Chapter 15: Functions of Several Variables

Chapter 15: Functions of Several Variables
Section 15.1 Elementary Examples Notation: Two Variables Example Notation: Three Variables Functions of Several Variables Examples from the Sciences Section 15.2 A Brief Catalogue of the Quadric Surfaces Quadric Surfaces Type of Surfaces The Ellipsoid The Hyperboloid of One Sheet The Hyperboloid of Two Sheets The Elliptic Cone The Elliptic Paraboloid The Hyperbolic Paraboloid The Parabolic Cylinder The Elliptic Cylinder The Hyperbolic Cylinder Projections Section 15.3 Graphs; Level Curves and Level Surfaces Level Curves Computer-Generated Graphs Level Surfaces Section 15.4 Partial Derivatives Functions of Two Variables Partial Derivatives (Two Variables) A Geometric Interpretation (y0-section) A Geometric Interpretation (x0-section) Partial Derivatives (Three Variables) Example Section 15.5 Open and Closed Sets Neighborhood of a Point The Interior of a Set The Boundary of a Set Open and Closed Sets Two-Dimensional Example Three-Dimensional Example Section 15.6 Limits and Continuity The Limit of a Function of Several Variables Continuity Examples of Continuous Functions The Continuity of Composite Functions Continuity in Each Variable Separately Derivatives of Higher Order Partial Derivatives and Continuity Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Elementary Examples Notation
Points P(x, y) of the xy-plane will be written (x, y) and points P(x, y, z) of three-space will be written (x, y, z). Let D be a nonempty subset of the xy-plane. A function f that assigns a real number f (x, y) to each point in D is called a real-valued function of two variables. The set D is called the domain of f, and the set of all values f (x, y) is called the range of f. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Elementary Examples Example
Take D as the open unit disk: D = {(x, y) : x2 + y2 < 1}. The set consists of all points which lie inside the unit circle x2 + y2 < 1; the circle itself is not part of the set. To each point (x, y) in D assign the number Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Elementary Examples Notation for Three-Space
Let D be a nonempty subset of three-space. A function f that assigns a real number f (x, y, z) to each point (x, y, z) in D is called a real-valued function of three variables. The set D is called the domain of f, and the set of all values f (x, y, z) is called the range of f. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Elementary Examples Functions of several variables arise naturally in very elementary settings. f (x, y) = gives the distance between (x, y) and the origin; f (x, y) = xy gives the area of a rectangle of dimensions x, y; and f (x, y) = 2(x + y) gives the perimeter. f (x, y, z) = gives the distance between (x, y, z) and the origin; f (x, y, z) = xyz gives the volume of a rectangular solid of dimensions x, y, z; f (x, y, z) = 2(xy + xz + yz) gives the total surface area. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Elementary Examples A mass M exerts a gravitational force on a mass m. According to the law of universal gravitation, if M is located at the origin of our coordinate system and m is located at (x, y, z), then the magnitude of the gravitational force is given by the function (3 variables: x, y, z) where G is the universal gravitational constant. According to the ideal gas law, the pressure P of a gas enclosed in a container varies directly with the temperature T of the gas and varies inversely with the volume V of the container. Thus P is given by a function of the form (2 variables: T, V) An investment A0 is made at continuous compounding at interest rate r. Over time t the investment grows to have value A(t, r ) = A0ert (2 variables: r, t) Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

A Brief Catalogue of the Quadric Surfaces
The curves in the xy-plane defined by equations in x and y of the second degree are the conic sections: circle, ellipse, parabola, hyperbola. The surfaces in three-dimensional space defined by equations in x, y, z of the second degree, (∗) Ax2 + By2 + Cz2 + Dxy + Exz + Fyz + Hx + I y + Jz + K = 0, are called the quadric surfaces. Equation (∗) contains terms in xy, xz, yz. These terms can be eliminated by a suitable change of coordinates. Thus, for our purposes, the quadric surfaces are given by equations of the form Ax2 + By2 + Cz2 + Dx + Ey + Fz + H = 0 with A, B,C not all zero. (If A, B,C are all zero, we don’t have an equation of the second degree.) Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The quadric surfaces can be viewed as the three-space analogs of the conic sections. They fall into nine distinct types. 1. The ellipsoid. 2. The hyperboloid of one sheet. 3. The hyperboloid of two sheets. 4. The elliptic cone. 5. The elliptic paraboloid. 6. The hyperbolic paraboloid. 7. The parabolic cylinder. 8. The elliptic cylinder. 9. The hyperbolic cylinder. