Weber / Arfken Essential Mathematical Methods for Physicists, ISE

1. Auflage 2003
ISBN: 978-0-08-046985-0
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 932 Seiten

This new adaptation of Arfken and Weber's bestselling Mathematical Methods for Physicists, Fifth Edition, is the most comprehensive, modern, and accessible text for using mathematics to solve physics problems. Additional explanations and examples make it student-friendly and more adaptable to a course syllabus.
KEY FEATURES:
· This is a more accessible version of Arfken and Weber's blockbuster reference, Mathematical Methods for Physicists, 5th Edition
· Many more detailed, worked-out examples illustrate how to use and apply mathematical techniques to solve physics problems
· More frequent and thorough explanations help readers understand, recall, and apply the theory
· New introductions and review material provide context and extra support for key ideas
· Many more routine problems reinforce basic concepts and computations

Autoren/Hrsg.

Weber, Hans J.

Arfken, George B.

Leseproben

Chapter 1 Vector Analysis 1.1 Elementary Approach
In science and engineering we frequently encounter quantities that have only magnitude: mass, time, and temperature. This magnitude remains the same no matter how we orient the coordinate axes that we may use. These quantities we label scalar quantities. In contrast, many interesting physical quantities have magnitude or length and, in addition, an associated direction. Quantities with magnitude and direction are called vectors. Their length and the angle between any vectors remain unaffected by the orientation of coordinates we choose. To distinguish vectors from scalars, we identify vector quantities with boldface type (i.e., V). Vectors are useful in solving systems of linear equations (Chapter 3). They are not only helpful in Euclidean geometry but also indispensable in classical mechanics and engineering because force, velocity, acceleration, and angular momentum are vectors. Electrodynamics is unthinkable without vector fields such as electric and magnetic fields. Practical problems of mechanics and geometry, such as searching for the shortest distance between straight lines or parameterizing the orbit of a particle, will lead us to the differentiation of vectors and to vector analysis. Vector analysis is a powerful tool to formulate equations of motions of particles and then solve them in mechanics and engineering, or field equations of electrodynamics. In this section, we learn to add and subtract vectors geometrically and algebraically in terms of their rectangular components. A vector may be geometrically represented by an arrow with length proportional to the magnitude. The direction of the arrow indicates the direction of the vector, the positive sense of direction being indicated by the point. In this representation, vector addition
    (1.1)
consists of placing the rear end of vector B at the point of vector A (head to tail rule). 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 (sketch it), we see that Figure 1.1 Triangle Law of Vector Addition
    (1.2)
In words, vector addition is commutative. For the sum of three vectors

illustrated in Fig. 1.2, we first add A and B: Figure 1.2 Vector Addition Is Associative

Then this sum is added to C:

Alternatively, we may first add B and C:

Then

In terms of the original expression,

so that these alternative ways of summing three vectors lead to the same vector, or vector addition is associative. A direct physical example of the parallelogram addition law is provided by a weight suspended by two cords in Fig. 1.3. If the junction point is in equilibrium, the vector sum of the two forces F1 and F2 must cancel the downward force of gravity, F3. Here, the parallelogram addition law is subject to immediate experimental verification.1. Such a balance of forces is of immense importance for the stability of buildings, bridges, airplanes in flight, etc. Figure 1.3 Equilibrium of Forces: F1 + F2 = - F3 Subtraction is handled by defining the negative of a vector as a vector of the same magnitude but with reversed direction. Then

The graphical representation of vector A by an arrow suggests using coordinates as a second possibility. Arrow A (Fig. 1.4), starting from the origin,2terminates 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 rectangular or Cartesian coordinates (Ax, Ay, Az) of the arrow head. Figure 1.4 Components and Direction Cosines of A Although A could have represented any vector quantity (momentum, electric field, etc.), one particularly important vector quantity, the distance from the origin to the point (x, y, z), is denoted by the special symbol r. We then have a choice of referring to the displacement as either the vector r or the collection (x, y, z), the coordinates of its end point:
    (1.3)
Defining the magnitude r of vector r as its geometrical length, we find that Fig. 1.4 shows that the end point coordinates and the magnitude are related by
    (1.4)
cos a, cos ß, and cos ? are called the direction cosines, where a is the angle between the given vector and the positive x-axis, and so on. The (Cartesian) components Ax, Ay, and Az can also be viewed as the projections of A on the respective axes. Thus, any vector A may be resolved into its components (or projected onto the coordinate axes) to yield Ax = A cos a, etc., as in Eq. (1.4). We 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 or an algebraic representation. The geometric “arrow in space” often aids in visualization. The algebraic set of components is usually more suitable for precise numerical or algebraic calculations. (This is illustrated in Examples 1.1.1-1.1.3 and also applies to Exercises 1.1.1, 1.1.3, 1.1.5, and 1.1.6.) Vectors enter physics in two distinct forms: • 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. • Vector A may be defined over some extended region; that is, A and its components may be functions of position: Ax = Ax(x, y, z), and so on. Imagine a vector A attached to each point (x, y, z), whose length and direction change with position. Examples include the velocity of air around the wing of a plane in flight varying from point to point and electric and magnetic fields (made visible by iron filings). Thus, vectors defined at each point of a region are usually characterized as a vector field. The concept of the vector defined over a region and being a function of position will be extremely important in Section 1.2 and in later sections in which we differentiate and integrate vectors. A unit vector has length 1 and may point in any direction. Coordinate unit vectors are implicit in the projection of A onto the coordinate. axes to define its Cartesian components. Now, we define explicitly as a vector of unit magnitude pointing in the positive x-direction, y as a vector of unit magnitude in the positive y-direction, and as a vector of unit magnitude in the positive z-direction. Then is a vector with magnitude equal to Ax and in the positive x-direction; that is, the projection of A onto the x-direction, etc. By vector addition
    (1.5)
which states that a vector equals the vector sum of its components or projections. Note that if A vanishes, all of its components must vanish individually; that is, if

Finally, by the Pythagorean theorem, the length of vector A is
    (1.6)
This resolution of a vector into its components can be carried out in a variety of coordinate systems, as demonstrated in Chapter 2. Here, we restrict ourselves to Cartesian coordinates, where the unit vectors have the coordinates and Equation (1.5) means that the three unit vectors , y, and span the real three-dimensional space: Any constant vector may be written as a linear combination of , y, and Since , y, and are linearly independent (no one is a linear combination of the other two), they form a basis for the real three-dimensional space. Complementary to the geometrical technique, algebraic addition and subtraction of vectors may now be...