Essentials of Mathematical Methods in Science and Engineering

By Selçuk S. Bayin

A complete introduction to the multidisciplinary applications of mathematical methods

In order to work with varying levels of engineering and physics research, it is important to have a firm understanding of key mathematical concepts such as advanced calculus, differential equations, complex analysis, and introductory mathematical physics. Essentials of Mathematical Methods in Science and Engineering provides a comprehensive introduction to these methods under one cover, outlining basic mathematical skills while also encouraging students and practitioners to develop new, interdisciplinary approaches to their research.

The book begins with core topics from various branches of mathematics such as limits, integrals, and inverse functions. Subsequent chapters delve into the analytical tools that are commonly used in scientific and engineering studies, including vector analysis, generalized coordinates, determinants and matrices, linear algebra, complex numbers, complex analysis, and Fourier series. The author provides an extensive chapter on probability theory with applications to statistical mechanics and thermodynamics that complements the following chapter on information theory, which contains coverage of Shannon's theory, decision theory, game theory, and quantum information theory. A comprehensive list of references facilitates further exploration of these topics.

Throughout the book, numerous examples and exercises reinforce the presented concepts and techniques. In addition, the book is in a modular format, so each chapter covers its subject thoroughly and can be read independently. This structure affords flexibility for individualizing courses and teaching.

Providing a solid foundation and overview of the various mathematical methods and applications in multidisciplinary research, Essentials of Mathematical Methods in Science and Engineering is an excellent text for courses in physics, science, mathematics, and engineering at the upper-undergraduate and graduate levels. It also serves as a useful reference for scientists and engineers who would like a practical review of mathematical methods.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Jun 5, 2013
• ISBN: 9781118626160

Book preview

Ş.S.B.

CHAPTER 1

FUNCTIONAL ANALYSIS

A function is basically a rule that relates the members of one set of objects to the members of another set. In this regard, it has a very wide range of applications in both science and mathematics. Functional analysis is basically the branch of mathematics that deals with the functions of numbers. In this chapter, we confine ourselves to the real domain and introduce some of the most commonly used techniques in functional analysis.

1.1 CONCEPT OF FUNCTION

We start with a quick review of the basic concepts of set theory. Let S be a set of objects of any kind: points, numbers, functions, vectors, etc. When s is an element of the set S, we show it as

For finite sets we may define S by listing its elements as

For infinite sets, S is usually defined by a phrase describing the condition to be a member of the set, for example,

When there is no room for confusion, we may also write an infinite set as

When each member of a set A is also a member of set B, we say that A is a subset of B and write

The phrase B covers or contains A is also used. The union of two sets,

consists of the elements of both A and B. The intersection of two sets, A and B, is defined as

When two sets have no common element, their intersection is called the null set or the empty set, which is usually shown by ϕ. The neighborhood of a point, (x1, y1), in the xy-plane is the set of all points, (x, y), inside a circle centered at (x1, y1) and with the radius δ:

An open set is defined as the set of points with neighborhoods entirely within the set. The interior of a circle defined by

is an open set. A boundary point is a point whose every neighborhood contains at least one point in the set and at least one point that does not belong to the set. The boundary of the set in Equation (1.9) is the set of points on the circumference, that is,

An open set plus its boundary is a closed set.

A function, f, is in general a rule, a relation that uniquely associates members of one set, A, with the members of another set, B. The concept of function is essentially the same as that of mapping, which in general is so broad that it allows mathematicians to work with them without any resemblance to the simple class of functions with numerical values. The set A that f acts upon is called the domain, and the set B composed of the elements that f can produce is called the range. For single-valued functions the common notation used is

Here f stands for the function or mapping that acts upon a single number x, which is an element of the domain, and produces f(x), which is an element of the range. In general, f refers to the function itself and f(x) refers to the value it returns. However, in practice, f(x) is also used to refer to the function itself. In this chapter we basically concern ourselves with functions that take numerical values as f(x), where the argument, x, is called the independent variable. We usually define a new variable, y, as

which is called the dependent variable.

