i Chapter II: General Coordinate Transformations Before beginning this chapter, please note the Cartesian coordinate system belowand the definitions of the angles θ and φ in the spherical coordinate system. In the spherical coordinate system, (r,θ,φ) we shall use: x = r sinθcosφ y = r sinθsinφ z = r cosΘ and in the cylindrical coordinate system (ρ,φ,z): x = ρ cosφ y = ρ sinφ ρ2 = x2 + y2 . Figure 1. The Cartesian coordinate system for 3- dimensional Euclidian space. II-1 Chapter II: General Coordinate Transformations Consider two coordinate systems in 3-dimensional Euclidian space: 1. a Cartesian system where a point is specified by (x1, x2, x3) (x,y,z) 2. a general "qi" system where a point is specified by (q1, q2, q3). Each point (x1, x2, x3) corresponds to a unique set of real numbers (q1, q2, q3). Further, each xi is a function of the qj, fi (q1,q2,q3), and each qj = hj (x1,x2,x3) where all first partial derivatives of fi and hj exist. Using only the chain rule for differentiation, the following equations can be obtained: f i f i f i dx i dq 1 dq 2 dq 3 q 1 q 2 q 3 i x j j dq q j also k q k x i x i q k using chain rule where we have used the notation x i (q1,q2,q3) f i (q1,q2,q3) and q j(x1,x2,x3) hj(x1,x2,x3) . Note that the differential, dxi, "transforms with" [ xi/qj ] and the partial derivative "transforms with" [ qj/xi]. So, using the summation notation (repeated indices ==> summation): i i x j i j dx dq A .j dq q j j q j Bi . x i x i q j q j Note the placement of the indices. II-2 i j j i i i In general, x /q ! q /x . This means that dx and /x transform differently under the coordinate transformation. The two transformations have been named contravariant and covariant, respectively. i i x A .j contravariant transformation matrix q j . j j q Bi. covariant transformation matrix x i Example: Transformation from Cartesian to spherical coordinates: x = rsinθcosφ x1 r = [x² + y² + z²]½ q1 y = rsinθsinφ x2 θ = cos-1(z/r) q2 z = rcosθ x3 φ = tan-1(y/x) q3 The differentials are given by: dx = sinθcosφ dr + rcosθcosφ dθ + -rsinθsinφ dφ dy = sinθsinφ dr + rcosθsinφ dθ + rsinθcosφ dφ dz = cos θ dr + -rsinθ dθ + 0 · dφ Thus, dx dx 1 sincos rcoscos rsinsin dr dy dx 2 sinsin rcossin rsincos d dz dx 3 cos rsin 0 d dr dq 1 d dq 2 d dq 3 II-3 1 3 In the above equation A 1 = sinθcosφ and A 2 = -rsinθ, etc. The elements of the contravariant transformation matrix are obtained from the expression for the differentials of dx, dy and dz. One can also show that: dr sinθϕ cos sin θϕ sin cos θ dx θ =− θϕ θϕ θ ⋅ d (coscos)/rrr (cossin)/ sin/ dy d − sinϕθϕθ / (rr sin ) cos / ( sin ) 0 dz And using the chain rule one finds: k k Mq j &1 k j dq ' dx ' [A ] .j dx Mx j m M Mx M &1 m M ' ' [B ]n. Mq n Mq n Mx m Mx m Note that My/Mθ = rcosθsinφ is not simply related to Mθ/My = (cosθsinφ)/r. One can see also from the above -1 T -1 T equations that A = B and B = A . In the above expressions different summation indices were used for each of the sums. This is a good habit to adopt. Try to do this in all your derivations. j j Sometimes you will find the notation: Ai for Bi . We won't use this notation since it is i j confusing. Keep in mind that A j … Ai . If you don't understand this comment look again carefully at the indices on the above equations. Exercise: Evaluate the A (contravariant) matrix for the transformation (from Cartesian to spherical coordinates) if r = 3, θ = π/4 and φ = π/2. -1 Exercise: Find the A matrix for the same values of r, θ, and φ in the above exercise. II-4 Rules for keeping track of indices: 1. Always use different index symbols for different summations; 2. Indices on the left side of the equation match with the "left-most" index on the right hand side; 3. "Superscripts" found in the "denominator" (such as the k in /xk) function as subscripts in the expression; i m 4. A j and Bk have four "positions", two "upper" and two "lower" (see below); i m 5. A "left" position on A j or Bk denotes a row index; the "right" position denotes column index; 6. Indices are always "matched" in an equation (look for the pairs); left index denotes a matrix "row" i m j k right index denotes a matrix "column" Contravariant and covariant vectors: A triplet of quantities in the Cartesian system, (V1,V2,V3), [such as (dx1,dx2,dx3) ] which transforms into its 1 2 3 i 1 2 