STUDY MATERIAL FOR B.TECH ,FIRST YEAR ,1ST SEMESTER

PAPER CODE –M-101 TOPICS-FUNCTION OF SEVEREL VARIABLES

LECTURE-1

FUNCTIONS OF SEVEREL VARIABLE

OBJECTIVE:

We shall discuss functions of two or more variables which are useful in many ways,particularly for evaluation of extream values of functions and for evaluation of surface and volume integrals. It will turn out that homogeneous function have a special character demonstrated by Euler’s theorem and its corollaries.Since most variables that we need to know depends not on one but severel variables in physical and social problems,the study of functions of severel variables is amust.

INTRODUCTION:

A variable z is said to be a function of two variables x and y if each pair (x,y) corresponds a value of z . This is expressed by z = f(x,y). for example z = x2+ y2 ,then z is a function of x and y.If z = f(x,y) ,then z is dependent variable and x,y are independent variables.The set of values (x,y) for which a function is defined ,is called the domain of the definition of the function.

LIMIT AND CONTINUITY

Recall that for a function of one variable, the mathematical statement

means that for x close enough to c, the difference between f(x) and L is "small". Very similar definitions exist for functions of two or more variables; however, as you can imagine, if we have a function of two or more independent variables, some complications can arise in the computation and interpretation of limits. Once we have a notion of limits of functions of two variables we can discuss concepts such as continuity and derivatives.

LIMIT

1. Double limit

The following definition and results can be easily generalized to functions of more than two variables. Let f be a function of two variables that is defined in some circular region around (x_0,y_0). The limit of f as x approaches (x_0,y_0) equals L if and only if for every epsilon>0 there exists a delta>0 such that f satisfies

whenever the distance between (x,y) and (x_0,y_0) satisfies

We will of course use the natural notation

when the limit exists. The usual properties of limits hold for functions of two variables: If the following hypotheses hold:

and if c is any real number, then we have the results:

Linearity 1:

Linearity 2:

Products of functions:

Quotients of functions:

(provided L is non-zero)

The linearity and product results can of course be generalized to any finite number of functions:

The limit of a sum of functions is the sum of the limits of the functions.

The limit of a product of functions is the product of the limits of the functions. It is important to remember that the limit of each individual function must exist before any of these results can be applied.

Notes on disproving limits:For a limit to exist, the function must approach that limit for every possible path of (x, y) approaching (a, b). Thus, it is usually very hard to prove a limit exists, and easier to show a limit does not exist. So, if a function f(x, y) approaches L1 as (x, y) approaches (a, b) along a path P1 and f(x, y) approaches L2 L1 as (x, y) approaches (a, b) along a different path P2,

then does not exist.

Some simple paths to try are the lines along x = a, y = b, or any other line through the point.

2. Repeated limit A limit of a function of several variables in which the passage to the limit is performed successively in the different variables. Let, for example, a function of

two variables and be defined on a set of the form , , , and let and be limit points of the sets and , respectively, or the symbol (if or

, and, respectively, may be infinities with signs: , ). If for any fixed the

limit

exists, and for the limit

exists, then this limit is called the repeated limit

of the function at the point . Similarly one defines the repeated limit

If the (finite or infinite) double limit

exists, and if for any fixed the finite limit (1) exists, then the repeated limit (2) also exists, and it is equal to the double limit (4).

If for each the finite limit (1) exists, for each the finite limit

exists, and for the function tends to a limit function uniformly in , then both the repeated limits (2) and (3) exist and are equal to one another.

Continuity

A function f of two variables is continuous at a point (x0,y0) if

1. f(x0,y0) is defined

2. exists

3.

This definition is a direct generalization of the concept of continuity of functions of one variable. The three requirements ensure that f does not oscillate wildly near the point, does not become infinite at the point, or have a jump discontinuity at the point. These are all familiar properties of continuous functions. As with functions of one variable, functions of two or more variables are continuous on an interval if they are continuous at each point in the interval.

Continuous functions of two variables satisfy all of the usual properties familiar from single variable calculus:

The sum of a finite number of continuous functions is a continuous function. The product of a finite number of continuous functions is a continuous function. The quotient of two continuous functions is a continuous function wherever the

denominator is non-zero.

ILLUSTRATIVE EXAMPLES

1.Prove that both the repeated limits exist but double limit does not exit for the function

and

.

Solution: Here . Also

Both the repeated limits exist and are equal.

Now the double limit of f(x,y) as (x,y) (0,0) in any manner.

Let (x,y) (0,0) along X axis (i.e, y=0)

Then

Let (x,y) (0,0) along y=x Then

Thus for two different modes of approach of (x,y) (0,0) the limits are different and

hence the double limit, does not exist.

2. For the function show whether the repeated limits and the

double limit exist and are equal.Examine whether the function continuous at(0,0) or not.

Solution: ( )

Also,

Both the repeated limits exist and are equal.

Now the double of f(x,y) as (x,y) (0,0) will exist if (x,y) (0,0) along any path . Let (x,y) (0,0) along the path x-y = mx3

Then = = =

= which depends on m.

Hence the limiting value is different for different values of m.

does not exist.Consequently ,f(x,y) is not continuous at (0,0).

MULTIPLE CHOICE QUESTIONS:

1. =

a) 0 b) 1 c) d) does not exist.