Functions with multiple variables, that is, multivariate functions, can also be defined. For example, for each point (x, y) in some region of the xy-plane we may assign a unique real number, f(x, y), according to the rule

We now say that f(x, y) is a function with two independent variables, x and y. In applications, f(x, y) may represent physical properties like the temperature or the density distribution of a flat disc with negligible thickness. Definition of function can be extended to cases with several independent variables as

where n stands for the number of independent variables.

The term function is also used for the objects that associate more than one element in the domain to a single element in the range. Such objects are called multiple-to-one relations. For example,

Sometimes the term function is also used for relations that map a single point in its domain to multiple points in its range. As we shall discuss in Chapters 7 and 8, such functions are called multivalued functions, which are predominantly encountered in complex analysis.

1.2 CONTINUITY AND LIMITS

Similar to its usage in everyday language, the word continuity in mathematics also implies the absence of abrupt changes. In astrophysics, pressure and density distributions inside a solid neutron star are represented by continuous functions of the radial position: P(r) and ρ(r), respectively. This means that small changes in the radial position inside the star also result in small changes in the pressure and density. At the surface, r = R, where the star meets the outside vacuum, pressure has to be continuous. Otherwise, there will be a net force on the surface layer, which will violate the static equilibrium condition. In this regard, in static neutron star models pressure has to be a monotonic decreasing function of r, which smoothly drops to zero at the surface:

On the other hand, the density at the surface can change abruptly from a finite value to zero. This is also in line with our everyday experiences, where solid objects have sharp contours marked by density discontinuities. For gaseous stars, both pressure and density have to vanish continuously at the surface. In constructing physical models, deciding on which parameters are going to be taken as continuous at the boundaries requires physical reasoning and some insight. Usually, a collection of rules that have to be obeyed at the boundaries are called the junction conditions or the boundary conditions.

We are now ready to give a formal definition of continuity as follows:

Continuity: A numerically valued function f(x) defined in some domain D, is said to be continuous at the point x0 Є D if, for any positive number є > 0, there is a neighborhood N about x0 such that |f(x) – f(x0)| < є for every point common to both N and D, that is N ∩ D. If the function f(x) is continuous at every point of D, we say it is continuous in D.

We finally quote two theorems, proofs of which can be found in books on advanced calculus:

Theorem1.1. Let f(x) be a continuous function at x and let {xn} be a sequence of points in the domain of f(x) with the limit

then the following is true:

Theorem 1.2. For a function f(x) defined in D, if the limit

exists whenever xn Є D and

then the function f(x) is continuous at x. For the limit in Equation (1.18) to exist, it is sufficient to show that the right and the left limits agree, that is,

In practice, the second theorem is more useful in showing that a given function is continuous. If a function is discontinuous at a finite number of points in its interval of definition, [xa, xb], it is called piecewise continuous.

Generalization of these theorems to multivariate functions is easily accomplished by taking x to represent a point in a space with n independent variables as

However, with more than one independent variable one has to be careful. Consider the simple function

which is finite at the origin. Depending on the direction of approach to the origin, f(x, y) takes different values:

Hence the limit lim (x, y) → (0,0) f(x, y) does not exist and the function f(x, y) is not continuous at the origin.

Limits: Basic properties of limits, which we give for functions with two variables, also hold for a general multivariate function:

Let u = f(x, y) and υ = g(x, y) be two functions defined in the domain D of the xy-plane. If the limits

(1.24)

exist, then we can write

If the functions f(x, y) and g(x, y) are continuous at (x0, y0), then the functions

are also continuous at (x0, y0), provided that in the last case g(x, y) is different from zero at (x0, y0).

Let F(u, υ) be a continuous function defined in some domain D0 of the uv-plane and let F(f(x, y), g(x, y)) be defined for (x, y) in D. Then, if (f0, g0) is in D0, we can write

If f(x, y) and g(x, y) are continuous at (x0, y0), then so is F(f(x, y), g(x, y)).

In evaluating limits of functions that can be expressed as ratios, L’Hôpital’s rule is very useful.

