Handbook of mathematics for engineers and scienteists part 43 ppt

An ε-neighborhood of a point M0on the plane or in space is the set consisting of all points M resp., on the plane or in space such that ρM , M0 < ε, where it is assumed that ε >0.. An ε-

c= lim

n→∞ a n= limn→∞ b n, and the estimate

0 ≤c – a n≤ 1

2n (b – a)

is valid

The following two methods are more efficient

6.2.7-3 Regula falsi method (false position method)

Suppose that the derivatives f  (x) and f  (x) exist on the interval [a, b] and the inequalities

f  (x)≠ 0and f  (x)≠ 0hold for all x[a, b].

If f  (a)f  (a) > 0, then we take x0 = a for the zero approximation; the subsequent

approximations are given by the formulas

x n+1= x n– f (b) – f (x f (x n)

n)(b – x n), n=0, 1,

If f  (a)f  (a) < 0, then we take x0 = b for the zero approximation; the subsequent

approximations are given by the formulas

x n+1= x n– f (a) – f (x f (x n)

n)(a – x n), n=0, 1,

The regula falsi method has the first order of local convergence as n → ∞:

|x n+1– c| ≤ k|x n – c|, where k is a constant depending on f (x) and c is the root of equation (6.2.7.1).

The regula falsi method has a simple geometric interpretation The straight line (secant)

passing through the points (a, f (a)) and (b, f (b)) of the curve y = f (x) meets the abscissa axis at the point x1; the value x n+1is the abscissa of the point where the line passing through

the points (x0, f (x0)) and (x n , f (x n )) meets the x-axis (see Fig 6.7 a).

a

f a( )

f a( )

f b( )

y= ( )f x y= ( )f x

f b( )

Figure 6.7 Graphical construction of successive approximations to the root of equation (6.2.7.1) by the regula

falsi method (a) and the Newton–Raphson method (b).

6.2.7-4 Newton–Raphson method

Suppose that the derivatives f  (x) and f  (x) exist on the interval [a, b] and the inequalities

f  (x)≠ 0and f  (x)≠ 0hold for all x [a, b].

If f (a)f  (a) >0, then we take x0= a for the zero approximation; if f (b)f  (b) >0, then

x0= b The subsequent approximations are computed by the formulas

x n+1= x n– f f  (x (x n)

n), n=0, 1,

If the initial approximation x0is sufficiently close to the desired root c, then the Newton–

Raphson method exhibits quadratic convergence:

|x n+1– c| ≤ 2M

m |x n – c|2,

where M = max

a x b|f  (x)| and m = min

a x b|f  (x)|.

The Newton–Raphson method has a simple geometric interpretation The tangent to the

curve y = f (x) through the point (x n , f (x n )) meets the abscissa axis at the point x n+1(see

Fig 6.7 b).

The Newton–Raphson method has a higher order of convergence than the regula falsi method Hence the former is more often used in practice

6.3 Functions of Several Variables Partial Derivatives 6.3.1 Point Sets Functions Limits and Continuity

6.3.1-1 Sets on the plane and in space

The distance between two points A and B on the plane and in space can be defined as

follows:

ρ (A, B) =

(x A – x B)2+ (y A – y B)2 (on the plane),

ρ (A, B) =

(x A – x B)2+ (y A – y B)2+ (z A – z B)2 (in three-dimensional space),

ρ (A, B) =

(x1A – x1B)2+· · · + (x nA – x nB)2 (in n-dimensional space) where x A , y A and x B , y B , and x A , y A , z A and x B , y B , z B , and x1A , , x nA and

x1B , , x nBare Cartesian coordinates of the corresponding points.

An ε-neighborhood of a point M0(on the plane or in space) is the set consisting of all

points M (resp., on the plane or in space) such that ρ(M , M0) < ε, where it is assumed that ε >0 An ε-neighborhood of a set K (on the plane or in space) is the set consisting

of all points M (resp., on the plane or in space) such that inf

M0 K ρ (M , M0) < ε, where it is assumed that ε >0

An interior point of a set D is a point belonging to D, together with some neighborhood

of that point An open set is a set containing only interior points A boundary point of a set D is a point such that any of its neighborhoods contains points outside D A closed set

is a set containing all its boundary points A set D is called a bounded set if ρ(A, B) < C for any points A, B D , where C is a constant independent of A, B Otherwise (i.e., if there is no such constant), the set D is called unbounded.

