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Class: M. Sc. Subject: Mathematical Methods for Physics I have taken all course materials for Mathematical Methods for Physics from G. B. Arfken and H.J. Weber, Mathematical Methods for Physicists. Students can download this book form given web address; Web address: https://b-ok.cc/book/446170/6e805d All topics have been taken from Chapter-2 & Chapter-4 from above said book ( https://b- ok.cc/book/446170/6e805d ). I am sending pdf file of Chapter 2 and Chapter 4 . Name of Topics Vector Analysis in Curved Coordinates and Tensors, Orthogonal Coordinates in R3, Differential Vector Operators, Special Coordinate Systems, Circular Cylinder Coordinates, Spherical and Polar Coordinates. Tensor Analysis, Contraction, Direct Product, Quotient Rule, Pseudotensors, Dual Tensors, General Tensors, Tensor Derivative Operators. Group Theory, Introduction, Generators of Continuous Group, Orbital Angular Momentum, Angular Momentum Coupling, Homogeneous Lorentz Group, Discrete Groups CHAPTER 2 VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS In Chapter 1 we restricted ourselves almost completely to rectangular or Cartesian coordi- nate systems. A Cartesian coordinate system offers the unique advantage that all three unit vectors, x, y, and z, are constant in direction as well as in magnitude. We did introduce the radial distanceˆ ˆ r, butˆ even this was treated as a function of x, y, and z. Unfortunately, not all physical problems are well adapted to a solution in Cartesian coordinates. For instance, if we have a central force problem, F rF(r), such as gravitational or electrostatic force, Cartesian coordinates may be unusually= inappropriate.ˆ Such a problem demands the use of a coordinate system in which the radial distance is taken to be one of the coordinates, that is, spherical polar coordinates. The point is that the coordinate system should be chosen to fit the problem, to exploit any constraint or symmetry present in it. Then it is likely to be more readily soluble than if we had forced it into a Cartesian framework. Naturally, there is a price that must be paid for the use of a non-Cartesian coordinate system. We have not yet written expressions for gradient, divergence, or curl in any of the non-Cartesian coordinate systems. Such expressions are developed in general form in Sec- tion 2.2. First, we develop a system of curvilinear coordinates, a general system that may be specialized to any of the particular systems of interest. We shall specialize to circular cylindrical coordinates in Section 2.4 and to spherical polar coordinates in Section 2.5. 2.1 ORTHOGONAL COORDINATES IN R3 In Cartesian coordinates we deal with three mutually perpendicular families of planes: x constant, y constant, and z constant. Imagine that we superimpose on this system = = = 103 104 Chapter 2 Vector Analysis in Curved Coordinates and Tensors three other families of surfaces qi(x,y,z), i 1, 2, 3. The surfaces of any one family qi need not be parallel to each other and they need= not be planes. If this is difficult to visualize, the figure of a specific coordinate system, such as Fig. 2.3, may be helpful. The three new families of surfaces need not be mutually perpendicular, but for simplicity we impose this condition (Eq. (2.7)) because orthogonal coordinates are common in physical applications. This orthogonality has many advantages: Orthogonal coordinates are almost like Cartesian coordinates where infinitesimal areas and volumes are products of coordinate differentials. In this section we develop the general formalism of orthogonal coordinates, derive from the geometry the coordinate differentials, and use them for line, area, and volume elements in multiple integrals and vector operators. We may describe any point (x,y,z)as the inter- section of three planes in Cartesian coordinates or as the intersection of the three surfaces that form our new, curvilinear coordinates. Describing the curvilinear coordinate surfaces by q1 constant, q2 constant, q3 constant, we may identify our point by (q1,q2,q3) as well= as by (x,y,z)=: = General curvilinear coordinates Circular cylindrical coordinates q1,q2,q3 ρ,ϕ,z x x(q1,q2,q3) <x ρ cos ϕ< = −∞ = ∞ y y(q1,q2,q3) <y ρ sin ϕ< (2.1) = −∞ = ∞ z z(q1,q2,q3) <z z< = −∞ = ∞ specifying x, y, z in terms of q1, q2, q3 and the inverse relations 2 2 1/2 q1 q1(x,y,z) 0 ρ x y < = = + ∞ q2 q2(x,y,z) 0 ϕ arctan(y/x) < 2π (2.2) = = q3 q3(x,y,z) <z z< . = −∞ = ∞ As a specific illustration of the general, abstract q1, q2, q3, the transformation equations for circular cylindrical