L’Hôpital’s rule: Let f and g be differentiable functions on the interval a ≤ x < b with g’(x) ≠ 0 there, where the upper limit b could be finite or infinite. If f and g have the limits

or

and if the limit

exists, where L could be zero or infinity, then

1.3 PARTIAL DIFFERENTIATION

A necessary and sufficient condition for the derivative of f(x) to exist at x(x(x0), derivatives exist and be equal (Fig. 1.1), that is,

where

Figure 1.1 Derivative is the slop of the tangent line.

When the derivative exists, we always mean a finite derivative. If f(x) has derivative at x0, it means that it is continuous at that point. When the derivative of f(x) exists at every point in the interval (a, b), we say that f(x) is differentiable in (a, b) and write its derivative as

Geometrically, derivative at a point is the slope of the tangent line at that point:

When a function depends upon two variables:

the partial derivative with respect to x at (x0, y0) is defined as the limit

and we show it as in one of the following forms:

Similarly, the partial derivative with respect to y at (x0, y0) is defined as

A geometric interpretation of the partial derivative is that the section of the surface z = f(x, y) with the plane y = y0 is the curve z = f(x, yis the slope of the tangent line (Fig. 1.2) to z = f(x, y0) at (x0, yis the slope of the tangent line to the curve z = f(x0, y) at (x0, y0). For a multivariate function the partial derivative with respect to the ith independent variable is defined as

(1.43)

For a given f(x,y) the partial derivatives fx and fy are functions of x and y and they also have partial derivatives which are written as

(1.45)

When fxy and fyx are continuous at (x0, y0), then the relation

holds at (x0, y0). Under similar conditions this result can be extended to cases with more than two independent variables and to higher-order mixed partial derivatives.

1.4 TOTAL DIFFERENTIAL

When a function depends on two or more variables, we have seen that the limit at a point may depend on the direction of approach. Hence, it is important that we introduce a nondirectional derivative for functions with several variables. Given the function

for a displacement of Δr = (Δx, Δy, Δz) we can write its new value as

(1.50)

Figure 1.2 Partial derivative, fx, is the slope of the tangent line to z = f(x, y0).

where r stands for the point (x, y, z) and Δr is the displacement (Δx, Δy, Δz). For small. Δr the change in f(x, y, z) to first order can be written as

(1.51)

Considering that the first-order partial derivatives of f are given as

Equation (1.52) is nothing but

In general, if a function f(x,y,z) is differentiable at (x,y,z) in some domain D with the partial derivatives

then the change in f(x, y, z) in D to first order in (Δx, Δy, Δz) can be written as

Figure 1.3 Total differential gives a local approximation to the change in a function.

In the limit as Δr → 0 we can write Equation (1.56) as

which is called the total differential of f(x, y, z). In the case of a function with one variable, f(x), the differential reduces to

which gives the local approximation to the change in the function at the point x via the value of the tangent line (Fig. 1.3) at that point. The smaller the value of Δx, the better the approximation. In cases with several independent variables, Δf is naturally approximated by using the tangent plane at that point.

1.5 TAYLOR SERIES

The Taylor series of a function about x0, when it exists, is given as

To evaluate the coefficients, we differentiate repeatedly and set x = x0 to find

where

and the zeroth derivative is defined as the function itself, that is,

Hence the Taylor series of a function with a single variable is written as

This formula assumes that f(x) is infinitely differentiable in an open domain including x0. Functions that are equal to their Taylor series in the neighborhood of any point x0 in their domain are called analytic functions. Taylor series about x0 = 0 are called Maclaurin series.

Using the Taylor series, we can approximate a given differentiable function in the neighborhood of x0 to orders beyond the linear term in Equation (1.58). For example, to second order we obtain

(1.65)

(1.66)

Since x0 is any point in the open domain that the Taylor series exists, we can drop the subscript in x0 and write

where Δ(2)f denotes the differential of f to the second order. Higher-order differentials are obtained similarly.

The Taylor series of a function depending on several independent variables is also possible under similar conditions and in the case of two independent variables it is given as

(1.69)

where the derivatives are to be evaluated at (x0, y0). For functions with two independent variables and to second order in the neighborhood of (x, y), Equation (1.69) gives

(1.70)

which yields the differential, Δ(2)f(x, y) = f(x + Δx, y + Δy) = f(x,), as

(1.71)

(1.72)

For the higher-order terms note how the powers in Equation (1.69) are expanded. Generalization to n independent variables is obvious.

Example 1.1. Partial derivatives: Consider the function

Partial derivatives are written as

Example Taylor series: Using the partial derivatives obtained in the previous example, we can write the first two terms of the Taylor series [Eq (1.69)] of z = xy² + ex about the point(0, 1) First, the required derivatives at(0, 1) are evaluated as

Using these derivatives we can write the first two terms of the Taylor series about the point (0, 1) as

(1.87)

where the subscript 0 indicates that the derivatives are to be evaluated at the point (0, 1) To find Δ(2) z(0, 1), which is good to the second order, we first write

Figure 1.4 Maximum and minimum points of a function.

and then obtain

(1.91)

1.6 MAXIMA AND MINIMA OF FUNCTIONS

We are frequently interested in the maximum or the minimum values that a function, f(x), attains in a closed domain [a, b]. The absolute maximum, M1, is the value of the function at some point, x0, if the inequality

holds for all x in [a, b]. An absolute minimum is also defined similarly. In general we can quote the following theorem (Fig. 1.4):

Theorem 1.3. If a function, f(x), is continuous in the closed interval [a, b], then it possesses an absolute maximum, M1, and an absolute minimum, M2, in that interval.

Proof of this theorem requires a rather detailed analysis of the real number system, which can be found in books on advanced calculus. On the other hand, we are usually interested in the extremum values, that is, the local maximum or the minimum values of a function. Operationally, we can determine whether a given point, x0, corresponds to an extremum or not by

Figure 1.5 Analysis of critical points.

looking at the change or the variation in the function in the neighborhood of x0. The total differential introduced in the previous sections is just the tool needed for this. We have seen that in one dimension we can write the first, Δf(1), the second, Δ(2)f, and the third, Δf(3), differentials of a function with single independent variable as

(1.96)

Extremum points are defined as the points where the first differential vanishes, which means

In other words, the tangent line at an extremum point is horizontal (Fig. 1.5a,b). In order to decide whether an extremum point corresponds to a local maximum or minimum we look at the second differential:

For a local maximum the function decreases for small displacements about the extremum point (Fig. 1.5a), which implies Δ(2) f(x0).< 0. For a local minimum a similar argument yields Δ(2) f(x0).> 0. Thus we obtain the following criteria:

(1.99)

and

(1.100)

Figure 1.6 Plot of y(x) = x³

In cases where the second derivative also vanishes, we look at the third differential, Δ(3)f(x0). We now say that we have an inflection point; and depending on the sign of the third differential, we have either the third or the fourth shape in Figure 1.5.

Consider the function

where the first derivative, f’(x) = 3x², vanishes at x0 = 0. However, the second derivative, f"(x) = 6x, also vanishes there, thus making x0 = 0 a point of inflection. From the third differential:

we see that Δ(3)f(x0) > 0 for Δx > 0 and Δ(3)f(x0) < 0 for Δx < 0. Thus we choose the third shape in Figure 1.5 and plot f(x) = x³ as in Figure 1.6. Points where the first derivative of a function vanishes are called the critical points.

Usually the potential in one-dimensional conservative systems can be represented by a (scalar) function, V(x). Negative of the derivative of the potential gives the x component of the force on the system:

Thus the critical points of a potential function, V(x), correspond to the points where the net force on the system is zero. In other words, the critical points are the points where the system is in equilibrium. Whether an equilibrium is stable or unstable depends on whether the critical point is a minimum or maximum, respectively.