6.3.1-2 Functions of two or three variables

A (numerical) function on a set D is, by definition, a relation that sets up a correspondence between each point M D and a unique numerical value If D is a plane set, then each

point M D is determined by two coordinates x, y, and a function z = f (M ) = f (x, y)

is called a function of two variables If D belongs to a three-dimensional space, then one speaks of a function of three variables The set D on which the function is defined is called the domain of the function For instance, the function z =

1– x2– y2 is defined on the

closed circle x2+ y2 ≤ 1, which is its domain

The graph of a function z = f (x, y) is the surface formed by the points (x, y, f (x, y)) in three-dimensional space For instance, the graph of the function z = ax + by + c is a plane, and the graph of the function z =

1– x2– y2is a half-sphere

A level line of a function z = f (x, y) is a line on the plane x, y with the following property: the function takes one and the same value z = c at all points of that line Thus, the equation of a level line has the form f (x, y) = c A level surface of a function u = f (x, y, z)

is a surface on which the function takes a constant value, u = c; the equation of a level surface has the form f (x, y, z) = c.

A function f (M ) is called bounded on a set D if there is a constant C such that

|f (M )| ≤ C for all M D

6.3.1-3 Limit of a function at a point and its continuity

Let M be a point that comes infinitely close to some point M0, i.e., ρ = ρ(M0, M ) →0 It

is possible that the values f (M ) come close to some constant b.

One says that b is the limit of the function f (M ) at the point M0if for any (arbitrarily

small) ε >0, there is δ >0such that for all points M belonging to the domain of the function

and satisfying the inequality0< ρ(M0, M ) < δ, we have|f (M ) – b| < ε In this case, one

ρ(M,M0→0f (M ) = b.

A function f (M ) is called continuous at a point M0 if lim

ρ(M,M0 →0f (M ) = f (M0) A

function is called continuous on a set D if it is continuous at each point of D Any continuous function f (M ) on a closed bounded set is bounded on that set and attains its

smallest and its largest values on that set

6.3.2 Differentiation of Functions of Several Variables

For the sake of brevity, we consider the case of a function of two variables However, all

statements can be easily extended to the case of n variables.

6.3.2-1 Total and partial increments of a function Partial derivatives

A total increment of a function z = f (x, y) at a point (x, y) is

Δz = f(x + Δx, y + Δy) – f(x, y),

whereΔx, Δy are increments of the independent variables Partial increments in x and in

yare, respectively,

Δx z = f (x + Δx, y) – f(x, y),

Δy z = f (x, y + Δy) – f(x, y).

Partial derivatives of a function z in x and in y at a point (x, y) are defined as follows:

∂z

∂x = lim

Δx→0

Δx z

∂z

∂y = lim

Δy→0

Δy z

Δy

(provided that these limits exist) Partial derivatives are also denoted by z x and z y , ∂ x z

and ∂ y z , or f x (x, y) and f y (x, y).

6.3.2-2 Differentiable functions Differential.

A function z = f (x, y) is called differentiable at a point (x, y) if its increment at that point

can be represented in the form

(Δx)2+ (Δy)2,

where o(ρ) is a quantity of a higher order of smallness compared with ρ as ρ → 0 (i.e.,

o (ρ)/ρ → 0as ρ →0) In this case, there exist partial derivatives at the point (x, y), and

z 

x = A(x, y), z y  = B(x, y).

A function that has continuous partial derivatives at a point (x, y) is differentiable at that

point

The differential of a function z = f (x, y) is defined as follows:

dz = f x  (x, y) Δx + f y  (x, y) Δy.

Taking the differentials dx and dy of the independent variables equal to Δx and Δy,

respectively, one can also write dz = f x  (x, y) dx + f y  (x, y) dy.

The relation Δz = dz + o(ρ) for small Δx and Δy is widely used for approximate

calculations, in particular, for finding errors in numerical calculations of values of a function

Example 1 Suppose that the values of the arguments of the function z = x2y5are known with the error

x= 2 0 01, y =1 0 01 Let us calculate the approximate value of the function.

We find the increment of the function z at the point x =2, y =1 forΔx = Δy =0 01 , using the formula

Δz≈dz= 2 ⋅ 2 ⋅ 1 5 ⋅ 0 01 + 5 ⋅ 2 2 ⋅ 1 4 ⋅ 0 01 = 0 24 Therefore, we can accept the approximation z =4 0 24

If a function z = f (x, y) is differentiable at a point (x0, y0), then

f (x, y) = f (x0, y0) + f x  (x0, y0)(x – x0) + f y  (x0, y0)(y – y0) + o(ρ).

Hence, for small ρ (i.e., for x≈x0, y ≈y0), we obtain the approximate formula

f (x, y)≈f (x0, y0) + f x  (x0, y0)(x – x0) + f y  (x0, y0)(y – y0)

The replacement of a function by this linear expression near a given point is called

lin-earization.