coordinates (Section 2.4) are included in Eqs. (2.1) and (2.2). With each family of surfaces qi constant, we can associate a unit vector qi normal to the = ˆ surface qi constant and in the direction of increasing qi . In general, these unit vectors will depend= on the position in space. Then a vector V may be written V q1V1 q2V2 q3V3, (2.3) = ˆ + ˆ + ˆ but the coordinate or position vector is different in general, r q1q1 q2q2 q3q3, = ˆ + ˆ + ˆ as the special cases r rr for spherical polar coordinates and r ρρˆ zz for cylindri- = ˆ 2 = + ˆ cal coordinates demonstrate. The qi are normalized to q 1 and form a right-handed ˆ ˆ i = coordinate system with volume q1 (q2 q3)>0. Differentiation of x in Eqs. (2.1)ˆ · leadsˆ × toˆ the total variation or differential ∂x ∂x ∂x dx dq1 dq2 dq3, (2.4) = ∂q1 + ∂q2 + ∂q3 and similarly for differentiation of y and z. In vector notation dr ∂r dq . From i ∂qi i the Pythagorean theorem in Cartesian coordinates the square of the distance= between two neighboring points is ds2 dx2 dy2 dz2. = + + 2.1 Orthogonal Coordinates in R3 105 Substituting dr shows that in our curvilinear coordinate space the square of the distance element can be written as a quadratic form in the differentials dqi : 2 2 ∂r ∂r ds dr dr dr dqi dqj = · = = ∂qi · ∂qj ij 2 g11 dq g12 dq1 dq2 g13 dq1 dq3 = 1 + + 2 g21 dq2 dq1 g22 dq g23 dq2 dq3 + + 2 + 2 g31 dq3 dq1 g32 dq3 dq2 g33 dq + + + 3 gij dqi dqj , (2.5) = ij where nonzero mixed terms dqi dqj with i j signal that these coordinates are not or- = thogonal, that is, that the tangential directions qi are not mutually orthogonal. Spaces for ˆ which Eq. (2.5) is a legitimate expression are called metric or Riemannian. Writing Eq. (2.5) more explicitly, we see that ∂x ∂x ∂y ∂y ∂z ∂z ∂r ∂r gij (q1,q2,q3) (2.6) = ∂qi ∂qj + ∂qi ∂qj + ∂qi ∂qj = ∂qi · ∂qj ∂r are scalar products of the tangent vectors to the curves r for qj const., j i. These ∂qi = = coefficient functions gij , which we now proceed to investigate, may be viewed as speci- fying the nature of the coordinate system (q1,q2,q3). Collectively these coefficients are referred to as the metric and in Section 2.10 will be shown to form a second-rank sym- metric tensor.1 In general relativity the metric components are determined by the proper- ties of matter; that is, the gij are solutions of Einstein’s field equations with the energy– momentum tensor as driving term; this may be articulated as “geometry is merged with physics.” At usual we limit ourselves to orthogonal (mutually perpendicular surfaces) coordinate systems, which means (see Exercise 2.1.1)2 gij 0,ij, (2.7) = = and qi qj δij . (Nonorthogonal coordinate systems are considered in some detail in ˆ · ˆ = Sections 2.10 and 2.11 in the framework of tensor analysis.) Now, to simplify the notation, 2 we write gii h > 0, so = i 2 2 2 2 2 ds (h1 dq1) (h2 dq2) (h3 dq3) (hi dqi) . (2.8) = + + = i 1 The tensor nature of the set of gij ’s follows from the quotient rule (Section 2.8). Then the tensor transformation law yields Eq. (2.5). 2 In relativistic cosmology the nondiagonal elements of the metric gij are usually set equal to zero as a consequence of physical assumptions such as no rotation, as for dϕdt,dθ dt. 106 Chapter 2 Vector Analysis in Curved Coordinates and Tensors The specific orthogonal coordinate systems are described in subsequent sections by spec- ifying these (positive) scale factors h1, h2, and h3. Conversely, the scale factors may be conveniently identified by the relation ∂r dsi hi dqi, hiqi (2.9) = ∂qi = ˆ for any given dqi , holding all other q constant. Here, dsi is a differential length along the direction qi . Note that the three curvilinear coordinates q1, q2, q3 need not be lengths. The ˆ scale factors hi may depend on q and they may have dimensions. The product hi dqi must have a dimension of length. The differential distance vector dr may be written dr h1 dq1 q1 h2 dq2 q2 h3 dq3 q3 hi dqi qi. = ˆ + ˆ + ˆ = ˆ i Using this curvilinear component form, we find that a line integral becomes V dr Vihi dqi. · = i From Eqs. (2.9) we may immediately develop the area and volume elements dσij dsi dsj hihj dqi dqj (2.10) = = and dτ ds1 ds2 ds3 h1h2h3 dq1 dq2 dq3. (2.11) = = The expressions in Eqs. (2.10) and (2.11) agree, of course, with the results of using the transformation equations, Eq. (2.1), and Jacobians (described shortly; see also Exer- cise 2.1.5). From Eq. (2.10) an area element may be expanded: dσ ds2 ds3 q1 ds3 ds1 q2 ds1 ds2 q3 = ˆ + ˆ + ˆ h2h3 dq2 dq3 q1 h3h1 dq3 dq1 q2 = ˆ + ˆ h1h2 dq1 dq2 q3.
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