x^2-y^2}$	$[0,4]$	largest at $(0,0)$
13	range of $y^2-x^2$	$\mathbb R$	$y^2$ alone, $-x^2$ alone
15	$y^2-x^2=4$	a hyperbola	the saddle at $c=4$
17	$4-x-y$ at $c=0,4$	$x+y=4$ ; $x+y=0$	lines
19	$2x-y$ at $c=0,-2,2$	$2x-y=0,\ -2,\ 2$	three parallel lines
23	$e^{xy}$ at $c=\frac12,3$	$e^{xy}=\frac12$ , $e^{xy}=3$	hyperbolas $xy=-\ln 2$ , $xy=\ln 3$
27	$\ln(y/x^2)$ at $c=-2,0,2$	$y=e^{-2}x^2$ , $y=x^2$ , $y=e^2x^2$	parabolas
29	$(y+2)/x^2$ , any $c$	$y=cx^2-2$	parabolas through $(0,-2)$
31	$3x+y^3$ at $x=1$	$z=3+y^3$	trace lab, preset Exercise 31
33	domain of $\sqrt{100-4x^2-25y^2}$	$\frac{x^2}{25}+\frac{y^2}{4}\le 1$	domain lab, preset Exercise 33
35	domain of $1/\sqrt{36-4x^2-9y^2-z^2}$	$\frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}<1$	root in the bottom: strict
37	domain of $\sqrt[3]{16-x^2-y^2-z^2}$	all of space	cube roots take negatives
47	contours of $x^2+y^2-2x-2y$	circles	$(x-1)^2+(y-1)^2=c+2$
49	$x^2+y^2+z^2=9$	a sphere of radius 3	level-surface lab, preset Exercise 49
51	$x^2+y^2-z^2=4$	a hyperboloid of one sheet	$c>0$
53	$1-4x^2-y^2$ through $P(0,1)$	$4x^2+y^2=1$	$f(0,1)=0$
55	$e^{xy}(x^2+y^2)$ through $P(1,0)$	$1=e^{xy}(x^2+y^2)$	$g(1,0)=1$
57	$T$ inversely proportional to distance squared	$T=\dfrac{k}{x^2+y^2}$	—
59	level curves $T=40$ , $T=100$	$x^2+y^2=\frac{k}{40}$ , $x^2+y^2=\frac{k}{100}$	circles of radius $\frac{\sqrt{10k}}{20}$ , $\frac{\sqrt k}{10}$

Functions of Several Variables

A function of two variables sends each point $(x,y)$ to exactly one number $z$

The domain is every point where the formula makes sense

Drag a point across the edge: the formula works on one side only

Factor out the 9: $\sqrt{36-9x^2-9y^2}$ lives on a disk of radius 2

A domain keeps every condition the formula imposes

The graph of $z=f(x,y)$ is a surface: over each point, rise to height $z$

Fix one variable and a known curve appears: $x^2+y^2$ is a paraboloid

The profit function is a downward paraboloid, highest over $(3,2)$

Every profit below 16 is reached on a whole circle about $(3,2)$

A contour map joins points of equal height: bunched lines mean steep ground

A level curve is a horizontal slice of the surface, dropped to the floor

Squaring can invent a level curve: $c$ is in the range exactly when its level curve is not empty

Complete the square: the level curves of $\sqrt{8+8x-4y-4x^2-y^2}$ are ellipses about $(1,-2)$

$x^2+y^2-6x+2y=15$ is a circle of radius 5 about $(3,-1)$

Raise the plane $z=c$ : where it cuts the surface is the level curve below

A vertical trace is an upright slice: fix $x$ or $y$ , get a one-variable curve

The trace of $-x^2-y^2+2x+4y-1$ at $y=3$ is a downward parabola

Slide the upright plane: every cut is the graph of a one-variable function

One function, three views: the surface, its level curves, its traces

A function of three variables gives every point in space a number — too many dimensions for a graph

Same domain rules in space: the domain becomes a solid region

$(3t-6)\sqrt{y-4x^2+4}$ needs only $y\ge 4x^2-4$ ; $t$ is free

A level surface $f(x,y,z)=c$ is where a three-variable function keeps one value

Complete the square three times: the level surface is a sphere of radius 4 about $(1,-2,3)$

Levels of one function never meet: as $c$ drops through 0, one sheet becomes a cone, then two sheets

Change $c$ : the level surfaces of one function fill space like layers

Read a function of several variables by its domain, its surface and its level sets

Common mistakes

Practice and answers

Functions of Several Variables

A function of two variables sends each point (x,y)(x,y)(x,y) to exactly one number zzz

The domain is every point where the formula makes sense

Drag a point across the edge: the formula works on one side only

Factor out the 9: 36−9x2−9y2\sqrt{36-9x^2-9y^2}36−9x2−9y2​ lives on a disk of radius 2

A domain keeps every condition the formula imposes

The graph of z=f(x,y)z=f(x,y)z=f(x,y) is a surface: over each point, rise to height zzz

Fix one variable and a known curve appears: x2+y2x^2+y^2x2+y2 is a paraboloid

The profit function is a downward paraboloid, highest over (3,2)(3,2)(3,2)

Every profit below 16 is reached on a whole circle about (3,2)(3,2)(3,2)