3915.sagews

v = var("v")
r = var("r")
a = var("a")
theta = var("theta")
phi = var("phi")
A = (3-4*v)/r + a^2/(3*r^3)
B = (1+a^2/r^2)*(r^2/a^3)*sin(theta)*cos(phi)
u_x = A + B*sin(theta)*cos(phi)
u_y = B*sin(theta)*sin(phi)
u_z = B*cos(theta) 
M = matrix(SR, [[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)],[cos(theta)*cos(phi), cos(theta)*sin(phi), -sin(theta)],[-sin(phi), cos(phi), 0]])
u = matrix(SR, [[u_x],[u_y],[u_z]])

u_r = (M*u)[0]
u_theta = (M*u)[1]
u_phi = (M*u)[2]

e_r_r = u_r.diff(r)
e_theta_theta = (1/r)*u_theta.diff(theta)+u_r/r
e_phi_phi = (1/(r*sin(theta)))*u_phi.diff(phi)+(u_r/r)+u_theta*(cot(theta)/r)
e_r_phi = ((1/(r*sin(theta)))*u_r.diff(phi)-u_phi/r+u_phi.diff(r))/2
e_r_theta = (u_r.diff(theta)/r-u_theta/r+u_theta.diff(r))/2
e_phi_theta = (u_phi.diff(theta)/r-(u_phi*cot(theta))/r+u_theta.diff(phi)/(r*sin(theta)))/2

E1 = var("E1")
E2 = var("E2")
t_r_r = E1*((1-v)*e_r_r+v*(e_theta_theta))
t_theta_theta = E1*((1-v)*e_theta_theta+v*(e_r_r+e_phi_phi))
t_phi_phi = E1*((1-v)*e_phi_phi+v*(e_r_r+e_theta_theta))
t_r_theta = E2*e_r_theta
t_theta_phi = E2*e_phi_theta
t_r_phi = E2*e_r_phi

t_r_r

(-1/3*(3*(v - 1)*(2*(a^2/r^2 + 1)*r*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 2*sin(phi)^2*sin(theta)^3*cos(phi)/(a*r) + 2*(a^2/r^2 + 1)*r*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (2*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)^2/a^3 - 2*sin(theta)^2*cos(phi)^2/(a*r) + (4*v - 3)/r^2 - a^2/r^4)*sin(theta)*cos(phi) - 2*sin(theta)*cos(phi)*cos(theta)^2/(a*r)) - ((3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r - (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)^3*cos(theta)^2/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^3*cos(phi)/a^3 + 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r)*v)*E1)

t_theta_theta

(-1/3*((v - 1)*((3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r - (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)^3*cos(theta)^2/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^3*cos(phi)/a^3 + 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r) - (6*(a^2/r^2 + 1)*r*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 6*sin(phi)^2*sin(theta)^3*cos(phi)/(a*r) + 6*(a^2/r^2 + 1)*r*sin(theta)*cos(phi)*cos(theta)^2/a^3 + 3*(2*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)^2/a^3 - 2*sin(theta)^2*cos(phi)^2/(a*r) + (4*v - 3)/r^2 - a^2/r^4)*sin(theta)*cos(phi) - 6*sin(theta)*cos(phi)*cos(theta)^2/(a*r) + (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi)*cos(theta))*cot(theta)/r + (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^3/a^3 - (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi))/(r*sin(theta)))*v)*E1)

t_phi_phi

(-1/3*((v - 1)*((3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi)*cos(theta))*cot(theta)/r + (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^3/a^3 - (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi))/(r*sin(theta))) - (6*(a^2/r^2 + 1)*r*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 6*sin(phi)^2*sin(theta)^3*cos(phi)/(a*r) + 6*(a^2/r^2 + 1)*r*sin(theta)*cos(phi)*cos(theta)^2/a^3 + 3*(2*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)^2/a^3 - 2*sin(theta)^2*cos(phi)^2/(a*r) + (4*v - 3)/r^2 - a^2/r^4)*sin(theta)*cos(phi) - 6*sin(theta)*cos(phi)*cos(theta)^2/(a*r) + (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r - (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^3*cos(phi)/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)^3*cos(theta)^2/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^3*cos(phi)/a^3 + 6*(a^2/r^2 + 1)*r^2*sin(theta)*cos(phi)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(theta)*cos(phi))/r)*v)*E1)

t_r_theta

(1/6*(6*(a^2/r^2 + 1)*r*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 - 6*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/(a*r) - 6*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)*cos(theta)/a^3 + 3*(2*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)^2/a^3 - 2*sin(theta)^2*cos(phi)^2/(a*r) + (4*v - 3)/r^2 - a^2/r^4)*cos(phi)*cos(theta) + 6*sin(theta)^2*cos(phi)*cos(theta)/(a*r) - (3*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi)*cos(theta))/r + (9*(a^2/r^2 + 1)*r^2*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 + 6*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^3*cos(theta)/a^3 - 6*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)*cos(theta)/a^3 + 3*(a^2/r^2 + 1)*r^2*cos(phi)*cos(theta)^3/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*cos(phi)*cos(theta))/r)*E2)