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Ellipsoid The ellipsoid is centered at the origin and is symmetric about the three coordinate axes. It intersects the coordinate axes at six points: (±a, 0, 0), (0,±b, 0), (0, 0,±c). These points are called the vertices. The surface is bounded, being contained in the ball x2 + y2 + z2 ≤ a2 + b2 + c2. All three traces are ellipses; thus, for example, the trace in the xy-plane (the set z = 0) is the ellipse Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Hyperboloid of One Sheet The surface is unbounded. It is centered at the origin and is symmetric about the three coordinate planes. The surface intersects the coordinate axes at four points: (±a, 0, 0), (0,±b, 0). The trace in the xy-plane (set z = 0) is the ellipse Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Hyperboloid of Two Sheets The surface intersects the coordinate axes only at the two vertices (0, 0,±c). The surface consists of two parts: one for which z ≥ c, another for which z ≤ −c. We can see this by rewriting the equation as The equation requires Each of the two parts is unbounded. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Elliptic Cone The surface intersects the coordinate axes only at the origin. The surface is unbounded. Once again there is symmetry about the three coordinate planes. The trace in the xz plane is a pair of intersecting lines: z = ±x/a. The trace in the yz-plane is also a pair of intersecting lines: z = ±y/b. The trace in the xy-plane is just the origin. Sections parallel to the xy-plane are ellipses. If a = b, these sections are circles and we have a surface of revolution, what is commonly called a double circular cone or simply a cone. The upper and lower portions of the cone are called nappes. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Elliptic Paraboloid The surface does not extend below the xy-plane; it is unbounded above. The origin is called the vertex. Sections parallel to the xy-plane are ellipses; sections parallel to the other coordinate planes are parabolas. Hence the term “elliptic paraboloid.” The surface is symmetric about the xz-plane and about the yz-plane. It is also symmetric about the z-axis. If a = b, then the surface is a paraboloid of revolution. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Hyperbolic Paraboloid Here there is symmetry about the xz-plane and yz-plane. Sections parallel to the xy plane are hyperbolas; sections parallel to the other coordinate planes are parabolas. Hence the term “hyperbolic paraboloid.” The origin is a minimum point for the trace in the xz-plane but a maximum point for the trace in the yz-plane. The origin is called a minimax or saddle point of the surface. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Take any plane curve C. All the lines through C that are perpendicular to the plane of C form a surface. Such a surface is called a cylinder, the cylinder with base curve C. The perpendicular lines are known as the generators of the cylinder. The Parabolic Cylinder x2 = 4cy This surface is formed by all lines that pass through the parabola x2 = 4cy and are perpendicular to the xy-plane. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Elliptic Cylinder The surface is formed by all lines that pass through the ellipse and are perpendicular to the xy-plane. If a = b, we have the common right circular cylinder. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Hyperbolic Cylinder The surface has two parts, each generated by a branch of the hyperbola Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Projections Suppose that S1 : z = f (x, y) and S2 : z = g(x, y) are surfaces in three-space that intersect in a space curve C. The curve C is the set of all points (x, y, z) with z = f (x, y) and z = g(x, y). The set of all points (x, y, z) with f (x, y) = g(x, y) (Here z is unrestricted.) is the vertical cylinder that passes through C. The set of all points (x, y, 0) with f (x, y) = g(x, y) (Here z = 0.) is called the projection of C onto the xy-plane. In Figure it appears as the curve in the xy-plane that lies directly below C. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Graphs; Level Curves and Level Surfaces
Suppose that f is a nonconstant function defined on some portion of the xy-plane. If c is a value in the range of f, then we can sketch the curve f (x, y) = c. Such a curve is called a level curve for f. It can be obtained by intersecting the graph of f with the horizontal plane z = c and then projecting that intersection onto the xy-plane. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

One can try to visualize the behavior of a function of three variables, w = f (x, y, z), by examining the level surfaces of f. These are the subsets of the domain of f with equations of the form f (x, y, z) = c where c is a value in the range of f. Example For the function f (x, y, z) = Ax + By + Cz, the level surfaces are parallel planes Ax + By + Cz = c. Example For the function , the level surfaces are concentric spheres x2 + y2 + z2 = c2. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Partial Derivatives Functions of Two Variables
Let f be a function of x and y; take for example f (x, y) = 3x2 y − 5x cos πy. The partial derivative of f with respect to x is the function fx obtained by differentiating f with respect to x, keeping y fixed. In this case fx (x, y) = 6xy − 5 cos πy. The partial derivative of f with respect to y is the function obtained by differentiating f with respect to y, keeping x fixed. In this case fy (x, y) = 3x2 + 5πx sin πy. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Partial Derivatives A Geometric Interpretation