2.

a) 0 b)1 c) d) none of these

ASSIGNMENT:

1. Show that does not exist.

2. For the function show that both the

repeated limits as well as double limit exist at (0,0) and all equal to 0.

3. Show whether the following functions are continuous or not at the point (0,0)

i. ii.

LECTURE 2

PARTIAL DERIVATIVE

Let f(x,y) be a function with two variables. If we keep y constant and differentiate f (assuming f is differentiable) with respect to the variable x, we obtain what is called the partial derivative of f with respect to x which is denoted by

∂f

∂xor fx

Similarly If we keep x constant and differentiate f (assuming f is differentiable) with respect to the variable y, we obtain what is called the partial derivative of f with respect

to y which is denoted by

∂f

∂yor fy

We might also define partial derivatives of function f as follows:

∂f

∂x=

limh→0

f(x + h , y) - f(x , y)

h

∂f

∂y

=lim

k→0

f(x , y + k) - f(x , y)

k

We now present several examples with detailed solution on how to calculate partial derivatives.

Example 1: Find the partial derivatives fx and fy if f(x , y) is given by

f(x , y) = x2 y + 2x + y

Solution to Example 1:

Assume y is constant and differentiate with respect to x to obtain

fx =∂

∂x=

∂

∂x[ x2 y + 2x + y ]

=∂

∂x[ x2 y] +

∂

∂x[ 2 x ] +

∂

∂x[ y ] = [2 x y] + [ 2 ] + [ 0 ] = 2x y + 2

Now assume x is constant and differentiate with respect to y to obtain

fy =∂f

∂y=

∂

∂y[ x2 y + 2x + y ]

=∂

∂y[ x2 y] +

∂

∂y[ 2 x ] +

∂

∂y[ y ] = [ x2 ] + [ 0 ] + [ 1 ] = x2 + 1

Example 2: Find fx and fy if f(x , y) is given by

f(x , y) = sin(x y) + cos x

Solution to Example 2:

Differentiate with respect to x assuming y is constant

fx =∂f

∂x=

∂

∂x[ sin(x y) + cos x ] = y cos(x y) - sin x

Differentiate with respect to y assuming x is constant

fy =∂f

∂y=

∂

∂y[ sin(x y) + cos x ] = x cos(x y)

Example 3: Find fx and fy if f(x , y) is given by

f(x , y) = x ex y

Solution to Example 3:

Differentiate with respect to x assuming y is constant

fx =∂f

∂x=

∂

∂x[ x ex y ] = ex y + x y ex y = (x y + 1)ex y

Differentiate with respect to y

fy =∂f

∂y=

∂

∂y[ x ex y ] = (x) (x ex y) = x2 ex y

Example 4: Find fx and fy if f(x , y) is given by

f(x , y) = ln ( x2 + 2 y)

Solution to Example 4: Differentiate with respect to x to obtain

fx =∂f

∂x=

∂

∂x[ ln ( x2 + 2 y) ] =

2x

x2 + 2 y

Differentiate with respect to y

fy =∂f

∂y=

∂

∂y[ ln ( x2 + 2 y) ] =

2

x2 + 2 y

Example 5: Find fx(2 , 3) and fy(2 , 3) if f(x , y) is given by

f(x , y) = y x2 + 2 ySolution to Example 5:

We first find fx and fy

fx(x,y) = 2x y

fy(x,y) = x2 + 2

We now calculate fx(2 , 3) and fy(2 , 3) by substituting x and y by their given values

fx(2,3) = 2 (2)(3) = 12

fy(2,3) = 22 + 2 = 6

T O TA L D I F F E R E N T I A L

In this topic we motivate the definition of total differential for a function of two variables. First we recall the derivative and its meaning as the limit of the difference quotient; and then we explain the linear approximation formula for a function of one variable.  Then the total differential is explained and an example is studied in detail by asking the question:  by comparing the values of the total differential and the absolute change for a given two variable function, which is easier to work with?

    Recall, from the calculus of one variable: if where is a differentiable

function, then differential represents the amount that the tangent line rises or falls,

whereas represents the amount that the curve rises or

falls when changes by an amount Since we have

when is small. If we take then we have which says that the actual

change in is approximately equal to the differential If is a known number and

it is desired to calculate an approximate value for where is small, then

yields the approximation,

.We will use the total differential to do the same for functions of two or more variables.

Definition (Total Differential) For a function of two variables , if and are

given increments and , then the corresponding increment of is

The differentials and are independent variables; that is, they can be given any

values. Then the differential , also called the total differential, is defined by

Example (Total Differential) If , find the differential

Further, if changes from to and changes from to , compare the values of

and Which is easier to compute or ?

   

 Solution. By definition,

Putting , , , and , we get

The increment of z is

MULTIPLE CHOICE QUESTIONS:

1. then xux+yuy =

a) b) c) d)

2 If u=log(x2+y2) then the value of ux at (1,1) is

a) b)1 c) 0 d) none of these

3. If u= xy+yz+zx then ux+uy+uz at (1,1,1) is

a)2 b)3 c)6 d)0

ASSIGNMENT:

1 If u = y2+z -2 , v = z2+x -2 , w =x2+y-2 and V is a differentiable function of x,y,z then prove that xVX+yVy+zVz+2(uVu+vVv+wWw)=4(y2Vu+z2Vv+x2Vw)

2 If u=xf(y/x)+g(y/x) ,then show that x +y =xf(y/x)

3.If u= prove that x +y =tanu.

4 If f = ,show that