Vector Calculus by Michael Corral - HTML preview

(1)(2)

2

lim

=

=

( x, y)→(1,2) x 2 + y 2

12 + 22

5

since f ( x, y) = xy is properly defined at the point (1,2).

x 2+ y 2

The major difference between limits in one variable and limits in two or more variables

has to do with how a point is approached. In the single-variable case, the statement “x → a”

means that x gets closer to the value a from two possible directions along the real number

line (see Figure 2.1.2(a)). In two dimensions, however, ( x, y) can approach a point ( a, b) along an infinite number of paths (see Figure 2.1.2(b)).

y

x

x

( a, b)

x

0

a

x

0

(a) x → a in R

(b) ( x, y) → ( a, b) in R2

Figure 2.1.2

“Approaching” a point in different dimensions

68

CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES

Example 2.7.

x y

lim

does not exist

( x, y)→(0,0) x 2 + y 2

Note that we can not simply substitute ( x, y) = (0,0) into the function, since doing so gives an

indeterminate form 0/0. To show that the limit does not exist, we will show that the function

approaches different values as ( x, y) approaches (0, 0) along different paths in R2. To see this,

suppose that ( x, y) → (0,0) along the positive x-axis, so that y = 0 along that path. Then

x y

x 0

f ( x, y) =

=

= 0

x 2 + y 2

x 2 + 02

along that path (since x > 0 in the denominator). But if ( x, y) → (0,0) along the straight line

y = x through the origin, for x > 0, then we see that

x y

x 2

1

f ( x, y) =

=

=

x 2 + y 2

x 2 + x 2

2

which means that f ( x, y) approaches different values as ( x, y) → (0,0) along different paths.

Hence the limit does not exist.

Limits of real-valued multivariable functions obey the same algebraic rules as in the

single-variable case, as shown in the following theorem, which we state without proof.

Theorem 2.1. Suppose that

lim

f ( x, y) and

lim

g( x, y) both exist, and that k is

( x, y)→( a, b)

( x, y)→( a, b)

some scalar. Then:

(a)

lim

[ f ( x, y) ± g( x, y)] =

lim

f ( x, y) ±

lim

g( x, y)

( x, y)→( a, b)

( x, y)→( a, b)

( x, y)→( a, b)

(b)

lim

k f ( x, y) = k

lim

f ( x, y)

( x, y)→( a, b)

( x, y)→( a, b)

(c)

lim

[ f ( x, y) g( x, y)] =

lim

f ( x, y)

lim

g( x, y)

( x, y)→( a, b)

( x, y)→( a, b)

( x, y)→( a, b)

lim

f ( x, y)

f ( x, y)

( x, y)→( a, b)

(d)

lim

=

if

lim

g( x, y) = 0

( x, y)→( a, b) g( x, y)

lim

g( x, y)

( x, y)→( a, b)

( x, y)→( a, b)

(e) If | f ( x, y) − L| ≤ g( x, y) for all ( x, y) and if

lim

g( x, y) = 0, then

lim

f ( x, y) = L.

( x, y)→( a, b)

( x, y)→( a, b)

Note that in part (e), it suffices to have | f ( x, y)− L| ≤ g( x, y) for all ( x, y) “sufficiently close”

to ( a, b) (but excluding ( a, b) itself).

2.1 Functions of Two or Three Variables

69

Example 2.8. Show that

y 4

lim

= 0.

( x, y)→(0,0) x 2 + y 2

Since substituting ( x, y) = (0,0) into the function gives the indeterminate form 0/0, we need

an alternate method for evaluating this limit. We will use Theorem 2.1(e). First, notice that

y 4 =

y 2 4 and so 0 ≤ y 4 ≤

x 2 + y 2 4 for all ( x, y). But

x 2 + y 2 4 = ( x 2 + y 2)2. Thus, for

all ( x, y) = (0,0) we have

y 4

( x 2 + y 2)2

≤

= x 2 + y 2 → 0 as ( x, y) → (0,0).

x 2 + y 2

x 2 + y 2

y 4

Therefore

lim

= 0.

( x, y)→(0,0) x 2 + y 2

Continuity can be defined similarly as in the single-variable case.

Definition 2.2. A real-valued function f ( x, y) with domain D in R2 is continuous at the

point ( a, b) in D if

lim

f ( x, y) = f ( a, b). We say that f ( x, y) is a continuous function if

( x, y)→( a, b)

it is continuous at every point in its domain D.

Unless indicated otherwise, you can assume that all the functions we deal with are con-

tinuous. In fact, we can modify the function from Example 2.8 so that it is continuous on all

of R2.

Example 2.9. Define a function f ( x, y) on all of R2 as follows:



0

if ( x, y)



= (0,0)

f ( x, y) =

y 4





if ( x, y) = (0,0)

x 2 + y 2

Then f ( x, y) is well-defined for all ( x, y) in R2 (i.e. there are no indeterminate forms for any

( x, y)), and we see that

b 4

lim

f ( x, y) =

= f ( a, b) for ( a, b) = (0,0).

