Vectors Dated: August 25 2016) I. PROPERTIE OF UNIT ANTIYMMETRIC TENOR ɛijkɛ klm = δ il δ jm δ im δ jl 1) Here index k is dummy index summation index), which can be denoted by any symbol. For two repeating indices ɛijkɛ kmj = 2δ im 2) For three repeating indices ɛijkɛ ijk = 6 3) II. APPLICATION Triple scalar product ) ) A B C = A i B C = ɛ ijka i B j C k 4) i In terms of the determinant of a matrix: A ) 1 A 2 A 3 A B C = det B 1 B 2 B 3 C 1 C 2 C 3 5) Using the antisymmetry of ɛ ijk, or, equivalently, the properties of the determinant, we have: ) A B C = B ) C A = C ) B. 6) Triple vector product )]
2 The i-th component of the triple vector product is given by: )]i = ɛ ijka j ) umming over l and m: Hence, )] )] Differentiation of the cross products Angular momentum L For simple parabolic spectrum d L = d r p d r = p m k = ɛ ijka j ɛ klm B l C m = δ il δ jm δ jl δ im )A j B l C m 7) i = B ia j C j A j B j C i 8) = B ) A C C ) A B 9) L = r p 10) = d r d p p + r 11) d r p = 0 12) Therefore, taking into account second Newton s law d p = F, F is the force, we obtain d L = d r p = r F 13) In the special case of a central force F r, we have angular momentum conservation law d L = 0. 14) This is also Kepler s second law, usually formulated as the constancy of the areal velocity. da = 1 2 rdθ)r = 1 2 r2 dθ 15) da = 1 2 r2 d dθ = 1 2 r2 ω 16) L = mvr = mωr 2 17) da = L = Const 18) m Constancy of the areal velocity is a consequence of the conservation of the angular momentum.
3 III. CALAR, VECTOR AND TENOR FIELD A scalar field is a function Φ r), which assigns a scalar to each point in space r. Examples: temperature, density, electromagnetic scalar potential A vector field is a function A r), which assigns a vector to each point in space r. Examples: electric and magnetic field, velocity, coordinate, momentum A tensor of n-th rank is the matematical and physical) object, which transform like a product of n vectors upon the transformations of a coordinate system and particularly upon symmetry transformations). A tensor field ijk... n indices) is a function that assigns a tensor of n-th rank to each point in space r. Examples: strain, conductivity, metric tensor, electromagnetic field tensor F µν The familiar differential operators, grdient, divergence, curl and Laplacian act on these fields and produce other fields. IV. PHYICAL INTERPRETATION OF DIV, GRAD, CURL... Gradient: Given a scalar function ux, y, z) we define vector u = ˆx u x + ŷ u y + ẑ u z 19) A physical picture of gradient can be developed noting that the scalar change of u, differential u, du is given by where vector dr = dx, dy, dz) = ˆxdx + ŷdy + ẑdz. Therefore du = u u u dx + dy + x y z dz = u dr, 20) du = u dr cos theta, 21) where θ is the angle between u and dr. We see that du is maximum when u and dr point the same direction. ummary: If we move from r to r + dr, then the scalar function ux, y, x) will change by du. This change is maximal if u and dr point the same direction. Thus, direction, in which change of u is maximal is the direction of a gradient of u.
4 Example: For a mountain, we characterize each point P with coordinates x, y, z by a gravitational potential energy ux, y, x). Then up ) points along the fall line, which is the path of the steepest descent. If a ski came loose, this is the path it would take. V. DIVERGENCE Divergence acts on a vector field to produce a scalar. diva A A = ˆx x + ŷ y + ẑ ) ˆxA x + ŷa y + ẑa z ) = z = A x + A y + A z ia i 22) One important identity involving diva is Gauss s Theorem divadv = n Ad = A nd) = V where n is the normal to the surface surrounding volume V. A d, 23) VI. CURL: This operation acts on vector fields and produces another vector field. Notation: x = / x, etc. A = ˆx y A z z A y ) + ŷ z A x x A z ) + ẑ x A y y A x ) 24) In terms of determinant of the matrix ˆx ŷ ẑ A = det x y z A x A y A z 25) For components of curl: This representation is useful for proving identities such as: A) i = ɛ ijk j A k 26) A) = i A) i = ɛ ijk i j A k = 0 27) The last equality takes hold because ɛ ijk is antisymmetric changes sign) upon permutation of i and j, while the product i j is symmetric upon this permutation. However summation includes both ij and ji terms, clearly of opposite sign. Hence we have cancellation of these terms.
5 Y=a Y=0 FIG. 1. The consequence of this identity is that any field, and most importantly magnetic field B which can be expessed as a curl of some vector A, has zero divergence. For the magnetic field, B = curla, where A is the electromagnetic vector potential. Given a vector field, such as the velocity v r) in the example in the Fig.1, that field will have a non-vanishing curl at r if a minute paddle-wheel inserted at r will rotate.this can happen even if all field lines for v r) are straight.consider the flow of river in Fig.1. The velocity field is given by v x = v 0 ya y), v y = 0. 28) The velocity x-component vaishes at the shores, and is maximal in the middle of the stream. The y-derivative of the velocity is dv x /dy = v 0 a 2y), and the curl is in general nonzero, despite straight lines of velocity field: v) z = dv x /dyẑ = ẑ2y a). 29) However, in accord with symmetry considerations, the curl vanishes at y = a/2, i.e., in the middle of the stream. VII. TOKE THEOREM An important integral theorem, playing vital role in topological phenomena great interest in nowaday physics is the tokes Theorem: A) nd = A) d = C A r, 30) where C is the contour spanned by surface.
6 You will learn in quantum mechanics that for the particular case of A being the electromagnetic vector potential, this theorem relates magnetic flux spanning the electron trajectory and the quantum mechanical phase accrued by electron wavefunction while electron traverses this trajectory.