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livro para fisica matemática
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Acquisitions Editor Tom Singer Project Manager Simon Crump Marketing Manager Linda Beattie Cover Design Eric DeCicco Composition VTEX Typesetting Services Cover Printer Phoenix Color Interior Printer The Maple–Vail Book Manufacturing Group
Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK
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xii Preface
examples. They may continue their studies with linear algebra in Chapter 3, then perhaps tensors and symmetries (Chapters 2 and 4), and next real and complex analysis (Chap- ters 5–7), differential equations (Chapters 9, 10), and special functions (Chapters 11–13). In general, the core of a graduate one-semester course comprises Chapters 5–10 and 11–13, which deal with real and complex analysis, differential equations, and special func- tions. Depending on the level of the students in a course, some linear algebra in Chapter 3 (eigenvalues, for example), along with symmetries (group theory in Chapter 4), and ten- sors (Chapter 2) may be covered as needed or according to taste. Group theory may also be included with differential equations (Chapters 9 and 10). Appropriate relations have been included and are discussed in Chapters 4 and 9. A two-semester course can treat tensors, group theory, and special functions (Chap- ters 11–13) more extensively, and add Fourier series (Chapter 14), integral transforms (Chapter 15), integral equations (Chapter 16), and the calculus of variations (Chapter 17).
CHANGES TO THE SIXTH EDITION
Improvements to the Sixth Edition have been made in nearly all chapters adding examples and problems and more derivations of results. Numerous left-over typos caused by scan- ning into LaTeX, an error-prone process at the rate of many errors per page, have been corrected along with mistakes, such as in the Dirac γ -matrices in Chapter 3. A few chap- ters have been relocated. The Gamma function is now in Chapter 8 following Chapters 6 and 7 on complex functions in one variable, as it is an application of these methods. Dif- ferential equations are now in Chapters 9 and 10. A new chapter on probability has been added, as well as new subsections on differential forms and Mathieu functions in response to persistent demands by readers and students over the years. The new subsections are more advanced and are written in the concise style of the book, thereby raising its level to the graduate level. Many examples have been added, for example in Chapters 1 and 2, that are often used in physics or are standard lore of physics courses. A number of additions have been made in Chapter 3, such as on linear dependence of vectors, dual vector spaces and spectral decomposition of symmetric or Hermitian matrices. A subsection on the dif- fusion equation emphasizes methods to adapt solutions of partial differential equations to boundary conditions. New formulas have been developed for Hermite polynomials and are included in Chapter 13 that are useful for treating molecular vibrations; they are of interest to the chemical physicists.
ACKNOWLEDGMENTS
We have benefited from the advice and help of many people. Some of the revisions are in re- sponse to comments by readers and former students, such as Dr. K. Bodoor and J. Hughes. We are grateful to them and to our Editors Barbara Holland and Tom Singer who organized accuracy checks. We would like to thank in particular Dr. Michael Bozoian and Prof. Frank Harris for their invaluable help with the accuracy checking and Simon Crump, Production Editor, for his expert management of the Sixth Edition.
CHAPTER 1
VECTOR ANALYSIS
1.1 DEFINITIONS, ELEMENTARY APPROACH
In science and engineering we frequently encounter quantities that have magnitude and magnitude only: mass, time, and temperature. These we label scalar quantities, which re- main the same no matter what coordinates we use. In contrast, many interesting physical quantities have magnitude and, in addition, an associated direction. This second group includes displacement, velocity, acceleration, force, momentum, and angular momentum. Quantities with magnitude and direction are labeled vector quantities. Usually, in elemen- tary treatments, a vector is defined as a quantity having magnitude and direction. To dis- tinguish vectors from scalars, we identify vector quantities with boldface type, that is, V. Our vector may be conveniently represented by an arrow, with length proportional to the magnitude. The direction of the arrow gives the direction of the vector, the positive sense of direction being indicated by the point. In this representation, vector addition
C = A + B (1.1) consists in placing the rear end of vector B at the point of vector A. Vector C is then represented by an arrow drawn from the rear of A to the point of B. This procedure, the triangle law of addition, assigns meaning to Eq. (1.1) and is illustrated in Fig. 1.1. By completing the parallelogram, we see that
C = A + B = B + A , (1.2) as shown in Fig. 1.2. In words, vector addition is commutative. For the sum of three vectors
D = A + B + C , Fig. 1.3, we may first add A and B : A + B = E.