6.3.2-3 Composite function

Consider a function z = f (x, y) and let x = x(u, v), y = y(u, v) Suppose that for (u, v)D,

the functions x(u, v), y(u, v) take values for which the function z = f (x, y) is defined In this way, one defines a composite function on the set D, namely, z(u, v) = f x (u, v), y(u, v)

In this situation, f (x, y) is called the outer function and x(u, v), y(u, v) are called the inner

functions

Partial derivatives of a composite function are expressed by

∂z

∂u = ∂f

∂x

∂x

∂u + ∂f

∂y

∂y

∂u,

∂z

∂v = ∂f

∂x

∂x

∂v + ∂f

∂y

∂y

∂v

For z = z(t, x, y), let x = x(t), y = y(t) Thus, z is actually a function of only one variable t The derivative dz dt is calculated by

dz

dt = ∂z

∂t + ∂z

∂x

dx

dt + ∂z

∂y

dy

dt This derivative, in contrast to the partial derivative ∂z ∂t , is called a total derivative.

6.3.2-4 Second partial derivatives and second differentials.

The second partial derivatives of a function z = f (x, y) are defined as the derivatives of its

first partial derivatives and are denoted as follows:

∂2z

∂x2 = z xx≡(z x)x, ∂

2z

∂x ∂y = z xy ≡(z x)y,

∂2z

∂y ∂x = z yx≡(z y)x, ∂

2z

∂y2 = z yy ≡(z y)y

The derivatives z xy and z yx are called mixed derivatives If the mixed derivatives are continuous at some point, then they coincide at that point, z xy = z yx

In a similar way, one defines higher-order partial derivatives

The second differential of a function z = f (x, y) is the expression

d2z = d(dz) = (dz)

x Δx + (dz) y Δy = z xx(Δx)2+2z xy ΔxΔy + z yy(Δy)2

In a similar way, one defines d3z , d4z, etc

6.3.2-5 Taylor’s formula

If at some point (x, y) the function z = f (x, y) possesses partial derivatives up to the order

ninclusively, then its incrementΔz at that point can be expressed by

Δz = dz + d2!2z + d

3z

3! +· · · +

d n z

n! + o(ρ

n), where ρ =

(Δx)2+ (Δy)2.

6.3.2-6 Implicit functions and their differentiation

Consider the equation F (x, y) = 0 with a solution (x0, y0) Suppose that the derivative

F y (x, y) is continuous in a neighborhood of the point (x0, y0) and F y (x, y) ≠ 0 in that

neighborhood Then the equation F (x, y) =0defines a continuous function y = y(x) (called

an implicit function) of the variable x in a neighborhood of the point x0 Moreover, if in a

neighborhood of (x0, y0) there exists a continuous derivative F x, then the implicit function

y = y(x) has a continuous derivative expressed by dy

dx = –F x

F y. Consider the equation F (x, y, z) = 0that establishes a relation between the variables

x , y, z If F (x0, y0, z0) =0and in a neighborhood of the point (x0, y0, z0) there exist

contin-uous partial derivatives F x , F y , F z such that F z (x0, y0, z0)≠ 0, then equation F (x, y, z) =0,

in a neighborhood of (x0, y0), has a unique solution z = ϕ(x, y) such that ϕ(x0, y0) = z0;

moreover, the function z = ϕ(x, y) is continuous and has continuous partial derivatives

expressed by

∂z

∂x = –F x

F z,

∂z

∂y = –F y

F z.

Example 2 For the equation x sin y + z + e z= 0we have F z= 1+ e z≠ 0 Therefore, this equation defines

a function z = ϕ(x, y) on the entire plane, and its derivatives have the form ∂z

∂x = – 1sin y + e z,

∂z

∂y = –x1cos y + e z .

6.3.2-7 Jacobian Dependent and independent functions Invertible transformations.

1◦ Two functions f (x, y) and g(x, y) are called dependent if there is a function Φ(z) such

that g(x, y) = Φ(f(x, y)); otherwise, the functions f(x, y) and g(x, y) are called independent.

The Jacobian is the determinant of the matrix whose elements are the first partial derivatives of the functions f (x, y) and g(x, y):

∂ (f , g)

∂ (x, y) ≡





∂f

∂x ∂f ∂y

∂g

∂x ∂g ∂y





1) If the Jacobian (6.3.2.1) in a domain D is identically equal to zero, then the functions

f (x, y) and g(x, y) are dependent in D.

2) If the Jacobian (6.3.2.1) is separated from zero in D, then the functions f (x, y) and

g (x, y) are independent in D.

2◦ Functions f

k (x1, x2, , x n ), k = 1,2, , n, are called dependent in a domain D if there is a functionΦ(z1, z2, , z n) such that

Φ f1(x1, x2, , x n ), f2(x1, x2, , x n ), , f n (x1, x2, , x n)

otherwise, these functions are called independent

The Jacobian is the determinant of the matrix whose elements are the first partial

derivatives:

∂ (f1, f2, , f n)

∂ (x1, x2, , x n) ≡det



∂f i

∂x j



The functions f k (x1, x2, , x n ) are dependent in a domain D if the Jacobian (6.3.2.2) is identically equal to zero in D The functions f k (x1, x2, , x n ) are independent in D if the Jacobian (6.3.2.2) does not vanish in D.