Analysis of the extrema of functions depending on more than one variable follows the same line of reasoning. However, since we can now approach the critical point from infinitely many different directions, one has to be careful. Consider a continuous function

defined in some domain D. We say this function has a local maximum at (x0, y0) if the inequality

is satisfied for all points in some neighborhood of (x0, y0) and to have a local minimum if the inequality

is satisfied. In the following argument we assume that all the necessary partial derivatives exist. Critical points are now defined as the points where the first differential, Δ(1)f(x, y), vanishes:

Since the displacements Δx and Δy are arbitrary, the only way to satisfy this equation is to have both partial derivatives, fx and fy, vanish. Hence at the critical point (x0, y0), shown with the subscript 0, one has

To study the nature of these critical points, we again look at the second differential, Δ(2)f(x0, y0), which is now given as

(1.111)

For a local maximum the second differential has to be negative, Δ(2)f(x0, y0) < 0, and for a local minimum positive, Δ(2)f(x0, y0) > 0. Since we can approach the point (x0, y0) from different directions, we substitute (Fig. 1.7)

to write Equation (1.111) as

(1.113)

Figure 1.7 Definition of Δs.

where we have defined

Now the analysis of the nature of the critical points reduces to investigating the sign of Δ(2)f(x0, y0) [Eq. (1.113)]. We present the final result as a theorem (Kaplan).

Theorem 1.4. Let z = f(x, y) and its first and second partial derivatives be continuous in a domain D and let (x0, y0) be a point in D, vanish. Then, we have the following cases:

I. For B² – AC < 0 and A + C < 0 we have a local maximum at (x0, y0).

II. For B² – AC < 0 and A + C > 0 we have a local minimum at (x0, y0).

III. For B² – AC > 0, we have a saddle point at (x0, y0).

IV. For B² – AC = 0, the nature of the critical point is undetermined.

When B² – AC > 0 at (x0, y0) we have what is called a saddle point.

In this case for some directions Δ(2)f(x0, y0) is positive and negative for the others. When B² – AC = 0, for some directions Δ(2)f(x0, y0) will be zero, hence one must look at higher-order derivatives to study the nature of the critical point. When A, B, and C are all zero, then Δ(2)f(x0, y0) also vanishes. Hence we need to investigate the sign of Δ(3)f(x0, y0).

1.7 EXTREMA OF FUNCTIONS WITH CONDITIONS

A problem of significance is finding the critical points of functions while satisfying one or more conditions. Consider finding the extremums of

while satisfying the conditions

and

In principle the two conditions define two surfaces, the intersection of which can be expressed as

where we have used the variable x as a parameter. We can now substitute this parametric equation into w = f(x,y,z) and write it entirely in terms of x as

extremum points of which can now be found by the technique discussed in the previous section. Geometrically, this problem corresponds to finding the extremum points of w = f(x, y, z) on the curve defined by the intersection of g1(x, y, z) = 0 and g2(x, y, z) = 0. Unfortunately, this method rarely works to yield a solution analytically. Instead, we introduce the following method: At a critical point we have seen that the change in w to first order in the differentials Δx, Δy, and Δz is zero:

We also write the differentials of g1(x, y, z) and g2(x, y, z) as

and

We now multiply Equation (1.123) with λ1 and Equation (1.124) with λ2 and add to Equation (1.122) to write

(1.125)

Because of the given conditions in Equations (1.116) and (1.117), Δx, Δy, and Δz are not independent. Hence their coefficients in Equation (1.122) cannot be set to zero directly. However, the values of λ1 and λ2, which are called the Lagrange undetermined multipliers, can be chosen so that the coefficients of Δx, Δy, and Δz are all zero in Equation (1.125):

Along with the two conditions, g1(x, y,z) – 0 and g2(x, y, z) = 0, these three equations are to be solved for the five unknowns:

The values that λ1 and λ2 assume are used to obtain the x, y, and z values needed, which correspond to the locations of the critical points. Analysis of the critical points now proceeds as before. Note that this method is quite general and as long as the required derivatives exist and the conditions are compatible, it can be used with any number of conditions.

