If x = (x1 , x2 , … , xn) represents a point in a subset A of Rn, and f(x) is exactly one point in Rm, then we say that f is a function from (a domain in) Rn to Rm.

The function f is called a

scalar-valued function if m = 1vector-valued function if m > 1function of a single variable if n = 1function of several variables if n > 1

f(x) = 8 – x is a scalar-valued function of a single variable withdomain {x | x 8} and

range{y | y 0}.

f(x1 , x2 , x3) = x2/x1 + 8 – x32 is a scalar-valued function of three

variables with domain{(x1 , x2 , x3) | x1 0 , – < x2 < , |x3| 8}{y | – < y < }.and range

If x = (x1 , x2 , … , xn) represents a point in a subset A of Rn, and f(x) is exactly one point in Rm, then we say that f is a function from (a domain in) Rn to Rm.

The function f is called a

scalar-valued function if m = 1vector-valued function if m > 1function of a single variable if n = 1function of several variables if n > 1

f(x) = (3|x| + 5 , x2) is a vector-valued function of a single variable withdomain {x | – < x < } and range {(y1 , y2) | y1 5 , y2

0}.f(x1 , x2) = (x2/x1 , 8 – x12 , x1x2) is a vector-valued function of two

variables with domain{(x1 , x2) | 0 < |x1| 8 , – < x2 < } and{(y1 , y2 , y3) | – < y1 < , 0 y2 < 8 , – < y3 < }.range

Definition of the graph of a single-valued function (page 97)

Definition of level curves and level surfaces (page 99)

Example 2 (page 98)

z = f(x,y) = x + y + 2

x

y

The level curves are x + y + 2 = c

c = 0

c = 1

c = –1

x + y = – 2

c = 0

x + y = – 1

c = 1

c = –1

x + y = – 3

This is the graph of a plane.

x

y

z = f(x,y)

Example 2 (page 98)

z = f(x,y) = x + y + 2

This is a plane.

(0, 0, 2)

(0, –2, 0)

(–2, 0, 0)

x

y

z = f(x,y)

(0, 0, 2)

(–2, 0, 0)

(0, 2, 0)

z = f(x,y) = x – y + 2

This is a plane.

x

y

z = f(x,y)

(1, 0, 3)

z = f(x,y) = 3x

This is a plane.

(–1, 0, –3)

Example 3 (page 99)

z = f(x,y) = x2 + y2

x

y

The level curves are x2 + y2 = c

c = 0

c = 1

c = –1

c = 2

x2 + y2 = 0c = 0

c = 1

x2 + y2 = 1

x2 + y2 = –1 The level curve is empty.

x2 + y2 = 2

c = 2

Example 3 (page 99)

z = f(x,y) = x2 + y2

Each level curve resulting from letting z be a constant c > 0 is

Each level curve resulting from letting either x or y be a constant c is

This is the graph of

a circle of radius c centered at the origin.

a parabola.

a circular paraboloid (pictured in Figure 2.1.7).

Look at the Conic Sections Handout and the Quadric Surfaces Handout to see how various two-dimensional graphs and three-dimensional graphs can be identified.

Example 4 (page 100)

z = f(x,y) = x2 – y2

x

y

The level curves are x2 – y2 = c

c = 0

c = 1

c = –1

x2 – y2 = 0

c = 0

x2 – y2 = 1

c = 1x2 – y2 = –1c = –1

This is the graph of a hyperbolic paraboloid (pictured in Figure 2.1.10).

x2 + 4y2 – z2 = – 4

Each level curve resulting from letting z be a constant c, where |c|>2, isan ellipse.

a point.

no points at all.

a hyperbola.

a hyperbola.

a hyperboloid of two sheets.

Each level curve resulting from letting z be a constant c, where |c|=2, is

Each level curve resulting from letting z be a constant c, where |c|<2, is

Each level curve resulting from letting x be a constant c is

Each level curve resulting from letting y be a constant c is

This is the graph of

x2/9 + y2/16 + z2 = 1

Each level curve resulting from letting z be a constant |c|<1 is an ellipse.

an ellipse.

an ellipse.

an ellipsoid.

Each level curve resulting from letting x be a constant |c|<3 is

Each level curve resulting from letting y be a constant |c|<4 is

This is the graph of

x2 + y2 – z2 = 4

Each level curve resulting from letting z be a constant c is

a circle.

a hyperbola.

a hyperbola.

a hyperboloid of one sheet.

Each level curve resulting from letting x be a constant c is

Each level curve resulting from letting y be a constant c is

This is the graph of

x2 + y2 – z2 = 0

Each level curve resulting from letting z be a constant |c| > 0 is a circle.

the point (0 , 0).

a hyperbola.

The level curve resulting from letting z be the constant 0 is

Each level curve resulting from letting x be a constant |c| > 0 is

two straight perpendicular lines.

The level curve resulting from letting x be the constant 0 is

two straight perpendicular lines.

The level curve resulting from letting y be the constant 0 is

right cone.This is the graph of

a hyperbola.Each level curve resulting from letting y be a constant |c| > 0 is

z = – y2

In the yz plane, this graph is

In R3, this graph (extended parallel to the x axis) is

a parabola.

a right parabolic cylinder.

x2 + y2 = 25

In the xy plane, this graph is

In R3, this graph (extended parallel to the z axis) is

a circle (of radius 5 centered at the origin).

a right circular cylinder.

z = x2 + y2 – 4x – 6y + 13 = (x – 2)2 + (y – 3)2

Each level curve resulting from letting z be a constant c > 0 isa circle (of radius c entered at (2,3)).

This is the graph of a shifted circular paraboloid.

Describe each graph in R3.

x2 + 3y2 + z2 = 11

x2 + 3y2 + z2 = 0

z2 = 0

x2 + y2 = 9

x2 + y2 = 0

x2 + y2 + z2 = 10

x2 + y2 + z2 + 1 = 0

x2 – y2 = 0

ellipsoid

the xy plane

the line which is the z axis

no points at all

two planes (x + y = 0 and x – y = 0 )

the single point (0 , 0 , 0)

a circular cylinder of radius 3 centered at the origin

a sphere of radius 10 centered at the origin

Level curves help us picture graphs in R3. It is impossible to picture graphs in Rn for n > 3, but level surfaces can be used to give some insight into graphs in R4.

Example 5 (page 102)

w = f(x,y,z) = x2 + y2 + z2

Example 6 (page 103)

w = f(x,y,z) = x2 + y2 – z2

Look at each of the following: