Chapter 15: Functions of Several Variables

Section 15.1 Elementary Examples

a. Notation: Two Variables

b. Example

c. Notation: Three Variables

d. Functions of Several Variables

e. Examples from the Sciences

Section 15.2 A Brief Catalogue of the Quadric Surfaces

a. Quadric Surfaces

b. Type of Surfaces

c. The Ellipsoid

d. The Hyperboloid of One Sheet

e. The Hyperboloid of Two Sheets

f. The Elliptic Cone

g. The Elliptic Paraboloid

h. The Hyperbolic Paraboloid

i. The Parabolic Cylinder

j. The Elliptic Cylinder

k. The Hyperbolic Cylinder

l. Projections

Section 15.3 Graphs; Level Curves and Level Surfaces

a. Level Curves

b. Computer-Generated Graphs

c. Level Surfaces

Chapter 15: Functions of Several Variables

Section 15.4 Partial Derivatives

a. Functions of Two Variables

b. Partial Derivatives (Two Variables)

c. A Geometric Interpretation (y₀-section)

d. A Geometric Interpretation (x₀-section)

e. Partial Derivatives (Three Variables)

f. Example

Section 15.5 Open and Closed Sets

a. Neighborhood of a Point

b. The Interior of a Set

c. The Boundary of a Set

d. Open and Closed Sets

e. Two-Dimensional Example

f. Three-Dimensional Example

Section 15.6 Limits and Continuity

a. The Limit of a Function of Several Variables

b. Continuity

c. Examples of Continuous Functions

d. The Continuity of Composite Functions

e. Continuity in Each Variable Separately

f. Derivatives of Higher Order

g. Partial Derivatives and Continuity

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.

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

( )

(

)

f x y

x y

= − +

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.

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.

2 2

x + y

2 2 2

x + y + z

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

(

)

F x y z

x y z

= + +

(

)

P T V k

= V

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.)

A Brief Catalogue of the Quadric Surfaces

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.

A Brief Catalogue of the Quadric Surfaces

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

2 2 2

2 2 2 1

x y z

a + b + c =

2 2

2 2 1

x y

a + b =

A Brief Catalogue of the Quadric Surfaces

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

2 2 2

2 2 2 1

x y z

a + b − c =

2 2

2 2 1

x y

a + b =

A Brief Catalogue of the Quadric Surfaces

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

2 2 2

2 2 2 1

x y z

a + b − c = −

2 2 2

2 2 2 1

x y z

a + b = c − The equation requires

2

2 2

2 1 0, ,

z z c z c

c − ≥ ≥ ≥

Each of the two parts is unbounded.

A Brief Catalogue of the Quadric Surfaces

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.

2 2

2

2 2

x y

a + b = z

A Brief Catalogue of the Quadric Surfaces

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.

2 2

2 2

x y

a + b = z

A Brief Catalogue of the Quadric Surfaces

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.

The Hyperbolic Paraboloid

2 2

2 2

x y

a − b = z

A Brief Catalogue of the Quadric Surfaces

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.

A Brief Catalogue of the Quadric Surfaces

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.

2 2

2 2 1

x y

a + b =

2 2

2 2 1

x y

a + b =

A Brief Catalogue of the Quadric Surfaces

2 2

2 2 1

x y

a − b = The Hyperbolic Cylinder

The surface has two parts, each generated by a branch of the hyperbola

2 2

2 2 1

x y

a − b =

A Brief Catalogue of the Quadric Surfaces

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

Graphs; Level Curves and Level Surfaces

Level Curves

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.

Graphs; Level Curves and Level Surfaces

Computer-Generated Graphs

Graphs; Level Curves and Level Surfaces

Level Surfaces

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.

(

)

g x y z = x + y + z

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 f_x obtained by differentiating f with respect to x, keeping y fixed. In this case

f_x (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

f_y (x, y) = 3x² + 5πx sin πy.

fy

Partial Derivatives

Partial Derivatives

A Geometric Interpretation

In Figure 15.4.1 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.

Partial Derivatives

The number f_y (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)).

Partial Derivatives

Partial Derivatives

The number f_x (x₀, y₀, z₀) gives the rate of change of f (x, y₀, z₀) with respect to

x at x = x₀; f_y (x₀, y₀, z₀) gives the rate of change of f (x₀, y, z₀) with respect to y at y = y₀; f_z(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

f_x (x, y, z) = y², f_y (x, y, z) = 2xy − z², f_z(x, y, z) = −2yz.

The number f_x (1, 2, 3) = 4 gives the rate of change with respect to x of the function f (x, 2, 3) = 4x − 18 at x = 1;

f_y(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.

f_z(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.

Open and Closed Sets

Open and Closed Sets

Open and Closed Sets

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.

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 15.5.3. 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.

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).

( )

{ }

( )

{ }

( )

2 2

1

2 2

2 2

3

, , : , , :

, , :1

S x y z z x y

S x y z z x y

x y

S x y z

z

2

= > +

= ≥ +

 + 

=  ≥ 

 

Limits and Continuity

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.

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;

( )

f x y

x y

= −

+

( )

g x y

x y

= −

( )

h x y

x y

= −

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))| < ∈. δ1

δ1

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

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.

( ) ( )

( ) ( )

0 0

0 0

,

lim

, , ,

x y x y

f x y f x y

→

=

( ) ( )

0

0 0 0

lim , ,

x x

f x y f x y

→

= ( ) ( )

0

0 0 0

lim , ,

y y

f x y f x y

→

=

and then

Limits and Continuity

Derivatives of Higher Order; Equality of Mixed Partials

Suppose that f is a function of x and y with first partials f_x and f_y . These are again functions of x and y and may themselves possess partial derivatives:

( f_x )_x , ( f_x )_y , ( f_y )_x , ( f_y )_y . These functions are called the second-order partials.

( ) ( )

( ) ( )

2 2

2 2

2 2

2 2

2 2

2 2

xx x x

xy x y

yx y x

yy y y

f f z

f f

x x x x

f f z

f f

y x y x y x

f f z

f f

x y x y x y

f f z

f f

y y y y

∂ ∂  ∂ ∂

= = ∂  ∂  = ∂ = ∂

∂ ∂  ∂ ∂

= = ∂  ∂ = ∂ ∂ = ∂ ∂

 

∂ ∂ ∂ ∂

= = ∂ ∂ = ∂ ∂ = ∂ ∂

 

∂ ∂ ∂ ∂

= = ∂  ∂  = ∂ = ∂

If z = f (x, y), we use the following notations for second-order partials

Limits and Continuity