In Figure we have sketched a surface z = f (x, y) which you can take as everywhere defined. Through the surface we have passed a plane y = y0 parallel to the xz-plane. The plane y = y0 intersects the surface in a curve, the y0-section of the surface. The number fx (x0, y0) is thus the slope of the y0-section of the surface z = f (x, y) at the point P(x0, y0, f (x0, y0)). Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Partial Derivatives The number fy (x0, y0) is the slope of the x0-section of the surface z = f (x, y) at the point P(x0, y0, f (x0, y0)). Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Partial Derivatives The number fx (x0, y0, z0) gives the rate of change of f (x, y0, z0) with respect to x at x = x0; fy (x0, y0, z0) gives the rate of change of f (x0, y, z0) with respect to y at y = y0; fz(x0, y0, z0) gives the rate of change of f (x0, y0, z) with respect to z at z = z0. Example The function f (x, y, z) = xy2 − yz2 has partial derivatives fx (x, y, z) = y2, fy (x, y, z) = 2xy − z2, fz(x, y, z) = −2yz. The number fx (1, 2, 3) = 4 gives the rate of change with respect to x of the function f (x, 2, 3) = 4x − at x = 1; fy(1, 2, 3) = −5 gives the rate of change with respect to y of the function f (1, y, 3) = y2 − 9y at y = 2. fz(1, 2, 3) = −12 gives the rate of change with respect to z of the function f (1, 2, z) = 4 − 2z2 at z = 3. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Open and Closed Sets Thus
(1) A set S is open provided that each of its points is an interior point. (2) A set S is open provided that it contains no boundary points. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Open and Closed Sets Two-Dimensional Examples The sets
S1 = {(x, y) : 1 < x < 2, 1 < y < 2}, S2 = {(x, y) : 3 ≤ x ≤ 4, 1 ≤ y ≤ 2}, S3 = {(x, y) : 5 ≤ x ≤ 6, 1 < y < 2} are displayed in Figure S1 is the inside of the first square. S1 is open because it contains a neighborhood of each of its points. S2 is the inside of the second square together with the four bounding line segments. S2 is closed because it contains its entire boundary. S3 is the inside of the last square together with the two vertical bounding line segments. S3 is not open because it contains part of its boundary, and it is not closed because it does not contain all of its boundary. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Open and Closed Sets Three-Dimensional Examples
We now examine some three-dimensional sets: The boundary of each of these sets is the paraboloid of revolution z = x2 + y2. The first set consists of all points above this surface. This set is open because, if a point is above this surface, then all points sufficiently close to it are also above this surface. Thus the set contains a neighborhood of each of its points. The second set is closed because it contains all of its boundary. The third set is neither open nor closed. It is not open because it contains some boundary points; for example, it contains the point (1, 1, 2). It is not closed because it fails to contain the boundary point (0, 0, 0). Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Limits and Continuity Suppose now that x0 is an interior point of the domain of f. To say that f is continuous at x0 is to say that Another way to indicate that f is continuous at x0 is to write To say that f is continuous on an open set S is to say that f is continuous at all points of S. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Limits and Continuity Some Examples of Continuous Functions
Polynomials in several variables, for example, P(x, y) = x2 y + 3x3 y4 − x + 2y and Q(x, y, z) = 6x3z − yz3 + 2xyz are everywhere continuous. In the two-variable case, that means continuity at each point of the xy-plane, and in the three-variable case, continuity at each point of three-space. Rational functions (quotients of polynomials) are continuous everywhere except where the denominator is zero. Thus is continuous at each point of the xy-plane other than the origin (0, 0); is continuous except on the line y = x; is continuous except on the parabola y = x2; is continuous at each point of three-space other than the origin (0, 0, 0) Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Limits and Continuity PROOF We begin with  > 0. We must show that there exists a δ > 0 such that if ||x − x0|| < δ, then | f (g(x)) − f (g(x0))| < . From the continuity of f at g(x0), we know that there exists a > 0 such that if |u − g(x0)| < , then | f (u) − f (g(x0))| < . From the continuity of g at x0, we know that there exists a δ > 0 such that if ||x − x0|| < δ, then |g(x) − g(x0)| < δ1. This last δ obviously works; namely, if ||x − x0|| < δ, then |g(x) − g(x0)| < δ1, and therefore | f (g(x)) − f (g(x0))| < . Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Limits and Continuity Continuity in Each Variable Separately
A continuous function of several variables is continuous in each of its variables separately. In the two-variable case, this means that, if then and The converse is false. It is possible for a function to be continuous in each variable separately and yet fail to be continuous as a function of several variables. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Limits and Continuity Derivatives of Higher Order; Equality of Mixed Partials Suppose that f is a function of x and y with first partials fx and fy . These are again functions of x and y and may themselves possess partial derivatives: ( fx )x , ( fx )y , ( fy )x , ( fy )y . These functions are called the second-order partials. If z = f (x, y), we use the following notations for second-order partials Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Chapter 15: Functions of Several Variables

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