Functions of two variables: Definitions: a) A function of two variables is a rule g that assigns to each ordered pair (, ) xy in a set 2 R a unique number (, ) gxy . The set R is called the domain of the function and the corresponding values of (, ) gxy constitute the range of g . We refer to x and y as the independent variables. We usually define (, ) gxy as a formula and assume that the domain is the largest set of points in the plane for which the formula is defined and real-valued. b) The graph of the function (, ) z gxy , where 2 (, ) xy R , is the set 3 ( , , ) | ( , ) and ( , ) S xyz z gxy xy R . Geometrically, if (, ) z gxy measures the vertical distance from the point (, ) xy in the xy -plane, then z describes the points (, , (, )) xygxy that lie on a surface in space. Thus, the graph of g is a surface in 3 whose projection onto the xy -plane is the domain R (see figure below). c) In an analogous manner, we define a function of three variables as a rule f that assigns to each triple (, ,) xyz in a set 3 D a unique number (, ,) fxyz . We refer to (, ,) fxyz as a function of three independent variables , , xyz ; define it as a formula and assume that its domain is the largest set of points in space 3 for which the formula is defined and real-valued. Example 1: Let T denote the temperature and D a metal plate in the xy -plane. The function (, ) Txy then gives the temperature at a point (, ) xy D . If T varies not only on the plate but also with time , t then T is a function of three independent variables , ,. xyt Similarly, if the metal plate lies in space, then (,,,) Txyzt is a function of four independent variables ,,,. xyzt Example 2: A polynomial function (, ) pxy is a function of two independent variables x and . y It is a sum of the functions of the form m n Cx y with nonnegative integers m , n and
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Functions of two variables:
Definitions:
a) A function of two variables is a rule g that assigns to each ordered pair ( , )x y in a
set 2R a unique number ( , )g x y . The set R is called the domain of the function
and the corresponding values of ( , )g x y constitute the range of g . We refer to x and
y as the independent variables. We usually define ( , )g x y as a formula and assume
that the domain is the largest set of points in the plane for which the formula is
defined and real-valued.
b) The graph of the function ( , )z g x y , where 2( , )x y R , is the set
3( , , ) | ( , ) and ( , )S x y z z g x y x y R .
Geometrically, if ( , )z g x y measures the vertical distance from the point ( , )x y in
the xy -plane, then z describes the points ( , , ( , ))x y g x y that lie on a surface in space.
Thus, the graph of g is a surface in 3 whose projection onto the xy -plane is the
domain R (see figure below).
c) In an analogous manner, we define a function of three variables as a rule f that
assigns to each triple ( , , )x y z in a set 3D a unique number ( , , )f x y z . We refer
to ( , , )f x y z as a function of three independent variables , ,x y z ; define it as a formula
and assume that its domain is the largest set of points in space 3 for which the
formula is defined and real-valued.
Example 1: Let T denote the temperature and D a metal plate in the xy -plane. The function
( , )T x y then gives the temperature at a point ( , )x y D . If T varies not only on the plate but
also with time ,t then T is a function of three independent variables , , .x y t Similarly, if the
metal plate lies in space, then ( , , , )T x y z t is a function of four independent variables , , , .x y z t
Example 2: A polynomial function ( , )p x y is a function of two independent variables x
and .y It is a sum of the functions of the form m nCx y with nonnegative integers m , n and
C a constant; for instance 5 3 2( , ) 3 7 2 3 11p x y x y x y x y . A rational function is a
quotient of two polynomial functions.
Example 3: Let 2 2( , ) 9 4f x y x y . Describe the domain and range of f .
Solution: The domain of f is the set of all ordered pairs ( , )x y for which f is defined and
real-valued. So, we must have 2 29 4 0x y or equivalently, 2 24 9x y in order that
square root is defined and real. Since a simple closed curve in xy -plane divides the plane into
disjoint regions (inside and outside) with curve itself as the common boundary, and the point
(0,0) satisfies 2 24 9x y , we get that the domain is the set of all points ( , )x y that is inside
or on the ellipse 2 24 9x y . We call it an elliptical region. See figure below.
