262 LIMITS AND DERIVATIVES c = lim n→∞ a n = lim n→∞ b n , and the estimate 0 ≤ c – a n ≤ 1 2 n (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)andf  (x) exist on the interval [a, b] and the inequalities f  (x) ≠ 0 and f  (x) ≠ 0 hold for all x [a, b]. If f  (a)f  (a)>0,thenwetakex 0 = a for the zero approximation; the subsequent approximations are given by the formulas x n+1 = x n – f(x n ) f(b)–f(x n ) (b – x n ), n = 0, 1, If f  (a)f  (a)<0,thenwetakex 0 = b for the zero approximation; the subsequent approximations are given by the formulas x n+1 = x n – f(x n ) f(a)–f(x 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)andc 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 x 1 ;thevaluex n+1 is the abscissa of the point where the line passing through the points (x 0 , f(x 0 )) and (x n , f(x n )) meets the x-axis (see Fig. 6.7 a). xx yy OO ac c x x x x 1 1 2 2 abb fa() ()a ()b fa() fb() yfx=() yfx=() fb() 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)andf  (x) exist on the interval [a, b] and the inequalities f  (x) ≠ 0 and f  (x) ≠ 0 hold for all x [a, b]. 6.3. FUNCTIONS OF SEVERAL VARIABLES.PARTIAL DERIVATIVES 263 If f(a)f  (a)>0,thenwetakex 0 = a for the zero approximation; if f (b)f  (b)>0,then x 0 = b. The subsequent approximations are computed by the formulas x n+1 = x n – f(x n ) f  (x n ) , n = 0, 1, If the initial approximation x 0 is sufficiently close to the desired root c, then the Newton– Raphson method exhibits quadratic convergence: |x n+1 – c| ≤ M 2m |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)=  (x 1A – x 1B ) 2 + ···+(x nA – x nB ) 2 (in n-dimensional space). where x A , y A and x B , y B ,andx A , y A , z A and x B , y B , z B ,andx 1A , , x nA and x 1B , , x nB are Cartesian coordinates of the corresponding points. An ε-neighborhood of a point M 0 (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 , M 0 )<ε, where it is assumed that ε > 0.Anε-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 M 0 K ρ(M, M 0 )<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.Aclosed 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,whereC 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 264 LIMITS AND DERIVATIVES 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.IfD belongs to a three-dimensional space, then one speaks of a function of three variables.ThesetD on which the function is defined is called the domain of the function. For instance, the function z =  1 – x 2 – y 2 is defined on the closed circle x 2 + y 2 ≤ 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 – x 2 – y 2 is 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.Alevel 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 M 0 , i.e., ρ = ρ(M 0 , 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 M 0 if for any (arbitrarily small) ε > 0,thereisδ > 0 such that for all points M belonging to the domain of the function and satisfying the inequality 0 < ρ(M 0 , M)<δ,wehave|f(M)–b| < ε. In this case, one writes lim ρ(M,M 0 )→0 f(M)=b. A function f (M) is called continuous at a point M 0 if lim ρ(M,M 0 )→0 f(M)=f(M 0 ). 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), where Δx, Δy are increments of the independent variables. Partial increments in x and in y are, respectively, Δ x z = f (x + Δx, y)–f (x, y), Δ y z = f (x, y + Δy)–f(x, y). Partial derivatives of a function z in x and in y at a point (x, y)aredefined as follows: ∂z ∂x = lim Δx→0 Δ x z Δx , ∂z ∂y = lim Δy→0 Δ y 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)andf y (x, y). 6.3. FUNCTIONS OF SEVERAL VARIABLES.PARTIAL DERIVATIVES 265 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 Δz = A(x, y)Δx + B(x, y)Δy + o(ρ), ρ =  (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 → 0 as 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)isdefined 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)forsmallΔx and Δy is widely used for approximate calculations, in particular, for fi nding errors in numerical calculations of values of a function. Example 1. Suppose that the values of the arguments of the function z = x 2 y 5 are 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 for Δx = 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 (x 0 , y 0 ), then f(x, y)=f (x 0 , y 0 )+f  x (x 0 , y 0 )(x – x 0 )+f  y (x 0 , y 0 )(y – y 0 )+o(ρ). Hence, for small ρ (i.e., for x ≈ x 0 , y ≈ y 0 ), we obtain the approximate formula f(x, y) ≈ f (x 0 , y 0 )+f  x (x 0 , y 0 )(x – x 0 )+f  y (x 0 , y 0 )(y – y 0 ). 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)andletx = 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)isdefined. 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. 