3◦ Consider the transformation

y k = f k (x1, x2, , x n), k=1,2, , n (6.3.2.3)

Suppose that the functions f k are continuously differentiable and the Jacobian (6.3.2.2)

differs from zero at the point (x ◦ , x ◦ , , x ◦ n) Then, in a sufficiently small neighborhood of this point, equations (6.3.2.3) specify a one-to-one correspondence between the points of that

neighborhood and the set of points (y1, y2, , y n) consisting of the values of the functions

(6.3.2.3) in the corresponding neighborhood of the point (y ◦1, y2◦ , , y ◦ n) This means that

the system (6.3.2.3) is locally solvable in a neighborhood of the point (x ◦1, x ◦2, , x ◦ n), i.e., the following representation holds:

x k = g k (y1, y2, , y n), k=1,2, , n,

where g k are continuously differentiable functions in the corresponding neighborhood of

the point (y ◦ , y ◦ , , y ◦ n)

6.3.3 Directional Derivative Gradient Geometrical Applications

6.3.3-1 Directional derivative

One says that a scalar field is defined in a domain D if any point M (x, y) of that domain

is associated with a certain value z = f (M ) = f (x, y) Thus, a thermal field and a pressure

field are examples of scalar fields A level line of a scalar field is a level line of the function

that specifies the field (see Subsection 6.3.1) Thus, isothermal and isobaric curves are, respectively, level lines of thermal and pressure fields

In order to examine the behavior of a field z = f (x, y) at a point M0(x0, y0) in the

direction of a vector a ={a1, a2}, one should construct a straight line passing through M0in

the direction of the vector a (this line can be specified in terms of the parametric equations

x = x0+ a1t , y = y0+ a2t ) and study the function z(t) = f (x0+ a1t , y0+ a2t) The derivative

of the function z(t) at the point M0(i.e., for t =0) characterizes the change rate of the field

at that point in the direction a Dividing z (0) by|a|=

a2

1+ a22, we obtain the so-called

derivative in the direction a of the given field at the given point:

∂f

∂a = 1

|a|



a1f 

x (x0, y0) + a2f y  (x0, y0)



The gradient of the scalar field z = f (x, y) is, by definition, the vector-valued function

grad f = f x  (x, y)i + f y  (x, y)j, where i and j are unit vectors along the coordinate axes x and y At each point, the

gradient of a scalar field is orthogonal to the level line passing through that point The gradient indicates the direction of maximal growth of the field In terms of the gradient, the directional derivative can be expressed as follows:

∂f

∂a = a

|a| grad f The gradient is also denoted by ∇f = grad f

Remark The above facts for a plane scalar field obviously can be extended to the case of a spatial scalar field.

6.3.3-2 Geometrical applications of the theory of functions of several variables

The equation of the tangent plane to the surface z = f (x, y) at a point (x0, y0, z0), where

z0 = f (x0, y0), has the form

z = f (x0, y0) + f x (x0, y0)(x – x0) + f y (x0, y0)(y – y0)

The vector of the normal to the surface at that point is



n=5

–f x (x0, y0), –f y (x0, y0), 16

If a surface is defined implicitly by the equationΦ(x, y, z) =0, then the equation of its

tangent plane at the point (x0, y0, z0) has the form

Φx (x0, y0, z0)(x – x0) +Φy (x0, y0, z0)(y – y0) +Φz (x0, y0, z0)(z – z0) =0

The vector of the normal to the surface at that point is



n=5

Φx (x0, y0, z0), Φy (x0, y0, z0), Φz (x0, y0, z0)6

Consider a surface defined by the parametric equations

x = x(u, v), y = y(u, v), z = z(u, v)

or, in vector form, r =r(u, v), wherer ={x , y, z}, and let M0 x (u0, v0), y(u0, v0), z(u0, v0)

be the point of the surface corresponding to the parameter values u = u0, v = v0 Then the

vector of the normal to the surface at the point M0can be expressed by



n (u, v) = ∂r

∂u × ∂r

∂v =







i j k

x u y u z u

x v y v z v





,

where all partial derivatives are calculated at the point M0

calculations, in particular, for finding errors in numerical calculations of values of a function

Example... Second partial derivatives and second differentials.

The second partial derivatives of a function z = f (x, y) are defined as the derivatives of its

first partial derivatives and. .. graph of a function z = f (x, y) is the surface formed by the points (x, y, f (x, y)) in three-dimensional space For instance, the graph of the function z = ax + by + c is a plane, and the graph of