Consider for definiteness a three-dimensional vector space. Given three vectors ~b1, ~b2, ~b3 such that ? i~bi = 0 only if ? i = 0 for all i (1) then ~b1, ~b2, ~b3 are called linearly independent and they represent a basis of the threedimensional vector space, i.e., any vector ~x in this space can be represented in terms of three components x i as ~x = x i~bi (2) Note: In (1),(2) and throughout, the Einstein summation convention is used, i.e., if an index appears doubly, a sum over that index is implied, x i~bi = X i x i~bi (3) Any vector ~x can furthermore be normalized, ˆx denoting the unit vector in ~x-direction, xˆ = ~x/|~x| (4) One useful type of basis is an orthonormal basis {~e1, ~e2, ~e3} satisfying ~ei · ~ej = dij = ( 1 i = j 0 i 6= j (5) where dij is called the Kronecker d-symbol . For a general basis, the set of scalar products defines an object called the metric tensor,
notes1-6.dvi 1 Review of vector analysis 1.1 Fixed coordinate systems Consider for definiteness a three-dimensional vector space. Given three vectors ~b1,~b2,~b3 such that λi~bi = 0 only if λ i = 0 for all i (1) then ~b1,~b2,~b3 are called linearly independent and they represent a basis of the three- dimensional vector space, i.e., any vector ~x in this space can be represented in terms of three components xi as ~x = xi~bi (2) Note: In (1),(2) and throughout, the Einstein summation convention is used, i.e., if an index appears doubly, a sum over that index is implied, xi~bi ≡ ∑ i xi~bi (3) Any vector ~x can furthermore be normalized, x̂ denoting the unit vector in ~x-direction, x̂ = ~x/|~x| (4) One useful type of basis is an orthonormal basis {~e1, ~e2, ~e3} satisfying ~ei · ~ej = δij = { 1 i = j 0 i 6= j (5) where δij is called the Kronecker δ-symbol . For a general basis, the set of scalar products defines an object called the metric tensor, gij = ~bi ·~bj (6) in terms of which the scalar product between any two vectors ~x = xi~bi and ~y = y i~bi reads ~x · ~y = xigijyj (7) The components xi and yi are more specifically called the contravariant components of ~x and ~y. Defining also the covariant components xi = gijx j (8) the scalar product reduces to ~x · ~y = xiyi. In the case of an orthonormal basis, contravariant and covariant components of vec- tors are identical, one does not need to distinguish them, and it is customary to use 1 lower indices to enumerate them. For the time being, only orthonormal bases will be considered. The use of nontrivial metric tensors is crucial, e.g., in the theory of special relativity, which will be discussed as an example at a later stage. A coordinate system in three-dimensional space consists of a choice of origin and a three-dimensional basis. Note that the former is not already contained in the vec- tor space concept. It is very important to make a conceptual distinction between physical quantities (such as position, velocity, etc.) and their representations in terms of particular coordinate systems. Physical quanti- ties have meaning independent of the physicist describing them and the coordinate system he/she may have devised to represent them. Many different representations of any given physical quantity exist. The fact that they all must describe the same physical quantity implies powerful relations between different representations. To exploit these relations is the essence of tensor algebra, a subject which will be treated at a later stage in these lectures. At that stage, the tensor concept will be explained in detail; for the time being, the designation “metric tensor” introduced above can be viewed as being merely a name. It is furthermore useful to introduce the ǫ-symbol ǫijk = 1 if i, j, k is an even permutation of 1,2,3 -1 if i, j, k is an odd permutation of 1,2,3 0 otherwise (9) This is very useful for dealing with outer products of vectors in a succinct manner. The components of the outer product are (~x× ~y)i = ǫijkxjyk (10) More complicated combinations of such objects can be evaluated efficiently using the rules ǫijk = −ǫikj = ǫkij = −ǫjik = ǫjki = −ǫkji (11) ǫijkǫmnk = δimδjn − δinδjm (12) ǫijkǫmjk = δimδjj − δijδjm = δim(3− 1) = 2δim (13) ǫijkǫijk = 2δii = 6 (14) Furthermore, an orthonormal basis is called a right-handed orthonormal basis if it satisfies ~e1 × ~e2 = ~e3 (15) Together with a choice of origin, such a basis defines a Cartesian coordinate system. 2 1.2 Coordinate systems depending on parameters If one defines vector fields depending, e.g., on spatial position, it may be useful to adopt different coordinate systems at different positions to represent the vector field in question. In practice, the starting point for this is often a reparametrization of Cartesian coordinates xi ≡ xi(θ1, θ2, θ3) such as x1 = r cosϕ (16) x2 = r sinϕ (17) x3 = z (18) (where θ1, θ2, θ3 ≡ r, ϕ, z). This permits the definition of a new basis ~bi = ∂~x/∂θi |∂~x/∂θi| (19) Here, ~x is the position vector, represented in the original Cartesian basis as ~x = xi~ei. No summation over i is implied on the right hand side of (19). The particular example (16)-(18) defines cylindrical coordinates. One can convince oneself that the basis ~b1 ≡ ~er = ~e1 cosϕ+ ~e2 sinϕ (20) ~b2 ≡ ~eϕ = −~e1 sinϕ+ ~e2 cosϕ (21) ~b3 ≡ ~ez = ~e3 (22) is again orthonormal. The variation of the basis with spatial