Functions of Two Variables
Summary and Review
A function of two variables is a mapping from points (x,y)
in the xyplane to numbers z on the zaxis. If
f(x,y) is a function of two variables, then z = f(x,y)
is the graph of f and defines a surface in
3dimensional space. The domain of a functions of two
variables is a set in the xyplane that is open if
it contains none of its boundary points, is closed if it
contains all of its boundary points, is bounded if it
is contained in a circle with a finite radius, is unbounded if
it is not bounded, is connected if a path between any
two points in the set is also in the set, and is not connected otherwise.
If the limit of f( x,y) as ( x,y)
approaches a point ( p,q) is equal to a number L,
then we write

lim
( x,y) ®
( p,q)

f( x,y)
= L 

In order for the limit to exist, the limit of f( x,y)
along any path through ( p,q) must approach the same number
L. Limits allow us to define the first partial derivatives
of f with respect to x and y as
f_{x}( x,y)
= 
lim
h® 0


f( x+h,y)f(
x,y)
h

and f_{y}(
x,y) = 
lim
h® 0


f( x,y+h)
f( x,y)
h



Second partial derivatives are defined to be partial
derivatives of first partial derivatives, and higher derivatives are
similarly defined. Partial differential equations are
equations involving an unknown function and its partial derivatives. In
many instances, it is possible to find a solution to a partial
differential equation using separation of variables.
Limits are also used to define the concept of differentiability
of a function f( x,y) at a point ( p,q)
. If f(x,y) is differentiable at a point ( p,q)
, then it can be approximated by its linearization
L( x,y)
= f( p,q) + f_{x}(p,q)
(xp) + f_{y}(p,q)
( yp) 

for ( x,y) near ( p,q) . It follows that the
graph of L( x,y) is the tangent plane
to z = f( x,y) at the point ( p,q,f(
p,q) ) . The chain rule for functions of two
variables follows from the concept of differentiability, and when
written in vector form, the chain rule also introduces the concept of the
gradient of a function. In particular, the gradient Ñg
of a function g( x,y) is normal to the
level curves of g(x,y) and points in the direction
that g(x,y) is increasing most quickly. In addition,
if f( x,y) is differentiable everywhere, then extrema
of f( x,y) occur at points ( p,q) such
that Ñf( p,q) = á
0,0 ñ . Thus, points where f_{x}(
x,y) = 0 and f_{y}( x,y)
= 0 are called critical points of f. The discriminant
of f is defined
D = f_{xx}f_{yy}(
f_{xy}) ^{2} 

If D( p,q) > 0 at a critical point ( p,q)
of f, then either a maximum or a minimum occurs at ( p,q)
. If D( p,q) < 0, then f has a saddle
at ( p,q) . Finally, extrema of a function f( x,y)
subject to a constraint g( x,y) = k
which is a smooth closed curve must occur at solutions to the equations
where l is called a Lagrange multiplier.
There are other ideas and concepts introduced in this chapter, and you should
reread the individual sections in addition to this summary. The
review materials are based on both the ideas above and some of those in
the chapter not mentioned here. Review questions and solutions are
in web page form on the left and in pdf form on the right. For
maximum benefit, you should attempt to answer the questions before
you look at the solutions.
You will need Acrobat reader in order to open and print the pdf files.
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