1
1.1 Definitions, Elementary Approach 3
FIGURE 1.4 Equilibrium of forces: F 1 + F 2 = − F 3.
sum of the two forces F 1 and F 2 must just cancel the downward force of gravity, F 3. Here the parallelogram addition law is subject to immediate experimental verification.^1 Subtraction may be handled by defining the negative of a vector as a vector of the same magnitude but with reversed direction. Then
A − B = A + (− B ).
In Fig. 1.3,
A = E − B.
Note that the vectors are treated as geometrical objects that are independent of any coor- dinate system. This concept of independence of a preferred coordinate system is developed in detail in the next section. The representation of vector A by an arrow suggests a second possibility. Arrow A (Fig. 1.5), starting from the origin,^2 terminates at the point (Ax , Ay , Az). Thus, if we agree that the vector is to start at the origin, the positive end may be specified by giving the Cartesian coordinates (Ax , Ay , Az) of the arrowhead. Although A could have represented any vector quantity (momentum, electric field, etc.), one particularly important vector quantity, the displacement from the origin to the point
(^1) Strictly speaking, the parallelogram addition was introduced as a definition. Experiments show that if we assume that the forces are vector quantities and we combine them by parallelogram addition, the equilibrium condition of zero resultant force is satisfied. (^2) We could start from any point in our Cartesian reference frame; we choose the origin for simplicity. This freedom of shifting the origin of the coordinate system without affecting the geometry is called translation invariance.
4 Chapter 1 Vector Analysis
FIGURE 1.5 Cartesian components and direction cosines of A.
(x, y, z), is denoted by the special symbol r. We then have a choice of referring to the dis- placement as either the vector r or the collection (x, y, z), the coordinates of its endpoint: r ↔ (x, y, z). (1.3) Using r for the magnitude of vector r , we find that Fig. 1.5 shows that the endpoint coor- dinates and the magnitude are related by x = r cos α, y = r cos β, z = r cos γ. (1.4) Here cos α, cos β, and cos γ are called the direction cosines , α being the angle between the given vector and the positive x-axis, and so on. One further bit of vocabulary: The quan- tities Ax , Ay , and Az are known as the (Cartesian) components of A or the projections of A , with cos^2 α + cos^2 β + cos^2 γ = 1. Thus, any vector A may be resolved into its components (or projected onto the coordi- nate axes) to yield Ax = A cos α, etc., as in Eq. (1.4). We may choose to refer to the vector as a single quantity A or to its components (Ax , Ay , Az). Note that the subscript x in Ax denotes the x component and not a dependence on the variable x. The choice between using A or its components (Ax , Ay , Az) is essentially a choice between a geometric and an algebraic representation. Use either representation at your convenience. The geometric “arrow in space” may aid in visualization. The algebraic set of components is usually more suitable for precise numerical or algebraic calculations. Vectors enter physics in two distinct forms. (1) Vector A may represent a single force acting at a single point. The force of gravity acting at the center of gravity illustrates this form. (2) Vector A may be defined over some extended region; that is, A and its compo- nents may be functions of position: Ax = Ax (x, y, z), and so on. Examples of this sort include the velocity of a fluid varying from point to point over a given volume and electric and magnetic fields. These two cases may be distinguished by referring to the vector de- fined over a region as a vector field. The concept of the vector defined over a region and
6 Chapter 1 Vector Analysis
Exercises
1.1.1 Show how to find A and B , given A + B and A − B. 1.1.2 The vector A whose magnitude is 1.732 units makes equal angles with the coordinate axes. Find Ax , Ay , and Az. 1.1.3 Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes. 1.1.4 The velocity of sailboat A relative to sailboat B, v rel, is defined by the equation v rel = v A − v B , where v A is the velocity of A and v B is the velocity of B. Determine the velocity of A relative to B if
v A = 30 km/hr east v B = 40 km/hr north.