Example 1.3. Extremum problems: We now find the dimensions of a rectangular swimming pool with fixed volume V0 and minimal area of its base and sides. If we denote the dimensions of its base with x and y and its height with z, the fixed volume is

and the total area of the base and the sides is

Using the condition of fixed volume we write a as a function of x and y as

Now the critical points of a are determined from the equations

which give the following two equations:

or

If we subtract Equation (1.137) from Equation (1.136), we obtain

which when substituted back into Equation (1.136) gives the critical dimensions

where the final dimension is obtained from V0 = xyz. To assure ourselves that this corresponds to a minimum, we evaluate the second-order derivatives at the critical point,

and find

(1.145)

Thus the critical dimensions we have obtained [Eqs. (1.139) – (1.141)] are indeed for a minimum by Theorem 1.4.

Example 1.4. Lagrange undetermined multipliers: We now solve the above problem by using the method of Lagrange undetermined multipliers. The equation to be minimized is now

with the condition

The equations to be solved are obtained from Equations (1.126) – (1.128) as

Along with V0 = xyz, these give 4 equations to be solved for the critical dimensions x, y, z, and λ. Multiplying the first equation by x and the second one by y and then subtracting gives

Substituting this into the third equation [Eq. (1.150)] gives the value of the Lagrange undetermined multiplier as λ = 4/x, which when substituted into Equations (1.148) – (1.150) gives

Using the condition V0 = xyz and Equation (1.151) these three equations [Eqs. (1.152) – (1.154)] can be solved easily to yield the critical dimensions in terms of V0 as

Analysis of the critical point is done as in the previous example by using Theorem 1.4.

1.8 DERIVATIVES AND DIFFERENTIALS OF COMPOSITE FUNCTIONS

In what follows we assume that the functions are defined in their appropriate domains and have continuous first partial derivatives.

Chain rule: If z – f(x, y) and x = x(t), y = y(t), then

Similarly, if z = f(x,y) and x = g(u,υ) and y = h(u, υ), then

A better notation to use is

This notation is particularly useful in thermodynamics, where z may also be expressed with another choice of variables, such as

Hence, when we write the derivative

we have to clarify whether we are in the (x, y) or the (x, w) space by writing

These formulas can be extended to any number of variables. Using Equation (1.158) we can write the differential dz as

We now treat x, y and z as functions of (u, v) and write the differential dz as

(1.172)

Since x and y are also functions of u and v, we have the differentials

and

which allow us to write Equation (1.172) as

This result can be extended to any number of variables. In other words, any equation in differentials that is true in one set of independent variables is also true for another choice of variables. Formal proofs of these results can be found in books on advanced calculus (Apostol, Kaplan).

1.9 IMPLICIT FUNCTION THEOREM

A function given as

can be used to describe several functions of the form

For example,

can be used to define the function

or

both of which are defined in the domain x² + y² + z² ≥ 9. We say these functions are implicitly defined by Equation (1.179). In order to be able to define a differentiable function,

by the implicit function F(x,y,z) = 0, the partial derivatives

should exist in some domain so that we can write the differential

Using the implicit function F(x, y, z) – 0, we write

and

where

Comparing the two differentials [Eqs. (1.184) and (1.186)], we obtain the partial derivatives

Hence, granted that Fz ≠ 0, we can use the implicit function F(x, y, z) = 0 to define a function of the form z = f(x, y).

We now consider a more complicated case, in which we have two implicit functions:

Using these two equations in terms of four variables, we can solve, in principle, for two of the variables in terms of the remaining two as

For f(x, y) and g(x, y) to be differentiable, certain conditions must be met by F(x, y, z, w) and G(x, y, z, w). First we write the differentials

and rearrange them as

We now have a system of two linear equations for the differentials dz and dw to be solved simultaneously. We can either solve by elimination or use determinants and the Cramer’s rule to write

and

Using the properties of determinants, we can write these as

and

For differentiable functions, z = f(x, y) and w = g(x, y), with existing first-order partial derivatives we can write

Thus by comparison with Equations (1.199) and (1.200), we obtain the partial derivatives

and

where the determinants written as

(1.205)

are called the Jacobi determinants. In summary, given two implicit equations

we can define two differentiable functions

with the partial derivatives given as in Equations (1.203) – (1.204), provided that the Jacobian

is different from zero in the domain of definition.