3 i counterpart, (V' ,V' ,V' ), in the q system [such as (dq ,dq ,dq )] via the A j matrix is said to be a set of contravariant vector components. Contravariant vector components are denoted by superscripts. i i j V A .j V ' contravariant vector transformation A triplet of quantities in the Cartesian system, (V1,V2,V3), [such as ( / x, / y, / z) ] which transforms into its i 1 2 3 m counterpart, (V'1,V'2,V'3), in the q system [such as ( / q , / q , / q )] via the Bk matrix is said to be a set of covariant vector components. Covariant vector components are denoted by subscripts. n Vk Bk. V 'n covariant vector transformation II-5 The differentials, dxi, transform like contravariant vector components under a generalized (linear) coordinate transformation. The "linear" means that the transformation is linear in the differentials. The partial derivatives, /xk, transform like covariant vector components under a generalized (linear) coordinate transformation. In fact, any triplet which transforms "like the differentials" under a generalized linear coordinate transformation is said to be a contravariant vector. A triplet which transforms "like the partial derivatives" is said to be a covariant vector under the generalized linear coordinate transformation. Later we shall find out how to find the proper contravariant and covariant components of a physical vector. Two "physical" vectors are always simple to deal with: dr and f(x,y,z). ---------------------------------------------------------------------------------------------------------------------- Exercise: Find the contravariant components of the vector dr in the Cartesian coordinate system. Exercise: Find the contravariant components of the vector dr in the spherical coordinate system. Exercise: Find the covariant components of the vector ^x i (/xi) in the spherical coordinate system. Exercise: Find the covariant components of the vector ^x i (/xi) in the cylindrical coordinate system. II-6 Kronecker Delta Symbol and Other Notation j ! The Kronecker delta symbol is į i = { 0 if i j and 1 if i=j}. Generalized Kronecker delta: + * 0 if any two superscripts (or subscripts) are equal * or if {i,j,k,...m} ! {i',j',k',...m'} .. m i j k, * į i' j' k', ... m' = +1 if subscripts are even permutation of superscripts * -1 if subscripts are odd permutation of superscripts . 123 Examples: į 132 = -1 123 į 312 = +1 122 į 132 = 0 i ij Numerically, į j = įij = į . But in an expression, these can carry different information. PERMUTATION SYMBOLS (COMPLETELY ANTISYMMETRIC "TENSORS") + ijk..m ijk...m * İ į 123...n = 0 if any two superscripts are equal * or if superscripts are not same set as 1,2,3...n * +1 if the superscripts are an even permutation of 1,2,3,...n * -1 if the superscripts are an odd permutation of 1,2,3,...n . + 123...n * İijk...m įijk...m = 0 if any two subscripts are equal * or if subscripts are not same set as 1,2,3...n * +1 if the subscripts are an even permutation of 1,2,3,...n * -1 if the subscripts are an odd permutation of 1,2,3,...n . The above symbols are special cases of the generalized Kronecker delta and are often used with expressions involving permutations. The above symbols are "almost" tensors as we shall see in the chapter on tensors. Examples: İ123 = İ312 = İ231 =+1 İ213 = İ132 = İ321 =-1 İ112 = İ223 = İ323 = 0, etc. İ123 = İ312 = İ231 =+1 İ213 = İ132 = İ321 =-1 İ313 = İ322 = İ211 = 0, etc. Example: İ1234 = +1 İ0123 = 0 II-7 The Metric Tensor i The metric tensor, gjk, for the q system is defined as follows: j k ds² gjk dq dq . j,k .summation implied g metric tensor i where ds² = (dx)² + (dy)² + (dz)² = dx dxi (summation over i implied) What is gjk ? i ds² = dr·dr =dxdxi = [xi/qj]dqj · [xi/qk]dqk Summation over i = [xi/qj] [xi/qk] dqj dqk Summation over i j k = gjk dq dq i j i k Thus gjk = [ x / q ] [ x / q ] (Summation over i) The metric tensor is symmetric since i k i j i j i k [ x / q ] [ x / q ] = [ x / q ] [ x / q ] = gkj = gjk. Summation over i THE METRIC TENSOR IN CARTESIAN SYSTEMS: i -1 i k i k k dx = [R ] k dx' =[x / x' ] dx' -1 i -1 i Thus gjk = i [R ] j [R ] k -1 T i -1 i = i [R ] j [R ] k i -1 i = i Rj [R ] k -1 = i [RR ]jk = 1jk and gcartesian = 1 II-8 The Jacobian: i j i Jacobian det [ q / x ]= det [Bj ] = det(B) where [qi/xj] implies a matrix whose i,j component is qi/xj.