ANS. v rel = 50 km/hr, 53. 1 ◦^ south of east.
1.1.5 A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading of 40◦^ east of north. The sailboat is simultaneously carried along by a current. At the end of the hour the boat is 6.12 km from its starting point. The line from its starting point to its location lies 60◦^ east of north. Find the x (easterly) and y (northerly) components of the water’s velocity. ANS. veast = 2 .73 km/hr, vnorth ≈ 0 km/hr.
1.1.6 A vector equation can be reduced to the form A = B. From this show that the one vector equation is equivalent to three scalar equations. Assuming the validity of Newton’s second law, F = m a , as a vector equation, this means that ax depends only on Fx and is independent of Fy and Fz.
1.1.7 The vertices A, B, and C of a triangle are given by the points (− 1 , 0 , 2 ), ( 0 , 1 , 0 ), and ( 1 , − 1 , 0 ), respectively. Find point D so that the figure ABCD forms a plane parallel- ogram.
ANS. ( 0 , − 2 , 2 ) or ( 2 , 0 , − 2 ). 1.1.8 A triangle is defined by the vertices of three vectors A , B and C that extend from the origin. In terms of A , B , and C show that the vector sum of the successive sides of the triangle (AB + BC + CA) is zero, where the side AB is from A to B, etc.
1.1.9 A sphere of radius a is centered at a point r 1.
(a) Write out the algebraic equation for the sphere. (b) Write out a vector equation for the sphere.
ANS. (a) (x − x 1 )^2 + (y − y 1 )^2 + (z − z 1 )^2 = a^2. (b) r = r 1 + a , with r 1 = center. ( a takes on all directions but has a fixed magnitude a.)
1.2 Rotation of the Coordinate Axes 7
1.1.10 A corner reflector is formed by three mutually perpendicular reflecting surfaces. Show that a ray of light incident upon the corner reflector (striking all three surfaces) is re- flected back along a line parallel to the line of incidence. Hint. Consider the effect of a reflection on the components of a vector describing the direction of the light ray. 1.1.11 Hubble’s law. Hubble found that distant galaxies are receding with a velocity propor- tional to their distance from where we are on Earth. For the ith galaxy, v i = H 0 r i , with us at the origin. Show that this recession of the galaxies from us does not imply that we are at the center of the universe. Specifically, take the galaxy at r 1 as a new origin and show that Hubble’s law is still obeyed. 1.1.12 Find the diagonal vectors of a unit cube with one corner at the origin and its three sides lying along Cartesian coordinates axes. Show that there are four diagonals with length√
2 and determine their components.
1.2 ROTATION OF THE COORDINATE AXES 3
In the preceding section vectors were defined or represented in two equivalent ways: (1) geometrically by specifying magnitude and direction, as with an arrow, and (2) al- gebraically by specifying the components relative to Cartesian coordinate axes. The sec- ond definition is adequate for the vector analysis of this chapter. In this section two more refined, sophisticated, and powerful definitions are presented. First, the vector field is de- fined in terms of the behavior of its components under rotation of the coordinate axes. This transformation theory approach leads into the tensor analysis of Chapter 2 and groups of transformations in Chapter 4. Second, the component definition of Section 1.1 is refined and generalized according to the mathematician’s concepts of vector and vector space. This approach leads to function spaces, including the Hilbert space. The definition of vector as a quantity with magnitude and direction is incomplete. On the one hand, we encounter quantities, such as elastic constants and index of refraction in anisotropic crystals, that have magnitude and direction but that are not vectors. On the other hand, our naïve approach is awkward to generalize to extend to more complex quantities. We seek a new definition of vector field using our coordinate vector r as a prototype. There is a physical basis for our development of a new definition. We describe our phys- ical world by mathematics, but it and any physical predictions we may make must be independent of our mathematical conventions. In our specific case we assume that space is isotropic; that is, there is no preferred di- rection, or all directions are equivalent. Then the physical system being analyzed or the physical law being enunciated cannot and must not depend on our choice or orientation of the coordinate axes. Specifically, if a quantity S does not depend on the orientation of the coordinate axes, it is called a scalar.
(^3) This section is optional here. It will be essential for Chapter 2.