This useful technique can be generalized to a set of m equations in n + m number of unknowns:

We look for m differentiable functions in terms of n variables as

We write the differentials

(1.211)

and obtain a set of m linear equations to be solved for the m differentials, dyi, i = 1,... , m, of the dependent variables. Using Cramer’s rule, we can solve for dyi if and only if the determinant of the coefficients is different from zero, that is,

To obtain closed expressions for the partial derivatives,

we take partial derivatives of the Equations (1.209) to write

as

and similar expressions for the other partial derivatives can be obtained. In general, granted that the Jacobi determinant does not vanish, namely

as

where i = 1,... , m and j = 1,... n. We conclude this section by stating the implicit function theorem, a proof of which can be found in Kaplan:

Implicit function theorem: Let the functions

be defined in the neighborhood of the point

with continuous first-order partial derivatives existing in this neighborhood. If

then in an appropriate neighborhood of P0, there is a unique set of continuous functions

with continuous partial derivatives,

where i = 1,... , m and j = 1,... n, such that

and

(1.224)

in the neighborhood of P0. Note that if the Jacobi determinant [Eq. (1.120)] is zero at the point of interest, then we search for a different set of dependent variables to avoid the difficulty.

1.10 INVERSE FUNCTIONS

A pair of functions,

can be considered as a mapping from the xy space to the uυ space. Under certain conditions, this maps a certain domain Dxy in the xy space to a certain domain Duυ in the uυ space on a one-to-one basis. Under such conditions, an inverse mapping should also exist. However, analytically it may not always be possible to find the inverse mapping or the functions:

In such cases, we may consider Equations (1.225) and (1.226) as implicit functions and write them as

We can now use Equation (1.215) with y1 = u, y2 = v and x1 = x, x2 = y to write the partial derivatives of the inverse functions as

Similarly, the other partial derivatives can be obtained. As seen, the inverse function or the inverse mapping is well-defined only when the Jacobi determinant J is different from zero, that is,

where J is also called the Jacobian of the mapping. We will return to this point when we discuss coordinate transformations in Chapter 3. Note that the Jacobian of the inverse mapping is 1/J. In other words,

Example 1.5. Change of independent variable: We now transform the Laplace equation:

into polar coordinates, that is, to a new set of independent variables defined by the equations

where r Є (0, ∞) and ϕ Є [0, 2π]. We first write the partial derivatives of z = z(x, y):

which lead to

Solving for ∂z/∂x and ∂z/∂y, we obtain

We now repeat this process with ∂z/∂x to obtain the second derivative ∂²z/∂x² as

(1.245)

A similar procedure for ∂z/∂y yields ∂²z/∂y²:

(1.246)

Adding Equations (1.245) and (1.246), we obtain the transformed equation as

Since the Jacobian of the mapping is different from zero, that is,

the inverse mapping exists and it is given as

1.11 INTEGRAL CALCULUS AND THE DEFINITE INTEGRAL

Let f(x) be a continuous function in the interval [xa, Xb}- By choosing (n – 1) points in this interval, x1, x2,... , xn–1, we can subdivide it into n subintervals, ∆x1, ∆x2,…, ∆xn, which are not necessarily all equal in length. From

Figure 1.8 Upper (left) and lower (right) Darboux sums.

Theorem 1.3 we know that f(x) assumes a maximum, M, and a minimum, m, in [xa, xb,]. Let Mi represent the maximum and mi the minimum values that f(x) assumes in ∆xi. We now denote a particular subdivision by d and write the sum of the rectangles shown in Figure 1.8 (left) as

and in Figure 1.8 (right) as

The sums S(d) and s(d) are called the upper and the lower Darboux sums, respectively. Naturally, their values depend on the subdivision d. We pick the smallest of all S(d) and call it the upper integral of f(x) in [xa, xb]:

Similarly, the largest of all s(d) is called the lower integral of f(x) in [xa, xb]:

When these two integrals are equal, we say the definite integral of f(x) in the interval [xa, xb] exists and we write

Figure 1.9 Riemann integral

This definition of integral is also called the Riemann integral, and the function f(x) is called the integrand.