Operations with functions of two variables:
If ( , )f x y and ( , )g x y are functions of two variables with domain D , then
( )( , ) ( , ) ( , ),
( )( , ) ( , ) ( , ),
( )( , ) ( , ) ( , ),
( , )( , ) , ( , ) 0.
( , )
f g x y f x y g x y
f g x y f x y g x y
f g x y f x y g x y
f f x yx y g x y
g g x y
Trace and level curve of the graph of a function:
Definitions: Consider a function ( , )z f x y of two variables with domain 2R and a
plane z c parallel to xy -plane. The equation ( , )f x y c is a curve, which represents the
intersection of the surface ( , )z f x y with the plane z c .
1) This cross-section of the surface ( , )z f x y in the plane z c is called the trace
or the contour curve of the graph of f in the plane z c .
2) The projection of the trace of f onto the xy -plane is called a level curve of
height c . That is, the set 1 , | , ,f c x y R f x y c where c is a
constant, is a level curve of f .
Remarks:
1. As the constant c varies over the range of ,f a family of level curves is generated. If
c is outside the range of f , then the trace is empty and hence no level curve is
obtained. We usually think of trace as a “slice” of the surface at a particular location
(height) and the level curve is simply the projection of this slice on to the xy -plane.
Note that the level curves lie in the domain of f .
2. Similarly, we obtain the trace of the graph of f in the plane x d (respectively,
y e ) and the corresponding level curve onto the yz -plane (respectively, the level
curve onto the xz -plane) by taking the intersection of the surface ( , )z f x y in the
plane x d (respectively, y e ) and their projection onto yz -plane (respectively,
onto the xz -plane).
3. The level curves are used to draw a 2-dimensional “profile” of the surface
( , ),z f x y such as a mountain range. Such a profile, called topographical or contour
map, is obtained by sketching the family of level curves in the xy -plane and labeling
each curve to show the elevation to which it corresponds.
4. Note that the regions on a topographical map where the level curves are crowded
together correspond to steeper portions of the surface. Isotherms and isobars are the
level curves used to indicate the places of same temperature and pressure,
respectively, on the weather report in the news.
Example 1: Sketch the graph of the function 2 2( , )f x y x y .
Solution: We find the traces of the surface 2 2z x y in planes parallel to coordinate planes.
First, consider its trace in the plane z k : it is 2 2x y k . If 0k this equation has no real
solution, so there is no trace. If 0k then the graph of 2 2x y k is a circle of radius k
centered at the point (0,0, )k on the z -axis (figure (a) below). Thus, for nonnegative values
of k the traces parallel to the xy -plane form a family of circles, centered on the z-axis, whose
radii start at zero and increase with k. This suggests that the surface has the form shown in
figure (b) below:
To obtain more detailed information about the shape of this surface, we can examine the
traces of 2 2z x y in planes parallel to yz -planes. Such planes have equations of the form
x k so that we get 2 2z k y or 2 2y z k . For 0k this equation reduces to 2y z
which is a parabola in the plane 0x that has its vertex at the origin, opens in the positive z
-direction and is symmetric about z -axis (the blue parabola in figure (a)).
Note that the effect of 2k term in 2 2y z k is to translate the parabola 2y z in the
positive z -direction so its new vertex in the plane x k is 2( ,0, )k k . This is the red parabola
in fig (a). Thus, the traces in planes parallel to the yz -plane form a family of parabolas
whose vertices move upward as 2k increases (figure (b)). Similarly, the traces in planes
parallel to the xz -plane ( y k ) have equations of the form 2 2x z k , which again is a
family of parabolas whose vertices move upward as 2k increases (figure (c)). This graph of
2 2,z f x y x y is called a circular paraboloid.
Example 2: Sketch the level curves of the surface 2 2100z x y .