( x, y)→( a, b)

a 2 + b 2

So since

lim

f ( x, y) = 0 = f (0,0) by Example 2.8,

( x, y)→(0,0)

then f ( x, y) is continuous on all of R2.

70

CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES

Exercises

A

For Exercises 1-6, state the domain and range of the given function.

1

1. f ( x, y) = x 2 + y 2 − 1

2. f ( x, y) = x 2 + y 2

x 2 + 1

3. f ( x, y) =

x 2 + y 2 − 4

4. f ( x, y) =

y

5. f ( x, y, z) = sin( xyz)

6. f ( x, y, z) =

( x − 1)( yz − 1)

For Exercises 7-18, evaluate the given limit.

7.

lim

cos( x y)

8.

lim

exy

( x, y)→(0,0)

( x, y)→(0,0)

x 2 − y 2

x y 2

9.

lim

10.

lim

( x, y)→(0,0) x 2 + y 2

( x, y)→(0,0) x 2 + y 4

x 2 − 2 xy + y 2

x y 2

11.

lim

12.

lim

( x, y)→(1,−1)

x − y

( x, y)→(0,0) x 2 + y 2

x 2 − y 2

x 2 − 2 xy + y 2

13.

lim

14.

lim

( x, y)→(1,1)

x − y

( x, y)→(0,0)

x − y

y 4 sin( x y)

1

15.

lim

16.

lim

( x 2 + y 2)cos

( x, y)→(0,0)

x 2 + y 2

( x, y)→(0,0)

x y

x

17.

lim

1

18.

lim

cos

( x, y)→(0,0) y

( x, y)→(0,0)

x y

B

19. Show that f ( x, y) = 1 e−( x 2+ y 2)/2 σ 2, for σ > 0, is constant on the circle of radius r > 0

2 πσ 2

centered at the origin. This function is called a Gaussian blur, and is used as a filter in

image processing software to produce a “blurred” effect.

20. Suppose that f ( x, y) ≤ f ( y, x) for all ( x, y) in R2. Show that f ( x, y) = f ( y, x) for all ( x, y) in R2.

21. Use the substitution r =

x 2 + y 2 to show that

sin

x 2 + y 2

lim

= 1 .

( x, y)→(0,0)

x 2 + y 2

( Hint: You will need to use L’Hôpital’s Rule for single-variable limits. )

C

22. Prove Theorem 2.1(a) in the case of addition. ( Hint: Use Definition 2.1. )

23. Prove Theorem 2.1(b).

2.2 Partial Derivatives

71

2.2 Partial Derivatives

Now that we have an idea of what functions of several variables are, and what a limit of

such a function is, we can start to develop an idea of a derivative of a function of two or more

variables. We will start with the notion of a partial derivative.

Definition 2.3. Let f ( x, y) be a real-valued function with domain D in R2, and let ( a, b) be

a point in D. Then the partial derivative of f at ( a, b) with respect to x, denoted by

∂ f ( a, b), is defined as

∂x

∂ f

f ( a + h, b) − f ( a, b)

( a, b) = lim

(2.2)

∂x

h→0

h

∂ f

and the partial derivative of f at ( a, b) with respect to y, denoted by

( a, b), is defined

∂y

as

∂ f

f ( a, b + h) − f ( a, b)

( a, b) = lim

.

(2.3)

∂y

h→0

h

Note: The symbol ∂ is pronounced “del” .1

Recall that the derivative of a function f ( x) can be interpreted as the rate of change of

that function in the (positive) x direction. From the definitions above, we can see that the

partial derivative of a function f ( x, y) with respect to x or y is the rate of change of f ( x, y) in the (positive) x or y direction, respectively. What this means is that the partial derivative of

a function f ( x, y) with respect to x can be calculated by treating the y variable as a constant,

and then simply differentiating f ( x, y) as if it were a function of x alone, using the usual

rules from single-variable calculus. Likewise, the partial derivative of f ( x, y) with respect to

y is obtained by treating the x variable as a constant and then differentiating f ( x, y) as if it

were a function of y alone.

∂ f

∂ f

Example 2.10. Find

( x, y) and

( x, y) for the function f ( x, y) = x 2 y + y 3.

∂x

∂y

Solution: Treating y as a constant and differentiating f ( x, y) with respect to x gives

∂ f ( x, y) =2 xy

∂x

and treating x as a constant and differentiating f ( x, y) with respect to y gives

∂ f ( x, y) = x 2+3 y 2 .

∂y

1It is not a Greek letter. The symbol was first used by the mathematicians A. Clairaut and L. Euler around

1740, to distinguish it from the letter d used for the “usual” derivative.

72

CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES

∂ f

∂ f

∂ f

∂ f

We will often simply write

and

instead of

( x, y) and

( x, y).

∂x

∂y

∂x

∂y

∂ f

∂ f

sin( x y 2)

Example 2.11. Find

and

for the function f ( x,