Darboux sums are not very practical to work with. Instead, for a particular subdivision we write the sum

is an arbitrary point in ∆xi (Fig. 1.9). It is clear that the inequality

is satisfied. For a given subdivision the largest value of ∆xi is called the norm of d, which we will denote as n(d).

1.12 RIEMANN INTEGRAL

We now give the basic definition of the Riemann integral as follows:

Definition 1.1. Given a sequence of subdivisions d1, d2,... of the interval [xa, xb]such that the sequence of norms n(d1), n(d2),… has the limit

and if f(x) is integrable in [xa, xb], then the Riemann integral is defined as

where

Theorem 1.5. For the existence of the Riemann integral

where xa and xb are finite numbers, it is sufficient to satisfy one of the following conditions:

i) f (x) is continuous in [xa, xb]

ii) f (x) is bounded and piecewise continuous in [xa, xb] .

From these definitions we can deduce the following properties of Riemann integrals. Their formal proofs can be found in books on mathematical analysis such as Apostol:

I. If f1(x) and f2(x) are integrable in [xa, xb], then their sum is also integrable and we can write

(1.261)

II. If f(x) is integrable in [xa, xb], then the following are true:

(1.264)

III. If f(x) is continuous and f(x) ≥ 0 in [xa, xb], then

means f(x) ≡ 0.

IV. The average or the mean, 〈f〉, of f(x) in the interval [xa, xb,] is defined as

If f(x) is continuous, then there exist at least one point x* Є [xa, xb] such that

This is also called the mean value theorem or Rolle’s theorem.

V. If f(x) is integrable in [xa, xb] and if xa < xc < xb, then

VI. If f(x) ≥ g(x) in [xa, xb,], then

VII. Fundamental theorem of calculus: If f(x) is continuous in [xa, xb], then the function

is also a continuous function of x in [xa, xb]. The function F(x) is differentiable for every point in [xa, xb] and its derivative at x is f(x) :

F(x) is called the primitive or the antiderivative of f(x). Given a primitive, F(x), then

is also a primitive. If a primitive is known for [xa, xb,], then we can write

When the region of integration is not specified, we write the indefinite integral

where C is an arbitrary constant and F(x) is any function the derivative of which is f(x).

VIII. If f(x) is continuous and f(x) ≥ 0 in [xa, xb], then geometrically the integral

is the area under f(x) between xa and xb.

IX. A very useful inequality in deciding whether a given integral is convergent or not is the Schwarz inequality:

X. One of the most commonly used techniques in integral calculus is the integration by parts:

or

where the derivatives u’ and υ’ and u and υ are continuous in [xa, xb].

XI. In general the following inequality holds:

|f(x)| dx f(x) dx f(x) dx, is said to be absolutely convergent, |f(x)\dx also converges. Integrals that converge but do not converge absolutely are called conditionally convergent.

1.13 IMPROPER INTEGRALS

We introduced Riemann integrals for bounded functions with finite intervals. Improper integrals are basically their extension to cases with infinite range and to functions that are not necessarily bounded.

Definition 1.2. Consider the integral

which exists in the Riemann sense in the interval [a, c], where a < c < b. If the limit

exists, where the function f(x) could be unbounded in the left neighborhood of bf(x) dx exists, or converges, and write

Example 1.6. Improper integrals: Consider the improper integral

where the integrand, x/(1 – x)¹/², is unbounded at the end point x = 1. We write I1 as the limit

thereby obtaining the value of I1 as 4/3. We now consider the integral

which does not exist since

In this case we say the integral does not exist or is divergent, and for its value we give +∞.

A parallel argument is given if the integral

exists in the interval [c, b], where a < c < b. We now write the limit

where f(x) could be unbounded in the right neighborhood of a. If the limit

exists, we write

We now present another useful result from integral calculus:

Theorem 1.6. Let c be a point in the interval (a, b) and let f(x) be integrable in the intervals [a, a′] and [b′, b], where a < a′ < c < b′ < b. Furthermore, f(x) could be unbounded in the neighborhood of c. Then the integral

exists if the integrals

and

both exist and when they exist, their sum is equal to I