Introduction to Functions of Several Variables Philippe B. Laval KSU Today Philippe B. Laval (KSU) Functions of Several Variables Today 1 / 20
Introduction In this section, we extend the definition of a function of one variable to functions of two or more variables. In later sections, we will extend the notions of limits, continuity, differentiation and integration to functions of several variables. Recall that a function is a rule which assigns a unique value to each input value. It is similar for functions of two or more variables. The only difference is that the input is not a number anymore, it is a pair, a triple,... Here is a more formal definition. Philippe B. Laval (KSU) Functions of Several Variables Today 2 / 20
Definitions Definition Let D = {(x, y) x R and y R} be a subset of R 2. 1 A real-valued function f of two variables is a rule which assigns to each ordered pair (x, y) in D a unique real number denoted f (x, y). 2 The set D is called the domain of f. Usually, when defining a function, one must also specify its domain. When the domain is not specified, it is understood to be the largest possible set of input values that is the set of values of x and y for which f (x, y) is defined. 3 The set {f (x, y) (x, y) D} is called the range of f. In other words, the range if the set of output values. 4 Similarly, a real-valued function of three variables is a rule which assigns to each triple (x, y, z) a unique real number denoted f (x, y, z). We can extend this definition to as many variables as we wish. Philippe B. Laval (KSU) Functions of Several Variables Today 3 / 20
s Find the domain of f (x, y) = sin ( x 2 + y 2) x 2 + y 2 Find the domain of g (x, y) = sin x cos y x y Find the domain of h (x, y) = x ln ( y x 2) Philippe B. Laval (KSU) Functions of Several Variables Today 4 / 20
Closed and Bounded Sets In this section, we extend to two and higher dimensions the notion of closed interval. You will recall that a closed interval on the real line is an interval which contains its endpoints. So, [a, b] is a closed interval, but [a, b), (a, b] and (a, b) are not closed. There is a similar notion for subsets of R 2 and R 3. In this section, we will not present this material very thoroughly. This is usually done in an advanced calculus or real analysis class. The intent here is to give the reader an idea of what the notion of closed set in R 2 and R 3 is. Philippe B. Laval (KSU) Functions of Several Variables Today 5 / 20
Closed and Bounded Sets Definition We extend the notion of the end point of an interval to higher dimensions. Such points are called boundary points. 1 Let D be a subset of R 2. A boundary point of D is a point (a, b) such that every disk centered at (a, b) contains both points of D and points not in D. 2 Let D be a subset of R 3. A boundary point of D is a point (a, b, c) such that every sphere centered at (a, b, c) contains both points of D and points not in D. Philippe B. Laval (KSU) Functions of Several Variables Today 6 / 20
Closed and Bounded Sets Definition We give two definitions of an interior point. 1 Let D be a subset of R 2 or R 3. An interior point of D is a point in D which is not on the boundary of D. The set of interior points of a given set is called the interior of that set. 2 Let D be a subset of R 2. A point P of D is an interior point of D if there exists a disk containing P which is included in D. 3 Let D be a subset of R 3. A point P of D is an interior point of D if there exists a sphere containing P which is included in D. Philippe B. Laval (KSU) Functions of Several Variables Today 7 / 20
Closed and Bounded Sets Let us make the following remarks: 1 This agrees with our intuitive definition of a boundary. If you were on the boundary between two countries, stepping on one side would put you in one country, stepping on the other side would put you in the other country. Every disk around you would include parts of both countries. 2 The definition of a boundary point does not require the boundary point of a set be in the set. We will see in the examples the boundary points of a set are not always in the set. In fact, it is a special property a set has when it contains all its boundary points. 3 An interior point of a set is always in the set. 4 An interval on the real line is the equivalent of a disk in the plane and a sphere in space. They represent a region around a point. When we do not specify the dimension, we will use the term ball. Thus a ball can be an interval, a disk or a sphere, depending on which dimension we are in. The term ball is also used in higher dimensions. Philippe B. Laval (KSU) Functions of Several Variables Today 8 / 20
Closed and Bounded Sets What are the boundary and interior of the disk defined by x 2 + y 2 1? Same question for the disk defined by x 2 + y 2 < 1. You will note that the above two sets are different, yet they have the same boundary. The main difference is that the first set contains its boundary, the second does not. This is an important fact to remember. A boundary point of a set does not necessarily belong to the set. Philippe B. Laval (KSU) Functions of Several Variables Today 9 / 20
