Download Functions of Several Variables A function of two variables is a rule

Functions of Several Variables
A function of two variables is a rule that
assigns to each ordered pair of real numbers (x, y)
in a subset D of the plane a unique real number
denoted by f (x, y). The set D is the domain of f
and its range is the set of values that f takes on,
that is, {f (x, y) : (x, y) ∈ D}.
We often write z = f (x, y) to make explicit the
values taken on by f at the general point (x, y). The
variables x and y are independent variables and
z is the dependent variable.
Functions of more than two variables can be defined similarly.
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Example. Let
√
f (x, y) =
x+y+1
.
x−1
(1) Evaluate f (3, 2), f (2, −5), f (1, −1), and f (−1, 4).
(2) Find the domain of f .
√
Solution. (1) f (3, 2) = 6/2; f (2, −5) is not defined; f (1, −1) is not defined; f (−1, 4) = −1.
(2) The domain of f is
D = {(x, y) : x + y + 1 ≥ 0, x 6= 1}.
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Limits of Functions
Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b).
Then we say that the limit of f (x, y) as (x, y) approaches (a, b) is L and we write
lim
f (x, y) = L
(x,y)→(a,b)
if for every number ε > 0 there is a corresponding
number δ > 0 such that
|f (x, y) − L| < ε
whenever
(x, y) ∈ D
and 0 <
p
3
(x − a)2 + (y − b)2 < δ.
Continuous Functions
A function f of two variables is called continuous at (a, b) if
lim(x,y)→(a,b) f (x, y) = f (a, b).
We say f is continuous on D if f is continuous at
every point (a, b) in D.
Let f and g be two functions defined on D, and
let (a, b) be a point in D. If f and g are continuous at (a, b), then both f + g and f g are continuous
at (a, b). Moreover, f /g is continuous at (a, b), provided g(a, b) 6= 0.
A polynomial of two variables is a sum of terms
of the form cxm y n , where c is a real number and m
and n are nonnegative integers. Any polynomial is
continuous on the whole plane IR2 .
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Graphs
If f is a function of two variables with domain
D, then the graph of f is the set of all points (x, y, z)
in IR3 such that z = f (x, y) and (x, y) is in D.
Example. The graph of the function
z = x2 + y 2
is called a paraboloid.
z
y
O
x
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Example. Let us consider the function
p
z = 9 − x2 − y 2 .
Its domain is the disk with radius 3 and center
at the origin:
{(x, y) : x2 + y 2 ≤ 9}.
Since z ≥ 0 and x2 + y 2 + z 2 = 32 , its graph is an
upper hemisphere.
z
y
O
x
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