Solution: The domain of f is the entire xy -plane and the range of f is the set of real
numbers less than or equal to 100. It‟s a circular paraboloid opening downwards and vertex at
(0,0,100) . In the plane 75z , the trace or the contour curve of f is the circle ( , ) 75f x y
or 2 2100 75x y . It is the circle in the plane 75z with canter at (0,0,75) and radius 5 .
Therefore, the level curve is the circle 2 2 25x y whose center is at the origin and radius 5.
In the xy -plane ( 0)z , the level curve ( , ) 0f x y or equivalently 2 2 100x y is the circle
with center at origin and radius 10. Similarly, the level curve ( , ) 51f x y is the circle in the
xy -plane of radius 7 and center at the origin: 2 2 2 2100 51 or equivalently 49.x y x y
The level curve 2 2( , ) 100 or equivalently 0f x y x y consists of the origin alone, i.e. it is
a point circle. The graph of the function and these few level curves of the surface 2 2100z x y are shown in the figure below.
Traces of the surface 2 2100z x y in planes parallel to yz and zx planes are shown
below, and the corresponding level curves are parabolas opening downwards and with
vertices on z -axis.
Example 3: Describe the level curves of the function 2 2( , )f x y x y .
Solution: The graph of the function 2 2( , )f x y x y is a hyperbolic paraboloid (saddle
surface) as shown in the figure (a) below:
The level curves have equations of the form 2 2y x k where k is a constant. For 0k
these curves are hyperbolas opening along lines parallel to y -axis; for 0k these curves are
hyperbolas opening along lines parallel to x -axis, and for 0k the level curves consists of
the intersecting lines 0y x and 0y x . See figure (b).
Example 4: Describe the level surfaces of the function 2 2 2( , , )f x y z x y z .
Solution: The value of f is the distance from the origin to the point ( , , )x y z . Each level
surface 2 2 2 , 0x y z c c , is a sphere of radius c centered at the origin. For 0c , the
level surface 2 2 2 0x y z consists of the origin alone. Of course, there is no level curve
for 0.c Thus, the level surfaces of the function are concentric spheres, as shown below.
Example 5: Describe the level surfaces of the function 2 2 2( , , )f x y z z x y .
Solution: The level surfaces have equations of the form 2 2 2z x y c . This equation
represents a cone if 0c , a hyperboloid of two sheets of 0c , and a hyperboloid of one
sheet if 0c (see figure below). For detailed proof, see remarks below.
Remarks:
The level surfaces of a function of three independent variables , ,x y z , namely
2 2 2( , , )f x y z Ax By Cz Dxy Exz Fyz Gx Hy Iz J
where , , , , , , , , ,A B C D E F G H I J are all constants and at least one of , , , , ,A B C D E F is not
zero, are called quadric surfaces or quadrics. Six common types of quadric surfaces are
1) Ellipsoids.
2) Hyperboloids of one sheet.
3) Hyperboloids of two sheets.
4) Elliptic cones.
5) Elliptic paraboloids.
6) Hyperbolic paraboloids
Their standard forms of equations, along with their graphs, are shown in the figures below.
Their traces in the planes parallel to the coordinate planes have also been indicated.
Special Cases:
a. If 0A B C D E F , then ( , , )f x y z Gx Hy Iz J , whose level surface
represents a plane, with direction ratios of the normal to the plane as , ,G H I .
b. In the case of ellipsoid, if 1a b c , then the resulting surface represents a sphere
of radius 1 and centered at the origin.
c. In the case of elliptic cone, if 1a b , then the resulting surface represents a
circular cone with axis as the z -axis and vertex at the origin.
Note that there is no minus sign and constant term is 1.
Note that there is one minus sign and constant term is 1.
Note that there are two minus signs and constant term is 1.
Note that there is no linear term and constant term is 0.
Note that there is one linear term and two quadratic terms with the same
sign. Also, the constant term is 0.
Note that there is one linear term and two quadratic terms with opposite
signs. Also, the constant term is 0.