Closed and Bounded Sets Definition We extend the notion of a closed interval to higher dimensions. Let D be a subset of R 2 or R 3. D is said to be closed if it contains all its boundary points. Definition We extend the notion of an open interval to higher dimensions. Let D be a subset of R 2 or R 3. D is said to be open if every point of D is an interior point of D. Is the disk defined by x 2 + y 2 1 closed, open, neither? Is the disk defined by x 2 + y 2 < 1 closed, open, neither? Philippe B. Laval (KSU) Functions of Several Variables Today 10 / 20
Closed and Bounded Sets Definition Let D be a subset of R 2. D is said to be bounded if it is contained within some disk of finite radius. A subset D of R 3 is said to be bounded if it is contained within some sphere of finite radius. In general, a subset D of R n is said to be bounded if it is contained within a ball of finite radius. Intuitively, this means a bounded set has finite extent. Is the disk defined by x 2 + y 2 1 bounded? Is the set { (x, y) R 2 2 x 2 } bounded? We will see that sets which are both closed and bounded have an important property related to finding extreme values later on in the chapter. Philippe B. Laval (KSU) Functions of Several Variables Today 11 / 20
Graphs of Functions of Two Variables Definition The graph of a function of two variables f (x, y) is the set of points in space {(x, y, z) : (x, y) is in the domain of f and z = f (x, y)}. Like in 2-D, the 3-D graph of a function of two variables is very helpful in the sense that it helps to visualize the behavior of f. The graph of a function of two variables is a surface in space. Unfortunately, graphing a function of two variables is far more diffi cult than a function of one variable. Fortunately for us, we have technology at which facilitates this task. Though we will not spend a lot of time graphing functions of two variables, we will explore some of the issues involved. Philippe B. Laval (KSU) Functions of Several Variables Today 12 / 20
Graphs of Functions of Two Variables We already know some simple 3-D surfaces. For example, we saw that the equation of a plane in space was of the form ax + by + cz + d = 0. If c 0, we can solve for z and rewrite the plane as a function of two variables then sketch its graph. Find the function f (x, y) so that the plane 2x + 3y z + 2 = 0 can be written as z = f (x, y). Sketch its graph using technology. If we cannot solve for z as we did above, we can still graph the corresponding function using an implicit graph. Many graphing programs have the capability of generating implicit graphs. s Graph x 2 4 + y 2 9 z2 16 = 1 Philippe B. Laval (KSU) Functions of Several Variables Today 13 / 20
Graphs of Functions of Two Variables If the graph is fairly simple, finding its intersection with the coordinate planes can be useful to help us visualize it. This is done by: 1 To find the intersection with the xy-plane, set z = 0 in the equation of the plane. 2 To find the intersection with the yz-plane, set x = 0 in the equation of the plane. 3 To find the intersection with the xz-plane, set y = 0 in the equation of the plane. Find the intersection of the plane 2x + 3y z + 2 = 0 with the coordinate axes. Philippe B. Laval (KSU) Functions of Several Variables Today 14 / 20
Graphs of Functions of Two Variables When a surface is more complicated to visualize, we do not limit ourselves to finding how it intersects the coordinate planes. We look how it intersects any plane parallel to one of the coordinate axes. The curves we obtain are called the traces or cross-sections of the surface. Definition The traces or cross-sections of a surface z = f (x, y) are the intersection of that surface with planes parallel to the coordinate planes, that is planes of the form x = C 1, y = C 2, z = C 3 where C 1, C 2 and C 3 are constants. Definition The curves obtained by finding the intersection of a surface z = f (x, y) with planes parallel to the xy-plane are also called contour curves. The projection of these curves onto the xy-plane are called level curves. A plot made of the contour curve is called a contour plot. Philippe B. Laval (KSU) Functions of Several Variables Today 15 / 20
Graphs of Functions of Two Variables The level curves are curves where the z value is constant. Level curves are uses for example in mapping, to indicate the altitude. The altitude is the same everywhere on a level curve. The next figure shows a topographic map of Kennesaw Mountain, one can clearly see the level curves indicating where the mountain is. The following figure shows a 3-D rendering of the same area. On weather maps, level curves represent isobars, that is areas where the atmospheric pressure is the same. Philippe B. Laval (KSU) Functions of Several Variables Today 16 / 20
Graphs of Functions of Two Variables Figure: Topographic map of Kennesaw Mountain Philippe B. Laval (KSU) Functions of Several Variables Today 17 / 20
Graphs of Functions of Two Variables Figure: 3-D map of Kennesaw Mountain Philippe B. Laval (KSU) Functions of Several Variables Today 18 / 20
Graphs of Functions of Two Variables Consider the surface z = f (x, y) = x 2 + y 2. Find its level curves and also the curves at which it intersects planes parallel to the xz-planes and the yz-planes. Same question for z = g (x, y) = 2x 2 + 3y 2. Philippe B. Laval (KSU) Functions of Several Variables Today 19 / 20
Exercises See the problems at the end of my notes on definitions and examples of functions of two or more variables. Review the notions of limits and continuity from Calculus I Philippe B. Laval (KSU) Functions of Several Variables Today 20 / 20