"Supplementary file accompanying paper 'Handling Cell-Parameter Errors in Crystallographic Data' " "Author: James Haestier" "Monoclinic Calculation: " " Notes: 1) beta refers to cell angle; Beta^ij refers to the ADP parameter 2) Bracketed indices (p),(q) refer to Atoms P and Q. Set Q = P for single atom results 3) (xp,yp,zp) are coordinates of atom P; simialar for Q 4) cov[A,B] refers to the covariance of parameter A with parameter B. 5) a, b and c are the cell lengths 6) Result are separated by a linebreak 7) Parameters with repect to the Cartesian frame are labelled '_Cart' 8) Labels for each result is given in quotation marks; the analytic expression follows immediately below 9) The results in this file are in the following sections: (i) Expression with respect to the Cartesian axes (ii) Expression with respect to the A' axes, with ADPs parameterised using U^ij (iii) Expression with respect to the A' axes, with ADPs parameterised using Beta^ij 9b)Each section is subdevided into Covariances of Cov[x^i,x^j]; Cov[x^i,U^jk]; Cov[U^ij,U^kl]; " "----------------- Cartesian Results -----------------" "Cov[ X_cart(p) , X_cart(q) ]" ((xq*zp + xp*zq)*Cos[beta]*cov[a, c])/2 + (a*c*Cos[beta]*cov[xp, zq])/2 + (a*c*Cos[beta]*cov[xq, zp])/2 + (xp*xq*cov[a^2]*(Cos[beta/2] + Sin[beta/2])^2)/2 + (a^2*cov[xp, xq]*(Cos[beta/2] + Sin[beta/2])^2)/2 + cov[a, beta]*((a*xp*xq*Cos[beta])/2 - (c*(xq*zp + xp*zq)*(Cos[beta/2] + Sin[beta/2])^2)/4) + cov[beta^2]*(-(a*c*(xq*zp + xp*zq)*Cos[beta])/8 + (c^2*zp*zq*(Cos[beta/2] + Sin[beta/2])^2)/8 - (a^2*xp*xq*(-1 + Sin[beta]))/ 8) + cov[beta, c]*(-(c*zp*zq*Cos[beta])/2 - (a*(xq*zp + xp*zq)*(-1 + Sin[beta]))/4) + (c^2*cov[zp, zq]*(1 - Sin[beta]))/2 - (zp*zq*cov[c^2]*(-1 + Sin[beta]))/2 "Cov[ Y_cart(p) , Y_cart(q) ]" yp*yq*cov[b^2] + b^2*cov[yp, yq] "Cov[ Z_cart(p) , Z_cart(q) ]" ((xq*zp + xp*zq)*Cos[beta]*cov[a, c])/2 + (a*c*Cos[beta]*cov[xp, zq])/2 + (a*c*Cos[beta]*cov[xq, zp])/2 + (zp*zq*cov[c^2]*(Cos[beta/2] + Sin[beta/2])^2)/2 + (c^2*cov[zp, zq]*(Cos[beta/2] + Sin[beta/2])^2)/2 + cov[beta, c]*((c*zp*zq*Cos[beta])/2 - (a*(xq*zp + xp*zq)*(Cos[beta/2] + Sin[beta/2])^2)/4) + cov[beta^2]*(-(a*c*(xq*zp + xp*zq)*Cos[beta])/8 + (a^2*xp*xq*(Cos[beta/2] + Sin[beta/2])^2)/8 - (c^2*zp*zq*(-1 + Sin[beta]))/ 8) + cov[a, beta]*(-(a*xp*xq*Cos[beta])/2 - (c*(xq*zp + xp*zq)*(-1 + Sin[beta]))/4) + (a^2*cov[xp, xq]*(1 - Sin[beta]))/2 - (xp*xq*cov[a^2]*(-1 + Sin[beta]))/2 "Cov[ X_cart(p) , Y_cart(q) ]" (yq*zp*cov[b, c]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (b*c*cov[yq, zp]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (xp*yq*cov[a, b]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (a*b*cov[xp, yq]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + cov[b, beta]*((a*xp*yq*(Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (c*yq*zp*(Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2])) "Cov[ X_cart(p) , Z_cart(q) ]" (xp*xq*Cos[beta]*cov[a^2])/2 + (zp*zq*Cos[beta]*cov[c^2])/2 + (a^2*Cos[beta]*cov[xp, xq])/2 + (c^2*Cos[beta]*cov[zp, zq])/2 + (a*c*cov[xp, zq]*(Cos[beta/2] + Sin[beta/2])^2)/2 + (a*c*cov[xq, zp]*(1 - Sin[beta]))/2 + cov[a, beta]*(-(c*(xq*zp - xp*zq)*Cos[beta])/4 - (a*xp*xq*Sin[beta])/2) + (cov[a, c]*(xq*zp + xp*zq - xq*zp*Sin[beta] + xp*zq*Sin[beta]))/2 + cov[beta, c]*(-(a*(xq*zp - xp*zq)*Cos[beta])/4 - (c*zp*zq*Sin[beta])/2) + cov[beta^2]*(-(a^2*xp*xq*Cos[beta])/8 - (c^2*zp*zq*Cos[beta])/8 + (a*c*(xq*zp + xp*zq + xq*zp*Sin[beta] - xp*zq*Sin[beta]))/8) "Cov[ Y_cart(p) , Z_cart(q) ]" (xq*yp*cov[a, b]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (a*b*cov[xq, yp]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (yp*zq*cov[b, c]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (b*c*cov[yp, zq]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + cov[b, beta]*((c*yp*zq*(Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (a*xq*yp*(Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2])) "Cov[ U^11_cart(p) , U^11_cart(q) ]" (Cot[beta]^2*cov[u11p, u33q]*Csc[beta]^2)/4 + (Cot[beta]^2*cov[u11q, u33p]*Csc[beta]^2)/4 + Cot[beta]^2*cov[u13p, u13q]*Csc[beta]^2 + (Cot[beta]*cov[u13p, u33q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (Cot[beta]*cov[u13q, u33p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u33p, u33q]*(Csc[beta/2] - Sec[beta/2])^4)/64 + (Cot[beta]*cov[u11p, u13q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (Cot[beta]*cov[u11q, u13p]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (cov[u11p, u11q]*(Csc[beta/2] + Sec[beta/2])^4)/64 + cov[beta^2]*(u11p*((u33q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u11q*(Csc[beta/2] + Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/64) + u13p*((u13q*(3 + Cos[2*beta])^2*Csc[beta]^6)/4 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + u33p*((u11q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8 + (u33q*(Csc[beta/2] - Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/64)) "Cov[ U^12_cart(p) , U^12_cart(q) ]" (Cot[beta]*cov[u12p, u23q]*Csc[beta])/2 + (Cot[beta]*cov[u12q, u23p]*Csc[beta])/2 + (cov[u23p, u23q]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u12p, u12q]*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[beta^2]*(u12p*((u23q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u12q*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/32) + u23p*((u12q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u23q*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/32)) "Cov[ U^13_cart(p) , U^13_cart(q) ]" (Cot[beta]^2*cov[u11p, u11q]*Csc[beta]^2)/4 + (Cot[beta]^2*cov[u11p, u33q]*Csc[beta]^2)/4 + (Cot[beta]^2*cov[u11q, u33p]*Csc[beta]^2)/4 + (Cot[beta]^2*cov[u33p, u33q]*Csc[beta]^2)/4 + (Cot[beta]*cov[u11p, u13q]*Csc[beta]^3)/2 + (Cot[beta]*cov[u11q, u13p]*Csc[beta]^3)/2 + (Cot[beta]*cov[u13p, u33q]*Csc[beta]^3)/2 + (Cot[beta]*cov[u13q, u33p]*Csc[beta]^3)/2 + cov[u13p, u13q]*Csc[beta]^4 + cov[beta^2]*(u13p*(4*u13q*Cot[beta]^2*Csc[beta]^4 + (u11q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5)/2 + (u33q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5)/2) + u11p*((u13q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5)/2 + (u11q*(3 + Cos[2*beta])^2*Csc[beta]^6)/16 + (u33q*(3 + Cos[2*beta])^2*Csc[beta]^6)/16) + u33p*((u13q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5)/2 + (u11q*(3 + Cos[2*beta])^2*Csc[beta]^6)/16 + (u33q*(3 + Cos[2*beta])^2*Csc[beta]^6)/16)) "Cov[ U^22_cart(p) , U^22_cart(q) ]" cov[u22p, u22q] "Cov[ U^23_cart(p) , U^23_cart(q) ]" (Cot[beta]*cov[u12p, u23q]*Csc[beta])/2 + (Cot[beta]*cov[u12q, u23p]*Csc[beta])/2 + (cov[u12p, u12q]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u23p, u23q]*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[beta^2]*(u23p*((u12q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u23q*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/32) + u12p*((u23q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u12q*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/32)) "Cov[ U^33_cart(p) , U^33_cart(q) ]" (Cot[beta]^2*cov[u11p, u33q]*Csc[beta]^2)/4 + (Cot[beta]^2*cov[u11q, u33p]*Csc[beta]^2)/4 + Cot[beta]^2*cov[u13p, u13q]*Csc[beta]^2 + (Cot[beta]*cov[u11p, u13q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (Cot[beta]*cov[u11q, u13p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u11p, u11q]*(Csc[beta/2] - Sec[beta/2])^4)/64 + (Cot[beta]*cov[u13p, u33q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (Cot[beta]*cov[u13q, u33p]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (cov[u33p, u33q]*(Csc[beta/2] + Sec[beta/2])^4)/64 + cov[beta^2]*(u33p*((u11q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u33q*(Csc[beta/2] + Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/64) + u13p*((u13q*(3 + Cos[2*beta])^2*Csc[beta]^6)/4 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + u11p*((u33q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8 + (u11q*(Csc[beta/2] - Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/64)) "Cov[ U^11_cart(p) , U^12_cart(q) ]" (Cot[beta]*cov[u13p, u23q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (2*Sqrt[2]) + (cov[u23q, u33p]*(Csc[beta/2] - Sec[beta/2])^3)/ (16*Sqrt[2]) + (Cot[beta]*cov[u12q, u13p]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + (cov[u12q, u33p]*(Csc[beta/2] - Sec[beta/2])^2*(Csc[beta/2] + Sec[beta/2]))/ (16*Sqrt[2]) + (cov[u11p, u23q]*(Csc[beta/2] - Sec[beta/2])* (Csc[beta/2] + Sec[beta/2])^2)/(16*Sqrt[2]) + (cov[u11p, u12q]*(Csc[beta/2] + Sec[beta/2])^3)/(16*Sqrt[2]) + cov[beta^2]* (u13p*((u23q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (16*Sqrt[2]) + (u12q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2])) + u33p*((u23q*(Csc[beta/2] - Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u12q*(-(Cot[beta/2]*Csc[beta/2]^3) + Cot[beta/2]^2*Csc[beta/2]^3 + Sec[beta/2]^3*(-1 + Tan[beta/2])*Tan[beta/2]))/(32*Sqrt[2])) + u11p*((u12q*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u23q*(Cot[beta/2]*Csc[beta/2]^3 + Cot[beta/2]^2*Csc[beta/2]^3 - Sec[beta/2]^3*Tan[beta/2]*(1 + Tan[beta/2])))/(32*Sqrt[2]))) "Cov[ U^11_cart(p) , U^13_cart(q) ]" (Cot[beta]^2*cov[u11q, u13p]*Csc[beta]^2)/2 + (Cot[beta]^2*cov[u13p, u33q]*Csc[beta]^2)/2 + Cot[beta]*cov[u13p, u13q]*Csc[beta]^3 + (Cot[beta]*cov[u11q, u33p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/16 + (Cot[beta]*cov[u33p, u33q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/16 + (cov[u13q, u33p]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2])^2)/8 + (Cot[beta]*cov[u11p, u11q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/16 + (Cot[beta]*cov[u11p, u33q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/16 + (cov[u11p, u13q]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[beta^2]*(u13p*(u13q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5 + (u11q*(3 + Cos[2*beta])^2*Csc[beta]^6)/8 + (u33q*(3 + Cos[2*beta])^2*Csc[beta]^6)/8) + u11p*((u13q*Cot[beta]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/4 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16) + u33p*((u13q*Cot[beta]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/4 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ U^11_cart(p) , U^22_cart(q) ]" Cot[beta]*cov[u13p, u22q]*Csc[beta] + (cov[u22q, u33p]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u11p, u22q]*(Csc[beta/2] + Sec[beta/2])^2)/8 "Cov[ U^11_cart(p) , U^23_cart(q) ]" (Cot[beta]*cov[u12q, u13p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (2*Sqrt[2]) + (cov[u12q, u33p]*(Csc[beta/2] - Sec[beta/2])^3)/ (16*Sqrt[2]) + (Cot[beta]*cov[u13p, u23q]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + (cov[u23q, u33p]*(Csc[beta/2] - Sec[beta/2])^2*(Csc[beta/2] + Sec[beta/2]))/ (16*Sqrt[2]) + (cov[u11p, u12q]*(Csc[beta/2] - Sec[beta/2])* (Csc[beta/2] + Sec[beta/2])^2)/(16*Sqrt[2]) + (cov[u11p, u23q]*(Csc[beta/2] + Sec[beta/2])^3)/(16*Sqrt[2]) + cov[beta^2]* (u13p*((u12q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (16*Sqrt[2]) + (u23q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2])) + u33p*((u12q*(Csc[beta/2] - Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u23q*(-(Cot[beta/2]*Csc[beta/2]^3) + Cot[beta/2]^2*Csc[beta/2]^3 + Sec[beta/2]^3*(-1 + Tan[beta/2])*Tan[beta/2]))/(32*Sqrt[2])) + u11p*((u23q*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u12q*(Cot[beta/2]*Csc[beta/2]^3 + Cot[beta/2]^2*Csc[beta/2]^3 - Sec[beta/2]^3*Tan[beta/2]*(1 + Tan[beta/2])))/(32*Sqrt[2]))) "Cov[ U^11_cart(p) , U^33_cart(q) ]" (Cot[beta]^2*cov[u11p, u11q]*Csc[beta]^2)/4 + Cot[beta]^2*cov[u13p, u13q]* Csc[beta]^2 + (Cot[beta]^2*cov[u33p, u33q]*Csc[beta]^2)/4 + (Cot[beta]*cov[u11q, u13p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (Cot[beta]*cov[u13q, u33p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u11q, u33p]*(Csc[beta/2] - Sec[beta/2])^4)/64 + (Cot[beta]*cov[u11p, u13q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (Cot[beta]*cov[u13p, u33q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/8 + (cov[u11p, u33q]*(Csc[beta/2] + Sec[beta/2])^4)/64 + cov[beta^2]*(u11p*((u11q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u33q*(Csc[beta/2] + Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/64) + u13p*((u13q*(3 + Cos[2*beta])^2*Csc[beta]^6)/4 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + u33p*((u33q*(7 + Cos[2*beta])*Cot[beta]^2*Csc[beta]^4)/8 + (u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8 + (u11q*(Csc[beta/2] - Sec[beta/2])^2*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/64)) "Cov[ U^12_cart(p) , U^13_cart(q) ]" (Cot[beta]*cov[u11q, u23p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (4*Sqrt[2]) + (Cot[beta]*cov[u23p, u33q]*Csc[beta]* (Csc[beta/2] - Sec[beta/2]))/(4*Sqrt[2]) + (cov[u13q, u23p]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (Cot[beta]*cov[u11q, u12p]*Csc[beta]*(Csc[beta/2] + Sec[beta/2]))/ (4*Sqrt[2]) + (Cot[beta]*cov[u12p, u33q]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(4*Sqrt[2]) + (cov[u12p, u13q]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + cov[beta^2]* (u12p*((u13q*Cot[beta]*Csc[beta]^2*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(2*Sqrt[2]) + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2]) + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + u23p*((u11q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (32*Sqrt[2]) + (u33q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (32*Sqrt[2]) + (u13q*Cot[beta]*Csc[beta]^2*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(2*Sqrt[2]))) "Cov[ U^12_cart(p) , U^22_cart(q) ]" (cov[u22q, u23p]*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (cov[u12p, u22q]*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) "Cov[ U^12_cart(p) , U^23_cart(q) ]" (Cot[beta]*cov[u12p, u12q]*Csc[beta])/2 + (Cot[beta]*cov[u23p, u23q]*Csc[beta])/2 + (cov[u12q, u23p]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u12p, u23q]*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[beta^2]*(u12p*((u12q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u23q*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/32) + u23p*((u23q*(15*Cos[beta] + Cos[3*beta])*Csc[beta]^4)/32 + (u12q*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/32)) "Cov[ U^12_cart(p) , U^33_cart(q) ]" (Cot[beta]*cov[u13q, u23p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (2*Sqrt[2]) + (cov[u11q, u23p]*(Csc[beta/2] - Sec[beta/2])^3)/ (16*Sqrt[2]) + (Cot[beta]*cov[u12p, u13q]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + (cov[u11q, u12p]*(Csc[beta/2] - Sec[beta/2])^2*(Csc[beta/2] + Sec[beta/2]))/ (16*Sqrt[2]) + (cov[u23p, u33q]*(Csc[beta/2] - Sec[beta/2])* (Csc[beta/2] + Sec[beta/2])^2)/(16*Sqrt[2]) + (cov[u12p, u33q]*(Csc[beta/2] + Sec[beta/2])^3)/(16*Sqrt[2]) + cov[beta^2]*(u12p*((u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2]) + (u33q*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u11q*(-(Cot[beta/2]*Csc[beta/2]^3) + Cot[beta/2]^2*Csc[beta/2]^3 + Sec[beta/2]^3*(-1 + Tan[beta/2])*Tan[beta/2]))/(32*Sqrt[2])) + u23p*((u13q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (16*Sqrt[2]) + (u11q*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u33q*(Cot[beta/2]*Csc[beta/2]^3 + Cot[beta/2]^2*Csc[beta/2]^3 - Sec[beta/2]^3*Tan[beta/2]*(1 + Tan[beta/2])))/(32*Sqrt[2]))) "Cov[ U^13_cart(p) , U^22_cart(q) ]" (Cot[beta]*cov[u11p, u22q]*Csc[beta])/2 + (Cot[beta]*cov[u22q, u33p]*Csc[beta])/2 + cov[u13p, u22q]*Csc[beta]^2 "Cov[ U^13_cart(p) , U^23_cart(q) ]" (Cot[beta]*cov[u11p, u12q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (4*Sqrt[2]) + (Cot[beta]*cov[u12q, u33p]*Csc[beta]* (Csc[beta/2] - Sec[beta/2]))/(4*Sqrt[2]) + (cov[u12q, u13p]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (Cot[beta]*cov[u11p, u23q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2]))/ (4*Sqrt[2]) + (Cot[beta]*cov[u23q, u33p]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(4*Sqrt[2]) + (cov[u13p, u23q]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + cov[beta^2]* (u11p*((u12q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (32*Sqrt[2]) + (u23q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + u33p*((u12q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (32*Sqrt[2]) + (u23q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + u13p*((u23q*Cot[beta]*Csc[beta]^2*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(2*Sqrt[2]) + (u12q*Cot[beta]*Csc[beta]^2*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(2*Sqrt[2]))) "Cov[ U^13_cart(p) , U^33_cart(q) ]" (Cot[beta]^2*cov[u11p, u13q]*Csc[beta]^2)/2 + (Cot[beta]^2*cov[u13q, u33p]*Csc[beta]^2)/2 + Cot[beta]*cov[u13p, u13q]*Csc[beta]^3 + (Cot[beta]*cov[u11p, u11q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/16 + (Cot[beta]*cov[u11q, u33p]*Csc[beta]*(Csc[beta/2] - Sec[beta/2])^2)/16 + (cov[u11q, u13p]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2])^2)/8 + (Cot[beta]*cov[u11p, u33q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/16 + (Cot[beta]*cov[u33p, u33q]*Csc[beta]*(Csc[beta/2] + Sec[beta/2])^2)/16 + (cov[u13p, u33q]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[beta^2]*(u13p*(u13q*(3 + Cos[2*beta])*Cot[beta]*Csc[beta]^5 + (u33q*Cot[beta]*Csc[beta]^2*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/4 + (u11q*Cot[beta]*Csc[beta]^2*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/4) + u11p*((u13q*(3 + Cos[2*beta])^2*Csc[beta]^6)/8 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16) + u33p*((u13q*(3 + Cos[2*beta])^2*Csc[beta]^6)/8 + (u33q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] + Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16 + (u11q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ U^22_cart(p) , U^23_cart(q) ]" (cov[u12q, u22p]*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (cov[u22p, u23q]*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) "Cov[ U^22_cart(p) , U^33_cart(q) ]" Cot[beta]*cov[u13q, u22p]*Csc[beta] + (cov[u11q, u22p]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (cov[u22p, u33q]*(Csc[beta/2] + Sec[beta/2])^2)/8 "Cov[ U^23_cart(p) , U^33_cart(q) ]" (Cot[beta]*cov[u12p, u13q]*Csc[beta]*(Csc[beta/2] - Sec[beta/2]))/ (2*Sqrt[2]) + (cov[u11q, u12p]*(Csc[beta/2] - Sec[beta/2])^3)/ (16*Sqrt[2]) + (Cot[beta]*cov[u13q, u23p]*Csc[beta]* (Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + (cov[u11q, u23p]*(Csc[beta/2] - Sec[beta/2])^2*(Csc[beta/2] + Sec[beta/2]))/ (16*Sqrt[2]) + (cov[u12p, u33q]*(Csc[beta/2] - Sec[beta/2])* (Csc[beta/2] + Sec[beta/2])^2)/(16*Sqrt[2]) + (cov[u23p, u33q]*(Csc[beta/2] + Sec[beta/2])^3)/(16*Sqrt[2]) + cov[beta^2]*(u23p*((u13q*Csc[beta]*(Cot[beta]^2 + Csc[beta]^2)* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2]) + (u33q*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u11q*(-(Cot[beta/2]*Csc[beta/2]^3) + Cot[beta/2]^2*Csc[beta/2]^3 + Sec[beta/2]^3*(-1 + Tan[beta/2])*Tan[beta/2]))/(32*Sqrt[2])) + u12p*((u13q*Csc[beta]^5*(19*Cos[beta/2] + 9*Cos[(3*beta)/2] + 3*Cos[(5*beta)/2] + Cos[(7*beta)/2] + 19*Sin[beta/2] - 9*Sin[(3*beta)/2] + 3*Sin[(5*beta)/2] - Sin[(7*beta)/2]))/ (16*Sqrt[2]) + (u11q*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2])^2)/(32*Sqrt[2]) + (u33q*(Cot[beta/2]*Csc[beta/2]^3 + Cot[beta/2]^2*Csc[beta/2]^3 - Sec[beta/2]^3*Tan[beta/2]*(1 + Tan[beta/2])))/(32*Sqrt[2]))) "--- Cov(x^k,U^ij) terms in Cartesian -------- " "Cov[ x_cart(p) , U^11_cart(q) ]" (c*Cot[beta]*cov[u13q, zp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (c*cov[u11q, zp]*(Csc[beta/2] + Sec[beta/2])^2*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2]) + (c*cov[u33q, zp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2])^3)/ (2*Sqrt[2]) + (a*Cot[beta]*cov[u13q, xp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (a*cov[u33q, xp]*(Csc[beta/2] - Sec[beta/2])^2*(Cos[beta/2] + Sin[beta/2]))/ (8*Sqrt[2]) + (a*cov[u11q, xp]*(Csc[beta/2] + Sec[beta/2])^2* (Cos[beta/2] + Sin[beta/2]))/(8*Sqrt[2]) + cov[a, beta]*(-(u33q*xp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) - (u11q*xp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta, c]*((u11q*zp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (u33q*zp*(Csc[beta/2] - Sec[beta/2])*(Cos[beta/2] - Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta^2]*(c*((u33q*zp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (16*Sqrt[2]) + (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) + (u11q*zp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2])) + a*((u11q*xp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/(16*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2]) - (u33q*xp*(Csc[beta/2] - Sec[beta/2])* (Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2]))) "Cov[ x_cart(p) , U^12_cart(q) ]" (c*Cot[beta]*cov[u12q, zp])/2 + (a*Cot[beta]*cov[u23q, xp])/2 + (c*cov[u23q, zp]*(-2 + Cot[beta/2] + Tan[beta/2]))/4 + (a*cov[u12q, xp]*(2 + Cot[beta/2] + Tan[beta/2]))/4 + cov[a, beta]*((u23q*xp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2* (-2 + Sin[beta]))/4 - (u12q*xp*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8) + cov[beta, c]*((u12q*zp*(1 + Csc[beta] - 2*Csc[beta]^2))/4 - (u23q*zp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + cov[beta^2]* (c*(-(u23q*zp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2*(-2 + Sin[beta]))/ 8 + (u12q*zp*(Cos[beta/2] + Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16) + a*((u12q*xp*(1 + Csc[beta] - 2*Csc[beta]^2))/8 - (u23q*xp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ x_cart(p) , U^13_cart(q) ]" cov[beta, c]*(-(Sqrt[2]*u13q*zp*Cot[beta]*Csc[beta]^2* (Cos[beta/2] - Sin[beta/2])) - (u11q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(4*Sqrt[2]) - (u33q*zp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2])) + (c*Cot[beta]*cov[u11q, zp]*Csc[beta]* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) + (c*Cot[beta]*cov[u33q, zp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/ (2*Sqrt[2]) + (c*cov[u13q, zp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2]))/ Sqrt[2] + (a*Cot[beta]*cov[u11q, xp]*Csc[beta]*(Cos[beta/2] + Sin[beta/2]))/ (2*Sqrt[2]) + (a*Cot[beta]*cov[u33q, xp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) + (a*cov[u13q, xp]*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + cov[a, beta]*(-(Sqrt[2]*u13q*xp*Cot[beta]*Csc[beta]^2* (Cos[beta/2] + Sin[beta/2])) - (u11q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) - (u33q*xp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] + Sin[beta/2]))/ (4*Sqrt[2])) + cov[beta^2]* (a*(-((u13q*xp*Cot[beta]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2]))/ Sqrt[2]) - (u11q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(8*Sqrt[2]) - (u33q*xp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2])) + c*((u13q*zp*Cot[beta]*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (u11q*zp*Csc[beta]^3*(6*Cos[beta/2] + Cos[(3*beta)/2] + Cos[(5*beta)/2] + 6*Sin[beta/2] - Sin[(3*beta)/2] + Sin[(5*beta)/2]))/ (16*Sqrt[2]) + (u33q*zp*Csc[beta]^3*(6*Cos[beta/2] + Cos[(3*beta)/2] + Cos[(5*beta)/2] + 6*Sin[beta/2] - Sin[(3*beta)/2] + Sin[(5*beta)/2]))/ (16*Sqrt[2]))) "Cov[ x_cart(p) , U^22_cart(q) ]" (c*cov[u22q, zp]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (a*cov[u22q, xp]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] "Cov[ x_cart(p) , U^23_cart(q) ]" (a*Cot[beta]*cov[u12q, xp])/2 + (c*Cot[beta]*cov[u23q, zp])/2 + (c*cov[u12q, zp]*(-2 + Cot[beta/2] + Tan[beta/2]))/4 + (a*cov[u23q, xp]*(2 + Cot[beta/2] + Tan[beta/2]))/4 + cov[a, beta]*((u12q*xp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2* (-2 + Sin[beta]))/4 - (u23q*xp*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8) + cov[beta, c]*((u23q*zp*(1 + Csc[beta] - 2*Csc[beta]^2))/4 - (u12q*zp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + cov[beta^2]* (c*(-(u12q*zp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2*(-2 + Sin[beta]))/ 8 + (u23q*zp*(Cos[beta/2] + Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16) + a*((u23q*xp*(1 + Csc[beta] - 2*Csc[beta]^2))/8 - (u12q*xp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ x_cart(p) , U^33_cart(q) ]" (c*Cot[beta]*cov[u13q, zp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (c*cov[u33q, zp]*(Csc[beta/2] + Sec[beta/2])^2*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2]) + (c*cov[u11q, zp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2])^3)/ (2*Sqrt[2]) + (a*Cot[beta]*cov[u13q, xp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (a*cov[u11q, xp]*(Csc[beta/2] - Sec[beta/2])^2*(Cos[beta/2] + Sin[beta/2]))/ (8*Sqrt[2]) + (a*cov[u33q, xp]*(Csc[beta/2] + Sec[beta/2])^2* (Cos[beta/2] + Sin[beta/2]))/(8*Sqrt[2]) + cov[a, beta]*(-(u11q*xp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) - (u33q*xp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta, c]*((u33q*zp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (u11q*zp*(Csc[beta/2] - Sec[beta/2])*(Cos[beta/2] - Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta^2]*(c*((u11q*zp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (16*Sqrt[2]) + (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) + (u33q*zp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2])) + a*((u33q*xp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/(16*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2]) - (u11q*xp*(Csc[beta/2] - Sec[beta/2])* (Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2]))) "Cov[ y_cart(p) , U^11_cart(q) ]" b*Cot[beta]*cov[u13q, yp]*Csc[beta] + (b*cov[u33q, yp]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (b*cov[u11q, yp]*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[b, beta]*(-(u13q*yp*(3 + Cos[2*beta])*Csc[beta]^3)/2 - (u11q*yp*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 - (u33q*yp*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) "Cov[ y_cart(p) , U^12_cart(q) ]" (b*cov[u23q, yp]*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (b*cov[u12q, yp]*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + cov[b, beta]*(-(u12q*yp*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/ (4*Sqrt[2]) - (u23q*yp*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2])) "Cov[ y_cart(p) , U^13_cart(q) ]" (b*Cot[beta]*cov[u11q, yp]*Csc[beta])/2 + (b*Cot[beta]*cov[u33q, yp]*Csc[beta])/2 + b*cov[u13q, yp]*Csc[beta]^2 + cov[b, beta]*(-2*u13q*yp*Cot[beta]*Csc[beta]^2 - (u11q*yp*(3 + Cos[2*beta])*Csc[beta]^3)/4 - (u33q*yp*(3 + Cos[2*beta])*Csc[beta]^3)/4) "Cov[ y_cart(p) , U^22_cart(q) ]" b*cov[u22q, yp] "Cov[ y_cart(p) , U^23_cart(q) ]" (b*cov[u12q, yp]*(Csc[beta/2] - Sec[beta/2]))/(2*Sqrt[2]) + (b*cov[u23q, yp]*(Csc[beta/2] + Sec[beta/2]))/(2*Sqrt[2]) + cov[b, beta]*(-(u23q*yp*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/ (4*Sqrt[2]) - (u12q*yp*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(4*Sqrt[2])) "Cov[ y_cart(p) , U^33_cart(q) ]" b*Cot[beta]*cov[u13q, yp]*Csc[beta] + (b*cov[u11q, yp]*(Csc[beta/2] - Sec[beta/2])^2)/8 + (b*cov[u33q, yp]*(Csc[beta/2] + Sec[beta/2])^2)/8 + cov[b, beta]*(-(u13q*yp*(3 + Cos[2*beta])*Csc[beta]^3)/2 - (u33q*yp*(Csc[beta/2] + Sec[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8 - (u11q*yp*(Csc[beta/2] - Sec[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) "Cov[ z_cart(p) , U^11_cart(q) ]" (a*Cot[beta]*cov[u13q, xp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (a*cov[u11q, xp]*(Csc[beta/2] + Sec[beta/2])^2*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2]) + (a*cov[u33q, xp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2])^3)/ (2*Sqrt[2]) + (c*Cot[beta]*cov[u13q, zp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (c*cov[u33q, zp]*(Csc[beta/2] - Sec[beta/2])^2*(Cos[beta/2] + Sin[beta/2]))/ (8*Sqrt[2]) + (c*cov[u11q, zp]*(Csc[beta/2] + Sec[beta/2])^2* (Cos[beta/2] + Sin[beta/2]))/(8*Sqrt[2]) + cov[beta, c]*(-(u33q*zp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) - (u11q*zp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[a, beta]*((u11q*xp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (u33q*xp*(Csc[beta/2] - Sec[beta/2])*(Cos[beta/2] - Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta^2]*(a*((u33q*xp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (16*Sqrt[2]) + (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) + (u11q*xp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2])) + c*((u11q*zp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/(16*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2]) - (u33q*zp*(Csc[beta/2] - Sec[beta/2])* (Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2]))) "Cov[ z_cart(p) , U^12_cart(q) ]" (a*Cot[beta]*cov[u12q, xp])/2 + (c*Cot[beta]*cov[u23q, zp])/2 + (a*cov[u23q, xp]*(-2 + Cot[beta/2] + Tan[beta/2]))/4 + (c*cov[u12q, zp]*(2 + Cot[beta/2] + Tan[beta/2]))/4 + cov[beta, c]*((u23q*zp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2* (-2 + Sin[beta]))/4 - (u12q*zp*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8) + cov[a, beta]*((u12q*xp*(1 + Csc[beta] - 2*Csc[beta]^2))/4 - (u23q*xp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + cov[beta^2]* (a*(-(u23q*xp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2*(-2 + Sin[beta]))/ 8 + (u12q*xp*(Cos[beta/2] + Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16) + c*((u12q*zp*(1 + Csc[beta] - 2*Csc[beta]^2))/8 - (u23q*zp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ z_cart(p) , U^13_cart(q) ]" cov[a, beta]*(-(Sqrt[2]*u13q*xp*Cot[beta]*Csc[beta]^2* (Cos[beta/2] - Sin[beta/2])) - (u11q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(4*Sqrt[2]) - (u33q*xp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2])) + (a*Cot[beta]*cov[u11q, xp]*Csc[beta]* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) + (a*Cot[beta]*cov[u33q, xp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/ (2*Sqrt[2]) + (a*cov[u13q, xp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2]))/ Sqrt[2] + (c*Cot[beta]*cov[u11q, zp]*Csc[beta]*(Cos[beta/2] + Sin[beta/2]))/ (2*Sqrt[2]) + (c*Cot[beta]*cov[u33q, zp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) + (c*cov[u13q, zp]*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + cov[beta, c]*(-(Sqrt[2]*u13q*zp*Cot[beta]*Csc[beta]^2* (Cos[beta/2] + Sin[beta/2])) - (u11q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) - (u33q*zp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] + Sin[beta/2]))/ (4*Sqrt[2])) + cov[beta^2]* (c*(-((u13q*zp*Cot[beta]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2]))/ Sqrt[2]) - (u11q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(8*Sqrt[2]) - (u33q*zp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2])) + a*((u13q*xp*Cot[beta]*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (u11q*xp*Csc[beta]^3*(6*Cos[beta/2] + Cos[(3*beta)/2] + Cos[(5*beta)/2] + 6*Sin[beta/2] - Sin[(3*beta)/2] + Sin[(5*beta)/2]))/ (16*Sqrt[2]) + (u33q*xp*Csc[beta]^3*(6*Cos[beta/2] + Cos[(3*beta)/2] + Cos[(5*beta)/2] + 6*Sin[beta/2] - Sin[(3*beta)/2] + Sin[(5*beta)/2]))/ (16*Sqrt[2]))) "Cov[z_cart(p) , U^22_cart(q) ]" (a*cov[u22q, xp]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (c*cov[u22q, zp]*(Cos[beta/2] + Sin[beta/2]))/Sqrt[2] string "Cov[ z_cart(p) , U^23_cart(q) ]" (c*Cot[beta]*cov[u12q, zp])/2 + (a*Cot[beta]*cov[u23q, xp])/2 + (a*cov[u12q, xp]*(-2 + Cot[beta/2] + Tan[beta/2]))/4 + (c*cov[u23q, zp]*(2 + Cot[beta/2] + Tan[beta/2]))/4 + cov[beta, c]*((u12q*zp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2* (-2 + Sin[beta]))/4 - (u23q*zp*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/8) + cov[a, beta]*((u23q*xp*(1 + Csc[beta] - 2*Csc[beta]^2))/4 - (u12q*xp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/8) + cov[beta^2]* (a*(-(u12q*xp*Csc[beta]^2*(Cos[beta/2] + Sin[beta/2])^2*(-2 + Sin[beta]))/ 8 + (u23q*xp*(Cos[beta/2] + Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/16) + c*((u23q*zp*(1 + Csc[beta] - 2*Csc[beta]^2))/8 - (u12q*zp*(Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/16)) "Cov[ z_cart(p) , U^33_cart(q) ]" (a*Cot[beta]*cov[u13q, xp]*Csc[beta]*(Cos[beta/2] - Sin[beta/2]))/Sqrt[2] + (a*cov[u33q, xp]*(Csc[beta/2] + Sec[beta/2])^2*(Cos[beta/2] - Sin[beta/2]))/ (8*Sqrt[2]) + (a*cov[u11q, xp]*Csc[beta]^2*(Cos[beta/2] - Sin[beta/2])^3)/ (2*Sqrt[2]) + (c*Cot[beta]*cov[u13q, zp]*Csc[beta]* (Cos[beta/2] + Sin[beta/2]))/Sqrt[2] + (c*cov[u11q, zp]*(Csc[beta/2] - Sec[beta/2])^2*(Cos[beta/2] + Sin[beta/2]))/ (8*Sqrt[2]) + (c*cov[u33q, zp]*(Csc[beta/2] + Sec[beta/2])^2* (Cos[beta/2] + Sin[beta/2]))/(8*Sqrt[2]) + cov[beta, c]*(-(u11q*zp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(2*Sqrt[2]) - (u33q*zp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[a, beta]*((u33q*xp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/ (8*Sqrt[2]) - (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] - Sin[beta/2]))/(2*Sqrt[2]) - (u11q*xp*(Csc[beta/2] - Sec[beta/2])*(Cos[beta/2] - Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(8*Sqrt[2])) + cov[beta^2]*(a*((u11q*xp*Cos[beta]*(Csc[beta/2]^3 + Sec[beta/2]^3))/ (16*Sqrt[2]) + (u13q*xp*(3 + Cos[2*beta])*Csc[beta]^3* (Cos[beta/2] + Sin[beta/2]))/(4*Sqrt[2]) + (u33q*xp*(Csc[beta/2] + Sec[beta/2])*(Cos[beta/2] + Sin[beta/2])* (Cot[beta/2]*Csc[beta/2] - Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2])) + c*((u33q*zp*Cos[beta]*(-Csc[beta/2]^3 + Sec[beta/2]^3))/(16*Sqrt[2]) - (u13q*zp*(3 + Cos[2*beta])*Csc[beta]^3*(Cos[beta/2] - Sin[beta/2]))/ (4*Sqrt[2]) - (u11q*zp*(Csc[beta/2] - Sec[beta/2])* (Cos[beta/2] - Sin[beta/2])*(Cot[beta/2]*Csc[beta/2] + Sec[beta/2]*Tan[beta/2]))/(16*Sqrt[2]))) "----------------- Monoclinic Calculation ----------------" "---------- The following results are all with repect to the A' frame -----------" "Cov[ X(p) , X(q) ]" (xp*xq*cov[a^2])/a^2 + cov[xp, xq] + cov[a, beta]*((xp*xq*Cot[beta])/a - (c*(xq*zp + xp*zq)*Csc[beta])/(2*a^2)) + cov[beta^2]*((xp*xq*Cot[beta]^2)/4 - (c*(xq*zp + xp*zq)*Cot[beta]*Csc[beta])/ (4*a) + (c^2*zp*zq*Csc[beta]^2)/(4*a^2)) "Cov[ Y(p) , Y(q) ]" (yp*yq*cov[b^2])/b^2 + cov[yp, yq] "Cov[ Z(p) , Z(q) ]" (zp*zq*cov[c^2])/c^2 + cov[zp, zq] + cov[beta, c]*((zp*zq*Cot[beta])/c - (a*(xq*zp + xp*zq)*Csc[beta])/(2*c^2)) + cov[beta^2]*((zp*zq*Cot[beta]^2)/4 - (a*(xq*zp + xp*zq)*Cot[beta]*Csc[beta])/ (4*c) + (a^2*xp*xq*Csc[beta]^2)/(4*c^2)) "Cov[ X(p) , Y(q) ]" (xp*yq*cov[a, b])/(a*b) + cov[xp, yq] + cov[b, beta]*((xp*yq*Cot[beta])/(2*b) - (c*yq*zp*Csc[beta])/(2*a*b)) "Cov[ X(p) , Z(q) ]" (xp*zq*cov[a, c])/(a*c) + cov[xp, zq] + cov[a, beta]*((xp*zq*Cot[beta])/(2*a) - (xp*xq*Csc[beta])/(2*c)) + cov[beta, c]*((xp*zq*Cot[beta])/(2*c) - (zp*zq*Csc[beta])/(2*a)) + cov[beta^2]*(-(a*xp*xq*Cot[beta]*Csc[beta])/(4*c) - (c*zp*zq*Cot[beta]*Csc[beta])/(4*a) + ((2*xq*zp + xp*zq + xp*zq*Cos[2*beta])*Csc[beta]^2)/8) "Cov[ Y(p) , Z(q) ]" (yp*zq*cov[b, c])/(b*c) + cov[yp, zq] + cov[b, beta]*((yp*zq*Cot[beta])/(2*b) - (a*xq*yp*Csc[beta])/(2*b*c)) "Cov[ U^11(p) , U^11(q) ]" cov[u11p, u11q] + cov[beta^2]* (u11p*(u11q*Cot[beta]^2 + u13q*Cot[beta]*Csc[beta]) + u13p*(u11q*Cot[beta]*Csc[beta] + u13q*Csc[beta]^2)) "Cov[ U^12(p) , U^12(q) ]" cov[u12p, u12q] + cov[beta^2]* (u12p*((u12q*Cot[beta]^2)/4 + (u23q*Cot[beta]*Csc[beta])/4) + u23p*((u12q*Cot[beta]*Csc[beta])/4 + (u23q*Csc[beta]^2)/4)) "Cov[ U^13(p) , U^13(q) ]" cov[u13p, u13q] + cov[beta^2]* (u13p*(u13q*Cot[beta]^2 + (u11q*Cot[beta]*Csc[beta])/2 + (u33q*Cot[beta]*Csc[beta])/2) + u11p*((u13q*Cot[beta]*Csc[beta])/2 + (u11q*Csc[beta]^2)/4 + (u33q*Csc[beta]^2)/4) + u33p*((u13q*Cot[beta]*Csc[beta])/2 + (u11q*Csc[beta]^2)/4 + (u33q*Csc[beta]^2)/4)) "Cov[ U^22(p) , U^22(q) ]" cov[u22p, u22q] "Cov[ U^23(p) , U^23(q) ]" cov[u23p, u23q] + cov[beta^2]* (u23p*((u23q*Cot[beta]^2)/4 + (u12q*Cot[beta]*Csc[beta])/4) + u12p*((u23q*Cot[beta]*Csc[beta])/4 + (u12q*Csc[beta]^2)/4)) "Cov[ U^33(p) , U^33(q) ]" cov[u33p, u33q] + cov[beta^2]* (u33p*(u33q*Cot[beta]^2 + u13q*Cot[beta]*Csc[beta]) + u13p*(u33q*Cot[beta]*Csc[beta] + u13q*Csc[beta]^2)) "Cov[ U^11(p) , U^12(q) ]" cov[u11p, u12q] + cov[beta^2]* (u11p*((u12q*Cot[beta]^2)/2 + (u23q*Cot[beta]*Csc[beta])/2) + u13p*((u12q*Cot[beta]*Csc[beta])/2 + (u23q*Csc[beta]^2)/2)) "Cov[ U^11(p) , U^13(q) ]" cov[u11p, u13q] + cov[beta^2]* (u11p*(u13q*Cot[beta]^2 + (u11q*Cot[beta]*Csc[beta])/2 + (u33q*Cot[beta]*Csc[beta])/2) + u13p*(u13q*Cot[beta]*Csc[beta] + (u11q*Csc[beta]^2)/2 + (u33q*Csc[beta]^2)/2)) "Cov[ U^11(p) , U^22(q) ]" cov[u11p, u22q] "Cov[ U^11(p) , U^23(q) ]" cov[u11p, u23q] + cov[beta^2]* (u11p*((u23q*Cot[beta]^2)/2 + (u12q*Cot[beta]*Csc[beta])/2) + u13p*((u23q*Cot[beta]*Csc[beta])/2 + (u12q*Csc[beta]^2)/2)) "Cov[ U^11(p) , U^33(q) ]" cov[u11p, u33q] + cov[beta^2]* (u11p*(u33q*Cot[beta]^2 + u13q*Cot[beta]*Csc[beta]) + u13p*(u33q*Cot[beta]*Csc[beta] + u13q*Csc[beta]^2)) "Cov[ U^12(p) , U^13(q) ]" cov[u12p, u13q] + cov[beta^2]* (u12p*((u13q*Cot[beta]^2)/2 + (u11q*Cot[beta]*Csc[beta])/4 + (u33q*Cot[beta]*Csc[beta])/4) + u23p*((u13q*Cot[beta]*Csc[beta])/2 + (u11q*Csc[beta]^2)/4 + (u33q*Csc[beta]^2)/4)) "Cov[ U^12(p) , U^22(q) ]" cov[u12p, u22q] "Cov[ U^12(p) , U^23(q) ]" cov[u12p, u23q] + cov[beta^2]* (u12p*((u23q*Cot[beta]^2)/4 + (u12q*Cot[beta]*Csc[beta])/4) + u23p*((u23q*Cot[beta]*Csc[beta])/4 + (u12q*Csc[beta]^2)/4)) "Cov[ U^12(p) , U^33(q) ]" cov[u12p, u33q] + cov[beta^2]* (u12p*((u33q*Cot[beta]^2)/2 + (u13q*Cot[beta]*Csc[beta])/2) + u23p*((u33q*Cot[beta]*Csc[beta])/2 + (u13q*Csc[beta]^2)/2)) "Cov[ U^13(p) , U^22(q) ]" cov[u13p, u22q] "Cov[ U^13(p) , U^23(q) ]" cov[u13p, u23q] + cov[beta^2]* (u13p*((u23q*Cot[beta]^2)/2 + (u12q*Cot[beta]*Csc[beta])/2) + u11p*((u23q*Cot[beta]*Csc[beta])/4 + (u12q*Csc[beta]^2)/4) + u33p*((u23q*Cot[beta]*Csc[beta])/4 + (u12q*Csc[beta]^2)/4)) "Cov[ U^13(p) , U^33(q) ]" cov[u13p, u33q] + cov[beta^2]* (u13p*(u33q*Cot[beta]^2 + u13q*Cot[beta]*Csc[beta]) + u11p*((u33q*Cot[beta]*Csc[beta])/2 + (u13q*Csc[beta]^2)/2) + u33p*((u33q*Cot[beta]*Csc[beta])/2 + (u13q*Csc[beta]^2)/2)) "Cov[ U^22(p) , U^23(q) ]" cov[u22p, u23q] "Cov[ U^22(p) , U^33(q) ]" cov[u22p, u33q] "Cov[ U^23(p) , U^33(q) ]" cov[u23p, u33q] + cov[beta^2]* (u23p*((u33q*Cot[beta]^2)/2 + (u13q*Cot[beta]*Csc[beta])/2) + u12p*((u33q*Cot[beta]*Csc[beta])/2 + (u13q*Csc[beta]^2)/2)) "---- Cov(x^k,U^ij) ----" "Cov[ x(p) , U^11(q) ]" cov[u11q, xp] + (cov[a, beta]*(-(u11q*xp*Cot[beta]) - u13q*xp*Csc[beta]))/a + cov[beta^2]*(-(u11q*xp*Cot[beta]^2)/2 - (u13q*xp*Cot[beta]*Csc[beta])/2 + (c*((u11q*zp*Cot[beta]*Csc[beta])/2 + (u13q*zp*Csc[beta]^2)/2))/a) "Cov[ x(p) , U^12(q) ]" cov[u12q, xp] + (cov[a, beta]*(-(u12q*xp*Cot[beta])/2 - (u23q*xp*Csc[beta])/2))/a + cov[beta^2]*(-(u12q*xp*Cot[beta]^2)/4 - (u23q*xp*Cot[beta]*Csc[beta])/4 + (c*((u12q*zp*Cot[beta]*Csc[beta])/4 + (u23q*zp*Csc[beta]^2)/4))/a) "Cov[ x(p) , U^13(q) ]" cov[u13q, xp] + (cov[a, beta]*(-(u13q*xp*Cot[beta]) - (u11q*xp*Csc[beta])/2 - (u33q*xp*Csc[beta])/2))/a + cov[beta^2]*(-(u13q*xp*Cot[beta]^2)/2 - (u11q*xp*Cot[beta]*Csc[beta])/4 - (u33q*xp*Cot[beta]*Csc[beta])/4 + (c*((u13q*zp*Cot[beta]*Csc[beta])/2 + (u11q*zp*Csc[beta]^2)/4 + (u33q*zp*Csc[beta]^2)/4))/a) "Cov[ x(p) , U^22(q) ]" cov[u22q, xp] "Cov[ x(p) , U^23(q) ]" cov[u23q, xp] + (cov[a, beta]*(-(u23q*xp*Cot[beta])/2 - (u12q*xp*Csc[beta])/2))/a + cov[beta^2]*(-(u23q*xp*Cot[beta]^2)/4 - (u12q*xp*Cot[beta]*Csc[beta])/4 + (c*((u23q*zp*Cot[beta]*Csc[beta])/4 + (u12q*zp*Csc[beta]^2)/4))/a) "Cov[ x(p) , U^33(q) ]" cov[u33q, xp] + (cov[a, beta]*(-(u33q*xp*Cot[beta]) - u13q*xp*Csc[beta]))/a + cov[beta^2]*(-(u33q*xp*Cot[beta]^2)/2 - (u13q*xp*Cot[beta]*Csc[beta])/2 + (c*((u33q*zp*Cot[beta]*Csc[beta])/2 + (u13q*zp*Csc[beta]^2)/2))/a) "Cov[ y(p) , U^11(q) ]" cov[u11q, yp] + (cov[b, beta]*(-(u11q*yp*Cot[beta]) - u13q*yp*Csc[beta]))/b "Cov[ y(p) , U^12(q) ]" cov[u12q, yp] + (cov[b, beta]*(-(u12q*yp*Cot[beta])/2 - (u23q*yp*Csc[beta])/2))/b "Cov[ y(p) , U^13(q) ]" cov[u13q, yp] + (cov[b, beta]*(-(u13q*yp*Cot[beta]) - (u11q*yp*Csc[beta])/2 - (u33q*yp*Csc[beta])/2))/b "Cov[ y(p) , U^22(q) ]" cov[u22q, yp] "Cov[ y(p) , U^23(q) ]" cov[u23q, yp] + (cov[b, beta]*(-(u23q*yp*Cot[beta])/2 - (u12q*yp*Csc[beta])/2))/b "Cov[ y(p) , U^33(q) ]" cov[u33q, yp] + (cov[b, beta]*(-(u33q*yp*Cot[beta]) - u13q*yp*Csc[beta]))/b "Cov[ z(p) , U^11(q) ]" cov[u11q, zp] + (cov[beta, c]*(-(u11q*zp*Cot[beta]) - u13q*zp*Csc[beta]))/c + cov[beta^2]*(-(u11q*zp*Cot[beta]^2)/2 - (u13q*zp*Cot[beta]*Csc[beta])/2 + (a*((u11q*xp*Cot[beta]*Csc[beta])/2 + (u13q*xp*Csc[beta]^2)/2))/c) "Cov[ z(p) , U^12(q) ]" cov[u12q, zp] + (cov[beta, c]*(-(u12q*zp*Cot[beta])/2 - (u23q*zp*Csc[beta])/2))/c + cov[beta^2]*(-(u12q*zp*Cot[beta]^2)/4 - (u23q*zp*Cot[beta]*Csc[beta])/4 + (a*((u12q*xp*Cot[beta]*Csc[beta])/4 + (u23q*xp*Csc[beta]^2)/4))/c) "Cov[ z(p) , U^13(q) ]" cov[u13q, zp] + (cov[beta, c]*(-(u13q*zp*Cot[beta]) - (u11q*zp*Csc[beta])/2 - (u33q*zp*Csc[beta])/2))/c + cov[beta^2]*(-(u13q*zp*Cot[beta]^2)/2 - (u11q*zp*Cot[beta]*Csc[beta])/4 - (u33q*zp*Cot[beta]*Csc[beta])/4 + (a*((u13q*xp*Cot[beta]*Csc[beta])/2 + (u11q*xp*Csc[beta]^2)/4 + (u33q*xp*Csc[beta]^2)/4))/c) "Cov[z(p) , U^22(q) ]" cov[u22q, zp] "Cov[ z(p) , U^23(q) ]" cov[u23q, zp] + (cov[beta, c]*(-(u23q*zp*Cot[beta])/2 - (u12q*zp*Csc[beta])/2))/c + cov[beta^2]*(-(u23q*zp*Cot[beta]^2)/4 - (u12q*zp*Cot[beta]*Csc[beta])/4 + (a*((u23q*xp*Cot[beta]*Csc[beta])/4 + (u12q*xp*Csc[beta]^2)/4))/c) "Cov[ z(p) , U^33(q) ]" cov[u33q, zp] + (cov[beta, c]*(-(u33q*zp*Cot[beta]) - u13q*zp*Csc[beta]))/c + cov[beta^2]*(-(u33q*zp*Cot[beta]^2)/2 - (u13q*zp*Cot[beta]*Csc[beta])/2 + (a*((u33q*xp*Cot[beta]*Csc[beta])/2 + (u13q*xp*Csc[beta]^2)/2))/c) "----------------- Monoclinic Calculation: VcV with ADPs parameterised by Beta(ij) ----------------" "Cov[ Beta^11(p) , Beta^11(q) ]" (4*u11p*u11q*cov[a^2])/a^2 + cov[u11p, u11q] + cov[a, beta]*((-2*c*u11q*u13p*Csc[beta])/a^2 + u11p*((4*u11q*Cot[beta])/a - (2*c*u13q*Csc[beta])/a^2)) + cov[beta^2]*(u11p*(u11q*Cot[beta]^2 - (c*u13q*Cot[beta]*Csc[beta])/a) + u13p*(-((c*u11q*Cot[beta]*Csc[beta])/a) + (c^2*u13q*Csc[beta]^2)/a^2)) "Cov[ Beta^12(p) , Beta^12(q) ]" (u12p*u12q*cov[a^2])/a^2 + (u12p*u12q*cov[b^2])/b^2 + (2*u12p*u12q*cov[a, b])/(a*b) + cov[u12p, u12q] + cov[a, beta]*(-(c*u12q*u23p*Csc[beta])/(2*a^2) + u12p*((u12q*Cot[beta])/a - (c*u23q*Csc[beta])/(2*a^2))) + cov[b, beta]*(-(c*u12q*u23p*Csc[beta])/(2*a*b) + u12p*((u12q*Cot[beta])/b - (c*u23q*Csc[beta])/(2*a*b))) + cov[beta^2]*(u12p*((u12q*Cot[beta]^2)/4 - (c*u23q*Cot[beta]*Csc[beta])/ (4*a)) + u23p*(-(c*u12q*Cot[beta]*Csc[beta])/(4*a) + (c^2*u23q*Csc[beta]^2)/(4*a^2))) "Cov[ Beta^13(p) , Beta^13(q) ]" (u13p*u13q*cov[a^2])/a^2 + (u13p*u13q*cov[c^2])/c^2 + (2*u13p*u13q*cov[a, c])/(a*c) + cov[u13p, u13q] + cov[beta, c]*(-(a*u11p*u13q*Csc[beta])/(2*c^2) - (u13q*u33p*Csc[beta])/(2*a) + u13p*((2*u13q*Cot[beta])/c - (a*u11q*Csc[beta])/(2*c^2) - (u33q*Csc[beta])/(2*a))) + cov[a, beta]*(-(u11p*u13q*Csc[beta])/(2*c) - (c*u13q*u33p*Csc[beta])/ (2*a^2) + u13p*((2*u13q*Cot[beta])/a - (u11q*Csc[beta])/(2*c) - (c*u33q*Csc[beta])/(2*a^2))) + cov[beta^2]*(u13p*(u13q*Cot[beta]^2 - (a*u11q*Cot[beta]*Csc[beta])/(2*c) - (c*u33q*Cot[beta]*Csc[beta])/(2*a)) + u11p*(-(a*u13q*Cot[beta]*Csc[beta])/(2*c) + (a^2*u11q*Csc[beta]^2)/ (4*c^2) + (u33q*Csc[beta]^2)/4) + u33p*(-(c*u13q*Cot[beta]*Csc[beta])/(2*a) + (u11q*Csc[beta]^2)/4 + (c^2*u33q*Csc[beta]^2)/(4*a^2))) "Cov[ Beta^22(p) , Beta^22(q) ]" (4*u22p*u22q*cov[b^2])/b^2 + cov[u22p, u22q] "Cov[ Beta^23(p) , Beta^23(q) ]" (u23p*u23q*cov[b^2])/b^2 + (u23p*u23q*cov[c^2])/c^2 + (2*u23p*u23q*cov[b, c])/(b*c) + cov[u23p, u23q] + cov[beta, c]*(-(a*u12p*u23q*Csc[beta])/(2*c^2) + u23p*((u23q*Cot[beta])/c - (a*u12q*Csc[beta])/(2*c^2))) + cov[b, beta]*(-(a*u12p*u23q*Csc[beta])/(2*b*c) + u23p*((u23q*Cot[beta])/b - (a*u12q*Csc[beta])/(2*b*c))) + cov[beta^2]*(u23p*((u23q*Cot[beta]^2)/4 - (a*u12q*Cot[beta]*Csc[beta])/ (4*c)) + u12p*(-(a*u23q*Cot[beta]*Csc[beta])/(4*c) + (a^2*u12q*Csc[beta]^2)/(4*c^2))) "Cov[ Beta^33(p) , Beta^33(q) ]" (4*u33p*u33q*cov[c^2])/c^2 + cov[u33p, u33q] + cov[beta, c]*((-2*a*u13p*u33q*Csc[beta])/c^2 + u33p*((4*u33q*Cot[beta])/c - (2*a*u13q*Csc[beta])/c^2)) + cov[beta^2]*(u33p*(u33q*Cot[beta]^2 - (a*u13q*Cot[beta]*Csc[beta])/c) + u13p*(-((a*u33q*Cot[beta]*Csc[beta])/c) + (a^2*u13q*Csc[beta]^2)/c^2)) "Cov[ Beta^11(p) , Beta^12(q) ]" (2*u11p*u12q*cov[a^2])/a^2 + (2*u11p*u12q*cov[a, b])/(a*b) + cov[u11p, u12q] + cov[b, beta]*((u11p*u12q*Cot[beta])/b - (c*u12q*u13p*Csc[beta])/(a*b)) + cov[a, beta]*(-((c*u12q*u13p*Csc[beta])/a^2) + u11p*((2*u12q*Cot[beta])/a - (c*u23q*Csc[beta])/a^2)) + cov[beta^2]*(u11p*((u12q*Cot[beta]^2)/2 - (c*u23q*Cot[beta]*Csc[beta])/ (2*a)) + u13p*(-(c*u12q*Cot[beta]*Csc[beta])/(2*a) + (c^2*u23q*Csc[beta]^2)/(2*a^2))) "Cov[ Beta^11(p) , Beta^13(q) ]" (2*u11p*u13q*cov[a^2])/a^2 + (2*u11p*u13q*cov[a, c])/(a*c) + cov[u11p, u13q] + cov[beta, c]*((u11p*u13q*Cot[beta])/c - (u13p*u13q*Csc[beta])/a) + cov[a, beta]*(-((c*u13p*u13q*Csc[beta])/a^2) + u11p*((3*u13q*Cot[beta])/a - (u11q*Csc[beta])/c - (c*u33q*Csc[beta])/a^2)) + cov[beta^2]*(u11p*(u13q*Cot[beta]^2 - (a*u11q*Cot[beta]*Csc[beta])/(2*c) - (c*u33q*Cot[beta]*Csc[beta])/(2*a)) + u13p*(-((c*u13q*Cot[beta]*Csc[beta])/a) + (u11q*Csc[beta]^2)/2 + (c^2*u33q*Csc[beta]^2)/(2*a^2))) "Cov[ Beta^11(p) , Beta^22(q) ]" (4*u11p*u22q*cov[a, b])/(a*b) + cov[u11p, u22q] + cov[b, beta]*((2*u11p*u22q*Cot[beta])/b - (2*c*u13p*u22q*Csc[beta])/(a*b)) "Cov[ Beta^11(p) , Beta^23(q) ]" (2*u11p*u23q*cov[a, b])/(a*b) + (2*u11p*u23q*cov[a, c])/(a*c) + cov[u11p, u23q] + u11p*cov[a, beta]*((u23q*Cot[beta])/a - (u12q*Csc[beta])/c) + cov[beta, c]*((u11p*u23q*Cot[beta])/c - (u13p*u23q*Csc[beta])/a) + cov[b, beta]*((u11p*u23q*Cot[beta])/b - (c*u13p*u23q*Csc[beta])/(a*b)) + cov[beta^2]*(u11p*((u23q*Cot[beta]^2)/2 - (a*u12q*Cot[beta]*Csc[beta])/ (2*c)) + u13p*(-(c*u23q*Cot[beta]*Csc[beta])/(2*a) + (u12q*Csc[beta]^2)/2)) "Cov[ Beta^11(p) , Beta^33(q) ]" (4*u11p*u33q*cov[a, c])/(a*c) + cov[u11p, u33q] + u11p*cov[a, beta]*((2*u33q*Cot[beta])/a - (2*u13q*Csc[beta])/c) + cov[beta, c]*((2*u11p*u33q*Cot[beta])/c - (2*u13p*u33q*Csc[beta])/a) + cov[beta^2]*(u11p*(u33q*Cot[beta]^2 - (a*u13q*Cot[beta]*Csc[beta])/c) + u13p*(-((c*u33q*Cot[beta]*Csc[beta])/a) + u13q*Csc[beta]^2)) "Cov[ Beta^12(p) , Beta^13(q) ]" (u12p*u13q*cov[a^2])/a^2 + (u12p*u13q*cov[a, b])/(a*b) + (u12p*u13q*cov[a, c])/(a*c) + (u12p*u13q*cov[b, c])/(b*c) + cov[u12p, u13q] + cov[beta, c]*((u12p*u13q*Cot[beta])/(2*c) - (u13q*u23p*Csc[beta])/(2*a)) + u12p*cov[b, beta]* ((u13q*Cot[beta])/b - (a*u11q*Csc[beta])/(2*b*c) - (c*u33q*Csc[beta])/(2*a*b)) + cov[a, beta]*(-(c*u13q*u23p*Csc[beta])/(2*a^2) + u12p*((3*u13q*Cot[beta])/(2*a) - (u11q*Csc[beta])/(2*c) - (c*u33q*Csc[beta])/(2*a^2))) + cov[beta^2]*(u12p*((u13q*Cot[beta]^2)/2 - (a*u11q*Cot[beta]*Csc[beta])/ (4*c) - (c*u33q*Cot[beta]*Csc[beta])/(4*a)) + u23p*(-(c*u13q*Cot[beta]*Csc[beta])/(2*a) + (u11q*Csc[beta]^2)/4 + (c^2*u33q*Csc[beta]^2)/(4*a^2))) "Cov[ Beta^12(p) , Beta^22(q) ]" (2*u12p*u22q*cov[b^2])/b^2 + (2*u12p*u22q*cov[a, b])/(a*b) + cov[u12p, u22q] + cov[b, beta]*((u12p*u22q*Cot[beta])/b - (c*u22q*u23p*Csc[beta])/(a*b)) "Cov[ Beta^12(p) , Beta^23(q) ]" (u12p*u23q*cov[b^2])/b^2 + (u12p*u23q*cov[a, b])/(a*b) + (u12p*u23q*cov[a, c])/(a*c) + (u12p*u23q*cov[b, c])/(b*c) + cov[u12p, u23q] + u12p*cov[a, beta]*((u23q*Cot[beta])/(2*a) - (u12q*Csc[beta])/(2*c)) + cov[beta, c]*((u12p*u23q*Cot[beta])/(2*c) - (u23p*u23q*Csc[beta])/(2*a)) + cov[b, beta]*(-(c*u23p*u23q*Csc[beta])/(2*a*b) + u12p*((u23q*Cot[beta])/b - (a*u12q*Csc[beta])/(2*b*c))) + cov[beta^2]*(u12p*((u23q*Cot[beta]^2)/4 - (a*u12q*Cot[beta]*Csc[beta])/ (4*c)) + u23p*(-(c*u23q*Cot[beta]*Csc[beta])/(4*a) + (u12q*Csc[beta]^2)/4)) "Cov[ Beta^12(p) , Beta^33(q) ]" (2*u12p*u33q*cov[a, c])/(a*c) + (2*u12p*u33q*cov[b, c])/(b*c) + cov[u12p, u33q] + u12p*cov[a, beta]*((u33q*Cot[beta])/a - (u13q*Csc[beta])/c) + u12p*cov[b, beta]*((u33q*Cot[beta])/b - (a*u13q*Csc[beta])/(b*c)) + cov[beta, c]*((u12p*u33q*Cot[beta])/c - (u23p*u33q*Csc[beta])/a) + cov[beta^2]* (u12p*((u33q*Cot[beta]^2)/2 - (a*u13q*Cot[beta]*Csc[beta])/(2*c)) + u23p*(-(c*u33q*Cot[beta]*Csc[beta])/(2*a) + (u13q*Csc[beta]^2)/2)) "Cov[ Beta^13(p) , Beta^22(q) ]" (2*u13p*u22q*cov[a, b])/(a*b) + (2*u13p*u22q*cov[b, c])/(b*c) + cov[u13p, u22q] + cov[b, beta]*((2*u13p*u22q*Cot[beta])/b - (a*u11p*u22q*Csc[beta])/(b*c) - (c*u22q*u33p*Csc[beta])/(a*b)) "Cov[ Beta^13(p) , Beta^23(q) ]" (u13p*u23q*cov[c^2])/c^2 + (u13p*u23q*cov[a, b])/(a*b) + (u13p*u23q*cov[a, c])/(a*c) + (u13p*u23q*cov[b, c])/(b*c) + cov[u13p, u23q] + u13p*cov[a, beta]*((u23q*Cot[beta])/(2*a) - (u12q*Csc[beta])/(2*c)) + cov[b, beta]*((u13p*u23q*Cot[beta])/b - (a*u11p*u23q*Csc[beta])/(2*b*c) - (c*u23q*u33p*Csc[beta])/(2*a*b)) + cov[beta, c]*(-(a*u11p*u23q*Csc[beta])/(2*c^2) - (u23q*u33p*Csc[beta])/(2*a) + u13p*((3*u23q*Cot[beta])/(2*c) - (a*u12q*Csc[beta])/(2*c^2))) + cov[beta^2]*(u13p*((u23q*Cot[beta]^2)/2 - (a*u12q*Cot[beta]*Csc[beta])/ (2*c)) + u33p*(-(c*u23q*Cot[beta]*Csc[beta])/(4*a) + (u12q*Csc[beta]^2)/4) + u11p*(-(a*u23q*Cot[beta]*Csc[beta])/(4*c) + (a^2*u12q*Csc[beta]^2)/(4*c^2))) "Cov[ Beta^13(p) , Beta^33(q) ]" (2*u13p*u33q*cov[c^2])/c^2 + (2*u13p*u33q*cov[a, c])/(a*c) + cov[u13p, u33q] + u13p*cov[a, beta]*((u33q*Cot[beta])/a - (u13q*Csc[beta])/c) + cov[beta, c]*(-((a*u11p*u33q*Csc[beta])/c^2) - (u33p*u33q*Csc[beta])/a + u13p*((3*u33q*Cot[beta])/c - (a*u13q*Csc[beta])/c^2)) + cov[beta^2]*(u13p*(u33q*Cot[beta]^2 - (a*u13q*Cot[beta]*Csc[beta])/c) + u33p*(-(c*u33q*Cot[beta]*Csc[beta])/(2*a) + (u13q*Csc[beta]^2)/2) + u11p*(-(a*u33q*Cot[beta]*Csc[beta])/(2*c) + (a^2*u13q*Csc[beta]^2)/ (2*c^2))) "Cov[ Beta^22(p) , Beta^23(q) ]" (2*u22p*u23q*cov[b^2])/b^2 + (2*u22p*u23q*cov[b, c])/(b*c) + cov[u22p, u23q] + u22p*cov[b, beta]*((u23q*Cot[beta])/b - (a*u12q*Csc[beta])/(b*c)) "Cov[ Beta^22(p) , Beta^33(q) ]" (4*u22p*u33q*cov[b, c])/(b*c) + cov[u22p, u33q] + u22p*cov[b, beta]*((2*u33q*Cot[beta])/b - (2*a*u13q*Csc[beta])/(b*c)) "Cov[ Beta^23(p) , Beta^33(q) ]" (2*u23p*u33q*cov[c^2])/c^2 + (2*u23p*u33q*cov[b, c])/(b*c) + cov[u23p, u33q] + u23p*cov[b, beta]*((u33q*Cot[beta])/b - (a*u13q*Csc[beta])/(b*c)) + cov[beta, c]*(-((a*u12p*u33q*Csc[beta])/c^2) + u23p*((2*u33q*Cot[beta])/c - (a*u13q*Csc[beta])/c^2)) + cov[beta^2]*(u23p*((u33q*Cot[beta]^2)/2 - (a*u13q*Cot[beta]*Csc[beta])/ (2*c)) + u12p*(-(a*u33q*Cot[beta]*Csc[beta])/(2*c) + (a^2*u13q*Csc[beta]^2)/(2*c^2))) "---- Cov(x^k,Beta^ij) ----" "Cov[ x(p) , Beta^11(q) ]" (2*u11q*xp*cov[a^2])/a^2 + cov[u11q, xp] + cov[a, beta]*((2*u11q*xp*Cot[beta])/a + (c*(-(u13q*xp*Csc[beta]) - u11q*zp*Csc[beta]))/a^2) + cov[beta^2]*((u11q*xp*Cot[beta]^2)/2 + (c^2*u13q*zp*Csc[beta]^2)/(2*a^2) + (c*(-(u13q*xp*Cot[beta]*Csc[beta])/2 - (u11q*zp*Cot[beta]*Csc[beta])/2))/a) "Cov[ x(p) , Beta^12(q) ]" (u12q*xp*cov[a^2])/a^2 + (u12q*xp*cov[a, b])/(a*b) + cov[u12q, xp] + cov[b, beta]*((u12q*xp*Cot[beta])/(2*b) - (c*u12q*zp*Csc[beta])/(2*a*b)) + cov[a, beta]*((u12q*xp*Cot[beta])/a + (c*(-(u23q*xp*Csc[beta])/2 - (u12q*zp*Csc[beta])/2))/a^2) + cov[beta^2]*((u12q*xp*Cot[beta]^2)/4 + (c^2*u23q*zp*Csc[beta]^2)/(4*a^2) + (c*(-(u23q*xp*Cot[beta]*Csc[beta])/4 - (u12q*zp*Cot[beta]*Csc[beta])/4))/a) "Cov[ x(p) , Beta^13(q) ]" (u13q*xp*cov[a^2])/a^2 + (u13q*xp*cov[a, c])/(a*c) + cov[u13q, xp] + cov[beta, c]*((u13q*xp*Cot[beta])/(2*c) - (u13q*zp*Csc[beta])/(2*a)) + cov[a, beta]*((3*u13q*xp*Cot[beta])/(2*a) - (u11q*xp*Csc[beta])/(2*c) + (c*(-(u33q*xp*Csc[beta])/2 - (u13q*zp*Csc[beta])/2))/a^2) + cov[beta^2]*((u13q*xp*Cot[beta]^2)/2 - (a*u11q*xp*Cot[beta]*Csc[beta])/ (4*c) + (u11q*zp*Csc[beta]^2)/4 + (c^2*u33q*zp*Csc[beta]^2)/(4*a^2) + (c*(-(u33q*xp*Cot[beta]*Csc[beta])/4 - (u13q*zp*Cot[beta]*Csc[beta])/2))/a) "Cov[ x(p) , Beta^22(q) ]" (2*u22q*xp*cov[a, b])/(a*b) + cov[u22q, xp] + cov[b, beta]*((u22q*xp*Cot[beta])/b - (c*u22q*zp*Csc[beta])/(a*b)) "Cov[ x(p) , Beta^23(q) ]" (u23q*xp*cov[a, b])/(a*b) + (u23q*xp*cov[a, c])/(a*c) + cov[u23q, xp] + cov[a, beta]*((u23q*xp*Cot[beta])/(2*a) - (u12q*xp*Csc[beta])/(2*c)) + cov[beta, c]*((u23q*xp*Cot[beta])/(2*c) - (u23q*zp*Csc[beta])/(2*a)) + cov[b, beta]*((u23q*xp*Cot[beta])/(2*b) - (c*u23q*zp*Csc[beta])/(2*a*b)) + cov[beta^2]*((u23q*xp*Cot[beta]^2)/4 - (a*u12q*xp*Cot[beta]*Csc[beta])/ (4*c) - (c*u23q*zp*Cot[beta]*Csc[beta])/(4*a) + (u12q*zp*Csc[beta]^2)/4) "Cov[ x(p) , Beta^33(q) ]" (2*u33q*xp*cov[a, c])/(a*c) + cov[u33q, xp] + cov[a, beta]*((u33q*xp*Cot[beta])/a - (u13q*xp*Csc[beta])/c) + cov[beta, c]*((u33q*xp*Cot[beta])/c - (u33q*zp*Csc[beta])/a) + cov[beta^2]*((u33q*xp*Cot[beta]^2)/2 - (a*u13q*xp*Cot[beta]*Csc[beta])/ (2*c) - (c*u33q*zp*Cot[beta]*Csc[beta])/(2*a) + (u13q*zp*Csc[beta]^2)/2) "Cov[ y(p) , Beta^11(q) ]" (2*u11q*yp*cov[a, b])/(a*b) + cov[u11q, yp] + cov[b, beta]*((u11q*yp*Cot[beta])/b - (c*u13q*yp*Csc[beta])/(a*b)) "Cov[ y(p) , Beta^12(q) ]" (u12q*yp*cov[b^2])/b^2 + (u12q*yp*cov[a, b])/(a*b) + cov[u12q, yp] + cov[b, beta]*((u12q*yp*Cot[beta])/(2*b) - (c*u23q*yp*Csc[beta])/(2*a*b)) "Cov[ y(p) , Beta^13(q) ]" (u13q*yp*cov[a, b])/(a*b) + (u13q*yp*cov[b, c])/(b*c) + cov[u13q, yp] + cov[b, beta]*((u13q*yp*Cot[beta])/b - (a*u11q*yp*Csc[beta])/(2*b*c) - (c*u33q*yp*Csc[beta])/(2*a*b)) "Cov[ y(p) , Beta^22(q) ]" (2*u22q*yp*cov[b^2])/b^2 + cov[u22q, yp] "Cov[ y(p) , Beta^23(q) ]" (u23q*yp*cov[b^2])/b^2 + (u23q*yp*cov[b, c])/(b*c) + cov[u23q, yp] + cov[b, beta]*((u23q*yp*Cot[beta])/(2*b) - (a*u12q*yp*Csc[beta])/(2*b*c)) "Cov[ y(p) , Beta^33(q) ]" (2*u33q*yp*cov[b, c])/(b*c) + cov[u33q, yp] + cov[b, beta]*((u33q*yp*Cot[beta])/b - (a*u13q*yp*Csc[beta])/(b*c)) "Cov[ z(p) , Beta^11(q) ]" (2*u11q*zp*cov[a, c])/(a*c) + cov[u11q, zp] + cov[a, beta]*((u11q*zp*Cot[beta])/a - (u11q*xp*Csc[beta])/c) + cov[beta, c]*((u11q*zp*Cot[beta])/c - (u13q*zp*Csc[beta])/a) + cov[beta^2]*((u11q*zp*Cot[beta]^2)/2 - (a*u11q*xp*Cot[beta]*Csc[beta])/ (2*c) - (c*u13q*zp*Cot[beta]*Csc[beta])/(2*a) + (u13q*xp*Csc[beta]^2)/2) "Cov[ z(p) , Beta^12(q) ]" (u12q*zp*cov[a, c])/(a*c) + (u12q*zp*cov[b, c])/(b*c) + cov[u12q, zp] + cov[a, beta]*((u12q*zp*Cot[beta])/(2*a) - (u12q*xp*Csc[beta])/(2*c)) + cov[b, beta]*((u12q*zp*Cot[beta])/(2*b) - (a*u12q*xp*Csc[beta])/(2*b*c)) + cov[beta, c]*((u12q*zp*Cot[beta])/(2*c) - (u23q*zp*Csc[beta])/(2*a)) + cov[beta^2]*((u12q*zp*Cot[beta]^2)/4 - (a*u12q*xp*Cot[beta]*Csc[beta])/ (4*c) - (c*u23q*zp*Cot[beta]*Csc[beta])/(4*a) + (u23q*xp*Csc[beta]^2)/4) "Cov[ z(p) , Beta^13(q) ]" (u13q*zp*cov[c^2])/c^2 + (u13q*zp*cov[a, c])/(a*c) + cov[u13q, zp] + cov[a, beta]*((u13q*zp*Cot[beta])/(2*a) - (u13q*xp*Csc[beta])/(2*c)) + cov[beta, c]*((3*u13q*zp*Cot[beta])/(2*c) - (u33q*zp*Csc[beta])/(2*a) + (a*(-(u13q*xp*Csc[beta])/2 - (u11q*zp*Csc[beta])/2))/c^2) + cov[beta^2]*((u13q*zp*Cot[beta]^2)/2 - (c*u33q*zp*Cot[beta]*Csc[beta])/ (4*a) + (a^2*u11q*xp*Csc[beta]^2)/(4*c^2) + (u33q*xp*Csc[beta]^2)/4 + (a*(-(u13q*xp*Cot[beta]*Csc[beta])/2 - (u11q*zp*Cot[beta]*Csc[beta])/4))/c) "Cov[z(p) , Beta^22(q) ]" (2*u22q*zp*cov[b, c])/(b*c) + cov[u22q, zp] + cov[b, beta]*((u22q*zp*Cot[beta])/b - (a*u22q*xp*Csc[beta])/(b*c)) "Cov[ z(p) , Beta^23(q) ]" (u23q*zp*cov[c^2])/c^2 + (u23q*zp*cov[b, c])/(b*c) + cov[u23q, zp] + cov[b, beta]*((u23q*zp*Cot[beta])/(2*b) - (a*u23q*xp*Csc[beta])/(2*b*c)) + cov[beta, c]*((u23q*zp*Cot[beta])/c + (a*(-(u23q*xp*Csc[beta])/2 - (u12q*zp*Csc[beta])/2))/c^2) + cov[beta^2]*((u23q*zp*Cot[beta]^2)/4 + (a^2*u12q*xp*Csc[beta]^2)/(4*c^2) + (a*(-(u23q*xp*Cot[beta]*Csc[beta])/4 - (u12q*zp*Cot[beta]*Csc[beta])/4))/c) "Cov[ z(p) , Beta^33(q) ]" (2*u33q*zp*cov[c^2])/c^2 + cov[u33q, zp] + cov[beta, c]*((2*u33q*zp*Cot[beta])/c + (a*(-(u33q*xp*Csc[beta]) - u13q*zp*Csc[beta]))/c^2) + cov[beta^2]*((u33q*zp*Cot[beta]^2)/2 + (a^2*u13q*xp*Csc[beta]^2)/(2*c^2) + (a*(-(u33q*xp*Cot[beta]*Csc[beta])/2 - (u13q*zp*Cot[beta]*Csc[beta])/2))/c)