position must be empha- sized. Consider, e.g., motion of a particle in the z = 0 plane. Its position is described by ~r = r~er (note that this form implies no further assumptions on the motion – any position in the z = 0 plane has this representation in cylindrical coordinates). Then the velocity is ~̇r = ṙ~er + r~̇er = ṙ~er + rϕ̇~eϕ (23) In the Cartesian and the cylindrical coordinate systems, a vector field ~F can be represented as ~F = Fi~ei = Fr~er + Fϕ~eϕ + Fz~ez (24) Using the above relation between the two bases, this permits expressing the Cartesian components Fi in terms of the cylindrical components Fr, Fϕ, Fz and vice versa. Note that, although the above discussion focused on the specific example of cylindrical coordinates, the definition (19) applies to general reparametrizations of space. The reader is encouraged to analogously calculate the spherical basis associated with the reparametrization x1 = r sinϑ cosϕ (25) x2 = r sinϑ sinϕ (26) x3 = r cosϑ (27) 3 1.3 Gradient and nabla operator Consider a scalar field U(~r). The gradient of U in a Cartesian coordinate system is defined as gradU ≡ ~ei ∂U ∂ri (28) Defining the nabla operator ~∇ ≡ ~ei ∂ ∂ri (29) the gradient can be written as gradU = ~∇U . It is a vector field which points into the direction of steepest ascent of the scalar field U . The nabla operator can be written in cylindrical and spherical coordinates, respectively, as ~∇ = ~er ∂ ∂r + ~eϕ 1 r ∂ ∂ϕ + ~ez ∂ ∂z (30) ~∇ = ~er ∂ ∂r + ~eϑ 1 r ∂ ∂ϑ + ~eϕ 1 r sinϑ ∂ ∂ϕ (31) The reader is invited to verify these representations starting from the Cartesian def- inition using the cylindrical and/or spherical bases introduced above. 1.4 Line integrals of vector fields Consider the line integral I = ∫ B A d~r · ~F (32) from a point A to a point B in three-dimensional space, along a line which may be parametrized as ~r(s) using a real parameter s. In terms of this parametrization, the infinitesimal line element d~r can be expressed as d~r = d~r ds ds (33) Its direction is always tangential to the line ~r(s), and the modulus |(d~r/ds)ds| corre- sponds to the length of the line element. The integral I = ∫ sB sA ds d~r ds · ~F (34) may, e.g., represent the work performed by moving a particle subject to the force ~F along the line in question. If the field ~F satisfies ~∇× ~F = 0 on a simply connected region in space, then a scalar field U(~r) (a potential) exists such that ~F = ~∇U and I = ∫ B A d~r · ~F = U(B)− U(A) (35) 4 is independent of the path which one uses to connect A to B. Note that this is nothing but the multi-dimensional analogue of ∫ b a dx df/dx = f(b)−f(a). The vector field ~∇× ~F ≡ rot ~F ≡ curl ~F (36) is called the rotation or the curl of the vector field ~F . Example: Integrate over a circle of radius R in the 1-2-plane. This can be parametrized in cylindrical coordinates as ~r(ϕ) = R~er , ϕ ∈ [0, 2π] (37) (note that ~er depends on ϕ). Consequently, d~r dϕ = R~eϕ (38) Now, consider integrating over the field ~F = r~er (39) In this case, ∮ d~r · ~F = ∫ 2π 0 dϕR~eϕ ·R~er = 0 (40) which is not surprising, since ~∇× ~F = ~er × ~er + ~eϕ × (r/r)~eϕ = 0 (41) In fact, the associated potential is simply U = r2/2. Consider on the other hand the field ~F = 1 r ~eϕ (42) Then, ∮ d~r · ~F = ∫ 2π 0 dϕR~eϕ · 1 R ~eϕ = 2π (43) which may at first sight be surprising, since ~∇× ~F = ~er × (−1/r2)~eϕ + ~eϕ × (1/r2)(−~er) = − 1 r2 (~er × ~eϕ + ~eϕ × ~er) = 0 (44) However, this is only true for r 6= 0. At r = 0, neither ~F nor ~∇× ~F are well-defined. As a consequence, there is no simply connected region in three-dimensional space which both contains the integration path and in which ~∇ × ~F = 0 holds. The field (42) is a vortex field. It describes an infinitely thin vortex, since the nontrivial curl is concentrated on the line r = 0. 5 1.5 Surface integrals Surface integrals are defined as sums over contributions associated with the infinites- imal surface elements d~f = n̂|d~f | (45) making up the surface. Here, n̂ denotes a unit vector normal to the surface element in question, and the modulus |d~f | is the area of the surface element. Note that two orientations of the surface element are possible, related by inverting the sign of n̂. A surface is called orientable if one can associate a surface element with every point on it in a continuous manner. Continuity is destroyed, e.g., when one is forced to invert the direction of n̂ at certain lines on the surface, such as on a Möbius strip. Given a parametrization of a surface ~r(u, v) in terms of two parameters u, v, the surface elements can be given as d~f = ( ∂~r ∂u × ∂~r ∂v ) du dv (46) Since the two partial derivatives yield vectors tangential to the surface, their outer product indeed is normal to the surface. The modulus |d~f | = ∣ ∣ ∣ ∣ ∣ ∂~r ∂u du ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∂~r ∂v dv ∣ ∣ ∣ ∣ ∣ sinφ , (47) where φ denotes the angle between the two partial derivatives, is the area of the parallelogram spanned by (∂~r/∂u)du and (∂~r/∂v)dv. Example: Consider integrating over a spherical surface S of radius R. In spherical coordinates, this can be parametrized as ~r(ϑ, ϕ) = R~er , ϑ ∈ [0, π], ϕ ∈ [0, 2π] (48) The surface elements d~f = ( ∂~r ∂ϑ × ∂~r ∂ϕ ) dϑ dϕ = (~eϑR)× (~eϕR sinϑ)dϑ dϕ = R2 sinϑ~erdϑ dϕ (49) indeed are