Functions of Two Variables

Summary and Review

A function of two variables is a mapping from points (x,y) in the xy-plane to numbers z on the z-axis.  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 3-dimensional space. The domain of a functions of two variables is a set in the xy-plane 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

fx( x,y) = lim h® 0   f( x+h,y)-f( x,y) h         and        fy( 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)  +  fx(p,q) (x-p) + fy(p,q) ( y-p)

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 = fxxfyy-( fxy) 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

Ñf = lÑg

where l is called a Lagrange multiplier.

There are other ideas and concepts introduced in this chapter, and you should re-read 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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