Limit of a function of two variables:
The definition of the limit of a function of two variables is completely analogous to
the definition for a function of a single variable. First, we define a two dimensional analog to
an open interval and a closed interval on the real line.
Open and closed disks: Using the formula for the distance between two points ( , )x y and
0 0( , )x y in the plane, an open disk D centered at 0 0( , )x y with radius 0 is defined by
2 2 2
0 0( , ) | ( ) ( )D x y x x y y (1)
Thus, an open disk D is the set of all the points in the plane that are enclosed by the circle of
positive radius centered at 0 0( , )x y but do not lie on the circle. We call this open disk D
the -neighborhood of the point 0 0( , ).x y The disk D is said to be a closed disk if the strict
inequality, in the set (1) is replaced by less than or equal to, . Thus, the set of points that
lie on the circle together with those enclosed by the circle is called the closed disk of radius
0 centered at 0 0( , )x y .
Interior and boundary points:
A point 0 0( , )x y is an interior point of a set R in the plane
2 if there is some open disk D
centered at 0 0( , )x y that is completely contained in R . If the set R is empty or if every point
of R is an interior point, then R is called an open subset of 2. A point
0 0( , )x y is called a
boundary point of a set R in the plane 2 if every open disk centered at 0 0( , )x y contains
both points that belong to R and points that do not belong to R . The collection of all
boundary points of R is called the boundary of R . The set R is said to be closed subset of 2 if it contains all of its boundary points. See figures below.
For instance, the open unit disk, its boundary and closed unit disk as subsets of the plane are
illustrated in the figures below.
Note that the empty set and 2 are both open and closed as subsets of
2.
Bounded and unbounded regions: A region (set) in the plane is bounded if it lies inside a
disk of fixed radius. A region is unbounded if it is not bounded. For example, bounded sets
in the plane include line segments, triangles, interiors of triangles, rectangles, circles, and
disks. Examples of unbounded sets in the plane include lines, coordinate axes, the graphs of
functions defined on infinite intervals, quadrants, half-planes, and the plane itself.
Definition: Let f be a function of two variables and assume that f is defined at all points
of some open disk centered at 0 0( , )x y , except possibly at
0 0( , )x y . The limit statement
0 0( , ) ( , )lim ( , )
x y x yf x y L
means that for every given number 0 , there exists a number 0 such that whenever
the distance between ( , )x y and 0 0( , )x y satisfies
2 2
0 00 ( ) ( )x x y y ,
( , )f x y satisfies
( , )f x y L .
Note that the definition says that given any desired degree of closeness 0 , we must be
able to find another number 0 so that all the points lying within a distance of 0 0( , )x y
are mapped by f to the points within distance of L on the real line.
Remarks:
When we consider the lim ( )x c
f x
of a function of one variable, we need to examine the
approach of x to c from two different directions, namely the left hand side and right hand
side of c on the real line. In fact, these two are the only possible directions in which x can
approach c on the real line (corresponding to left-hand limit and right-hand limit). However,
for a function of two variables, we write that 0 0( , ) ( , )x y x y to mean that the point ( , )x y is
allowed to approach 0 0( , )x y along any of the infinitely many different curves or paths in the
plane 2 passing through
0 0( , )x y .
Existence of limit:
The limit of a function of two variables ( , )f x y is said to exist and equal L , as
0 0( , ) ( , )x y x y , written symbolically as
0 0( , ) ( , )lim ( , )
x y x yf x y L
,
if the function ( , )f x y approaches L along every possible path that ( , )x y takes to approach
0 0( , )x y in the plane 2 within the domain of f .
Now, one obviously cannot check each path individually. This gives us a simple method for
determining that a limit does not exist.
Non-existence of limit:
If ( , )f x y approaches 1L as ( , )x y approaches 0 0( , )x y along a path 1P and ( , )f x y
approaches 2 1L L as ( , )x y approaches 0 0( , )x y along a path 2P , then the
0 0( , ) ( , )lim ( , ) .
x y x yf x y does not exist
Our first objective is, therefore, to define the limit of ( , )f x y as 0 0( , ) ( , )x y x y along a
path or a smooth curve C .
