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 (y*_{0}-section)

d. *A Geometric Interpretation (x*_{0}-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) : x*^{2} *+ y*^{2} *< 1}. The set consists*

*of all points which lie inside the unit circle x*^{2} *+ y*^{2} *< 1; the circle itself is not *
*part of the set. To each point (x, y) in D assign the number*

### ( )

^{,}

^{1}

### (

^{1}

^{2}

^{2}

### )

*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 A*_{0} *is made at continuous compounding at interest rate r. Over time*
*t the investment grows to have value*

### (

^{, ,}

### )

_{2}

^{GmM}_{2}

_{2}

*F x y z*

*x* *y* *z*

= + +

### (

^{,}

### )

^{T}*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,*

(∗) *Ax*^{2} *+ By*^{2} *+ Cz*^{2} *+ 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

*Ax*^{2} *+ By*^{2} *+ Cz*^{2} *+ 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 x*^{2} *+ y*^{2} *+ z*^{2} *≤ a*^{2} *+ b*^{2} *+ c*^{2}. 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**

*x*^{2} *= 4cy*

This surface is formed by all lines that pass through
*the parabola x*^{2} *= 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 S*_{1} *: z = f (x, y) and S*_{2} *: 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
*x*^{2} *+ y*^{2} *+ z*^{2} *= c*^{2}*.*

### (

^{, ,}

### )

^{2}

^{2}

^{2}

*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) = 3x*^{2} *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) = 3x*^{2} + 5*πx sin πy.*

*f**y*

## 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 = y*_{0} parallel to
*the xz-plane. The plane y = y*_{0} *intersects the surface in a curve, the y*_{0}-section of the
surface.

## Partial Derivatives

*The number f*_{y}*(x*_{0}*, y*_{0}*) is the slope of the x*_{0}*-section of the surface z = f (x, y) at *
*the point P(x*_{0}*, y*_{0}*, f (x*_{0}*, y*_{0})).

## Partial Derivatives

## Partial Derivatives

*The number f*_{x}*(x*_{0}*, y*_{0}*, z*_{0}*) gives the rate of change of f (x, y*_{0}*, z*_{0}*) with respect to*

*x at x = x*_{0}*; f*_{y}*(x*_{0}*, y*_{0}*, z*_{0}*) gives the rate of change of f (x*_{0}*, y, z*_{0}*) with respect to y at y = y*_{0};
*f*_{z}*(x*_{0}*, y*_{0}*, z*_{0}*) gives the rate of change of f (x*_{0}*, y*_{0}*, z) with respect to z at z = z*_{0}.

**Example**

*The function f (x, y, z) = xy*^{2} *− yz*^{2} has partial derivatives

*f*_{x}*(x, y, z) = y*^{2}*, f*_{y}*(x, y, z) = 2xy − z*^{2}*, 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) = y*^{2} *− 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 − 2z*^{2} *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

*S*_{1} *= {(x, y) : 1 < x < 2, 1 < y < 2},*
*S*_{2} *= {(x, y) : 3 ≤ x ≤ 4, 1 ≤ y ≤ 2},*
*S*_{3} *= {(x, y) : 5 ≤ x ≤ 6, 1 < y < 2}*

*are displayed in Figure 15.5.3. S*_{1} *is the inside of the first square. S*_{1} is open because it
*contains a neighborhood of each of its points. S*_{2} is the inside of the second square
*together with the four bounding line segments. S*_{2} is closed because it contains its
*entire boundary. S*_{3} is the inside of the last square together with the two vertical

*bounding line segments. S*_{3} 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 = x*^{2} *+ y*^{2}*. *
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 x**_{0} *is an interior point of the domain of f. To say that f is *
**continuous at x**_{0} is to say that

**Another way to indicate that f is continuous at x**_{0} 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) = x*^{2} *y + 3x*^{3} *y*^{4} *− x + 2y and Q(x, y, z) = 6x*^{3}*z − yz*^{3} *+ 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 = x*^{2};

### ( )

^{,}

^{2}

_{2}

^{x}

^{y}_{2}

*f x y*

*x* *y*

= −

+

### ( )

^{,}

^{x}^{4}

*g x y*

*x* *y*

= −

### ( )

^{,}

_{2}

^{1}

*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 − x**** _{0}**||

*0*

**< δ, then | f (g(x)) − f (g(x***))| <*∈

*.*

**From the continuity of f at g(x**_{0}*), we know that there exists a > 0 such that*
*if |u ***− g(x****0***)| < , then | f (u** ) − f (g(x*0

*))| <*∈

*.*

**From the continuity of g at x**_{0}, we know that there exists a *δ > 0 such that*
if ||**x ****− x****0**|| * < δ, then |g(x) − g(x*0)|

*< δ*1

*.*

This last *δ obviously works; namely,*

if ||**x − x**** _{0}**||

*0)|*

**< δ, then |g(x) − g(x***< δ*1

*,*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*