266 LIMITS AND DERIVATIVES 6.3.2-4. Second partial derivatives and second differentials. The second partial derivatives of a function z = f (x, y)aredefined as the derivatives of its first partial derivatives and are denoted as follows: ∂ 2 z ∂x 2 = z xx ≡ (z x ) x , ∂ 2 z ∂x∂y = z xy ≡ (z x ) y , ∂ 2 z ∂y ∂x = z yx ≡ (z y ) x , ∂ 2 z ∂y 2 = 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 d 2 z = 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 d 3 z, d 4 z,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 n inclusively, then its increment Δz at that point can be expressed by Δz = dz + d 2 z 2! + d 3 z 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 (x 0 , y 0 ). Suppose that the derivative F y (x, y) is continuous in a neighborhood of the point (x 0 , y 0 )andF y (x, y) ≠ 0 in that neighborhood. Then the equation F (x, y)=0 defines a continuous function y = y(x) (called an implicit function)ofthevariablex in a neighborhood of the point x 0 . Moreover, if in a neighborhood of (x 0 , y 0 ) 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)=0 that establishes a relation between the variables x, y, z.IfF (x 0 , y 0 , z 0 )=0 and in a neighborhood of the point (x 0 , y 0 , z 0 ) there exist contin- uous partial derivatives F x , F y , F z such that F z (x 0 , y 0 , z 0 ) ≠ 0, then equation F (x, y, z)=0, in a neighborhood of (x 0 , y 0 ), has a unique solution z = ϕ(x, y) such that ϕ(x 0 , y 0 )=z 0 ; 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 = 0 we 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 =– sin y 1 + e z , ∂z ∂y =– x cos y 1 + e z . 6.3. FUNCTIONS OF SEVERAL VARIABLES.PARTIAL DERIVATIVES 267 6.3.2-7. Jacobian. Dependent and independent functions. Invertible transformations. 1 ◦ . Two functions f (x, y)andg(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)andg(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)andg(x, y): ∂(f, g) ∂(x, y) ≡      ∂f ∂x ∂f ∂y ∂g ∂x ∂g ∂y      .(6.3.2.1) 1) If the Jacobian (6.3.2.1) in a domain D is identically equal to zero, then the functions f(x, y)andg(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 (x 1 , x 2 , , x n ), k = 1, 2, , n, are called dependent in a domain D if there is a function Φ(z 1 , z 2 , , z n ) such that Φ  f 1 (x 1 , x 2 , , x n ), f 2 (x 1 , x 2 , , x n ), , f n (x 1 , x 2 , , x n )  = 0 (in D); otherwise, these functions are called independent. The Jacobian is the determinant of the matrix whose elements are the first partial derivatives: ∂(f 1 , f 2 , , f n ) ∂(x 1 , x 2 , , x n ) ≡ det  ∂f i ∂x j  .(6.3.2.2) The functions f k (x 1 , x 2 , , 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 (x 1 , x 2 , , 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 (x 1 , x 2 , , 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 ◦ 1 , x ◦ 2 , , 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 (y 1 , y 2 , , y n ) consisting of the values of the functions (6.3.2.3) in the corresponding neighborhood of the point (y ◦ 1 , y ◦ 2 , , 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 (y 1 , y 2 , , y n ), k = 1, 2, , n, where g k are continuously differentiable functions in the corresponding neighborhood of the point (y ◦ 1 , y ◦ 2 , , 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 268 LIMITS AND DERIVATIVES 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 M 0 (x 0 , y 0 )inthe direction of a vector a = {a 1 , a 2 }, one should construct a straight line passing through M 0 in the direction of the vector a (this line can be specified in terms of the parametric equations x = x 0 +a 1 t, y = y 0 +a 2 t) and study the function z(t)=f(x 0 +a 1 t, y 0 +a 2 t). The derivative of the function z(t) at the point M 0 (i.e., for t = 0) characterizes the change rate of the fi eld at that point in the direction a. Dividing z  (0)by|a| =  a 2 1 + a 2 2 , we obtain the so-called derivative in the direction a of the given field at the given point: ∂f ∂a = 1 |a|  a 1 f  x (x 0 , y 0 )+a 2 f  y (x 0 , y 0 )  . 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 =gradf. 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 (x 0 , y 0 , z 0 ), where z 0 = f (x 0 , y 0 ), has the form z = f (x 0 , y 0 )+f x (x 0 , y 0 )(x – x 0 )+f y (x 0 , y 0 )(y – y 0 ). The vector of the normal to the surface at that point is n =  –f x (x 0 , y 0 ), –f y (x 0 , y 0 ), 1  . If a surface is defined implicitly by the equation Φ(x, y, z)=0, then the equation of its tangent plane at the point (x 0 , y 0 , z 0 )hastheform Φ x (x 0 , y 0 , z 0 )(x – x 0 )+Φ y (x 0 , y 0 , z 0 )(y – y 0 )+Φ z (x 0 , y 0 , z 0 )(z – z 0 )=0. The vector of the normal to the surface at that point is n =  Φ x (x 0 , y 0 , z 0 ), Φ y (x 0 , y 0 , z 0 ), Φ z (x 0 , y 0 , z 0 )  . 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), wherer ={x, y, z},andletM 0  x(u 0 , v 0 ), y(u 0 , v 0 ), z(u 0 , v 0 )  be the point of the surface corresponding to the parameter values u = u 0 , v = v 0 . Then the vector of the normal to the surface at the point M 0 can 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 M 0 . . o(ρ)forsmallΔx and Δy is widely used for approximate calculations, in particular, for fi nding errors in numerical calculations of values of a function. Example 1. Suppose that the values of the. 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. space). where x A , y A and x B , y B ,andx A , y A , z A and x B , y B , z B ,andx 1A , , x nA and x 1B , , x nB are Cartesian coordinates of the corresponding points. An ε-neighborhood of a point M 0 (on