Limit along a curve:
Let C be a smooth parametric curve in 2 that is represented by the equations
( ), ( ), [ , ]x x t y y t t a b .
If 0 0 0 0 0( ), ( ) for some [ , ]x x t y y t t a b , then the
0 0( , ) ( , )lim ( , )
x y x yf x y
along the curve C is
defined by
0 0 0( , ) ( , )(along )
lim ( , ) lim ( ( ), ( )). (2)x y x y t t
C
f x y f x t y t
In the right hand side of the formula (2), the limit of the function of t must be treated as a
one sided limit if 0 0( , )x y is an end point of C . A geometric interpretation of the limit along
a curve for a function of two variables is depicted in the figure below.
As the point ( ( ), ( ))x t y t moves along the curve C in the xy -plane towards 0 0( , )x y , the point
( ( ), ( ), ( ( ), ( )))x t y t f x t y t moves directly above (or below) it along the graph of ( , )z f x y
with ( ( ), ( ))f x t y t approaching the limiting value L.
Example 1: Consider a function of two variables defined by the formula
2 2( , ) , ( , ) (0,0)
xyf x y x y
x y
.
Find the limit of ( , )f x y as ( , ) (0,0)x y along the following paths:
(a) the x -axis,
(b) the y -axis,
(c) the line y x ,
(d) the line y x ,
(e) the parabola 2y x .
Solution: (a) The parametric equations of the x -axis are , 0,x t y t , with (0,0)
corresponding to 0t . So, we have
2( , ) (0,0) 0 0 0(along axis)
0lim ( , ) lim ( ,0) lim lim 0 0.
x y t t tx
f x y f tt
(b) The y -axis has parametric equations 0, ,x y t t , with (0,0) corresponding to
0t . So, we have
2( , ) (0,0) 0 0 0(along y axis)
0lim ( , ) lim (0, ) lim lim 0 0.
x y t t tf x y f t
t
(c) The line y x has the parametric equations , , ,x t y t t with (0,0)
corresponding to 0t . Thus, we get
2
2( , ) (0,0) 0 0 0(along y= )
1 1lim ( , ) lim ( , ) lim lim .
2 2 2x y t t tx
tf x y f t t
t
(d) The line y x has the parametric equations , , ,x t y t t with (0,0)
corresponding to 0t . Thus, we get
2
2( , ) (0,0) 0 0 0(along y=- )
1 1lim ( , ) lim ( , ) lim lim .
2 2 2x y t t tx
tf x y f t t
t
(e) The parabola 2y x has parametric equations 2, , ,x t y t t with (0,0)
corresponding to 0t . Thus, we have
2
32
2 4 2( , ) (0,0) 0 0 0
(along y )
lim ( , ) lim ( , ) lim lim 0.1x y t t t
x
t tf x y f t t
t t t
We conclude therefore that the limit of the function 2 2
( , )xy
f x yx y
does not exist. See
figures below for geometrical view of these limits.
At (0,0) , there is a sudden dip (hole) on the surface which supports our conclusion.
Example 2: Using the definition of limit, show that ( , ) ( , )
limx y a b
y b
and ( , ) ( , )
lim .x y a b
x a
Solution: We prove the first limit. The proof for the second limit is similar. Given any
number 0 , we must find another number 0 such that y b whenever
2 20 ( ) ( )x a y b . Note that
2 2 2( ) ( ) ( )x a y b y b y b
so that taking , we have that
2 2 2( ) ( ) ( )y b y b x a y b
whenever 2 20 ( ) ( )x a y b . This proves that ( , ) ( , )
lim .x y a b
y b
Algebra of limits:
Using the definition of limit, we can prove the following results:
1) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
lim ( , ) ( , ) lim ( , ) lim ( , ) .x y a b x y a b x y a b