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MT2503 Multivariate Calculus Chapter 03: Functions of Two or More Variables
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CHAPTER 03FUNCTIONS OF TWO OR MORE VARIABLES
ordered
y)
A real function of two variables floc takes the of real
, pair
numbers ( x ,y) and returns a
single real number
2- = fcx , y)
The DOMAIN of flx ,y ) is the area in the
icy plane on which f- is defined .
The RANGE of values domain
of f- Guy) is the set that f takes on its .
LIMITS & CONTINUITY
For function fcx y ) and P= ( xo yo) P
approach along
a
, a
point ,
,
we
may
several
If the ( and
different paths .
limit of f- is the same
along every path has value
2) we
say
:
Lim { flay) } =L OR fix , g) → L as IX. g) → (xo yo)
,
( x
, g) → ( Xo ,yo)
doesnt
The limit must be path independent otherwise limit exist and f is not continuous
-
Useful Strat convert to coordinates with rcoso and rsino
:
polar ✗ =
y=
( covers all
straight line
paths as r→o )
3.13
Example
ricosordlcoszots.in# r%
?
Find =
=
rcos @
=
"" → i%{¥ }
Iim
x'tyT
As r→O
,
roost → 0 ( for any value of 0 )
: A
unique limit exists 1=0
,
3. Is
Example
Definition Is
?yz
{
1797=110,0)
flu ,y) is continuous at Coco , yo) if f- IX. g) =
0 x=y=o
( x ,y )
Iim
→ ( xo ,yo)
{ flx.gl } =L = f- Geo yo) ,
continuous at the
origin ?
xy_ =
rZcosOsinO_ = cos @ Sino
> ' ?
steps : a +
y rzlcos @ +
sink
① Show limit exists
flxo ,y As r→0 cos @ Sino -1>0
② Show limit is equal to )
.
,
( i.e taken ]
Limit depends on 0 .
path
i. Not continuous at 10,0)
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