t_theta_phi

(-1/6*((3*(a^2/r^2 + 1)*r^2*sin(phi)*sin(theta)^2*cos(phi)^2/a^3 - (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(phi))*cot(theta)/r + (3*(a^2/r^2 + 1)*r^2*sin(phi)^3*sin(theta)^2*cos(theta)/a^3 - 3*(a^2/r^2 + 1)*r^2*sin(phi)*sin(theta)^2*cos(theta)/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(phi)*cos(theta))/(r*sin(theta)))*E2)

t_r_phi

(1/6*(6*(a^2/r^2 + 1)*r*sin(phi)*sin(theta)^2*cos(phi)^2/a^3 - 6*sin(phi)*sin(theta)^2*cos(phi)^2/(a*r) - 3*(2*(a^2/r^2 + 1)*r*sin(theta)^2*cos(phi)^2/a^3 - 2*sin(theta)^2*cos(phi)^2/(a*r) + (4*v - 3)/r^2 - a^2/r^4)*sin(phi) - (3*(a^2/r^2 + 1)*r^2*sin(phi)*sin(theta)^2*cos(phi)^2/a^3 - (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(phi))/r - (3*(a^2/r^2 + 1)*r^2*sin(phi)^3*sin(theta)^3/a^3 + 3*(a^2/r^2 + 1)*r^2*sin(phi)*sin(theta)*cos(theta)^2/a^3 + (3*(a^2/r^2 + 1)*r^2*sin(theta)^2*cos(phi)^2/a^3 - 3*(4*v - 3)/r + a^2/r^3)*sin(phi)*sin(theta))/(r*sin(theta)))*E2)

T_r_r = t_r_r.substitute(r=a)
T_theta_theta = t_theta_theta.substitute(r=a)
T_phi_phi = t_phi_phi.substitute(r=a)
T_r_theta = t_r_theta.substitute(r=a)
T_theta_phi = t_theta_phi(r=a)
T_r_phi = t_r_phi(r=a)

T_r_r

(-1/3*(3*(v - 1)*(2*sin(phi)^2*sin(theta)^3*cos(phi)/a^2 + (2*sin(theta)^2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta)*cos(phi) + 2*sin(theta)*cos(phi)*cos(theta)^2/a^2) - ((6*sin(phi)^2*sin(theta)^3*cos(phi)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a)/a - (6*sin(phi)^2*sin(theta)^3*cos(phi)/a - 12*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a - 12*sin(theta)*cos(phi)^3*cos(theta)^2/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) - 6*sin(theta)^3*cos(phi)/a + 12*sin(theta)*cos(phi)*cos(theta)^2/a)/a)*v)*E1)

T_theta_theta

(-1/3*((v - 1)*((6*sin(phi)^2*sin(theta)^3*cos(phi)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a)/a - (6*sin(phi)^2*sin(theta)^3*cos(phi)/a - 12*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a - 12*sin(theta)*cos(phi)^3*cos(theta)^2/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) - 6*sin(theta)^3*cos(phi)/a + 12*sin(theta)*cos(phi)*cos(theta)^2/a)/a) - (6*sin(phi)^2*sin(theta)^3*cos(phi)/a^2 + 3*(2*sin(theta)^2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a^2 + (6*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi)*cos(theta) - 6*sin(theta)^2*cos(phi)*cos(theta)/a)*cot(theta)/a + (6*sin(phi)^2*sin(theta)^3*cos(phi)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a)/a + (6*sin(theta)^2*cos(phi)^3/a - (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi))/(a*sin(theta)))*v)*E1)

T_phi_phi

(-1/3*((v - 1)*((6*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi)*cos(theta) - 6*sin(theta)^2*cos(phi)*cos(theta)/a)*cot(theta)/a + (6*sin(phi)^2*sin(theta)^3*cos(phi)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a)/a + (6*sin(theta)^2*cos(phi)^3/a - (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi))/(a*sin(theta))) - (6*sin(phi)^2*sin(theta)^3*cos(phi)/a^2 + 3*(2*sin(theta)^2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a^2 + (6*sin(phi)^2*sin(theta)^3*cos(phi)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) + 6*sin(theta)*cos(phi)*cos(theta)^2/a)/a - (6*sin(phi)^2*sin(theta)^3*cos(phi)/a - 12*sin(phi)^2*sin(theta)*cos(phi)*cos(theta)^2/a - 12*sin(theta)*cos(phi)^3*cos(theta)^2/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta)*cos(phi) - 6*sin(theta)^3*cos(phi)/a + 12*sin(theta)*cos(phi)*cos(theta)^2/a)/a)*v)*E1)

T_r_theta

(1/6*(6*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a^2 + 3*(2*sin(theta)^2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*cos(phi)*cos(theta) - 6*sin(theta)^2*cos(phi)*cos(theta)/a^2 - (6*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi)*cos(theta) - 6*sin(theta)^2*cos(phi)*cos(theta)/a)/a + (18*sin(phi)^2*sin(theta)^2*cos(phi)*cos(theta)/a + 12*sin(theta)^2*cos(phi)^3*cos(theta)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi)*cos(theta) - 12*sin(theta)^2*cos(phi)*cos(theta)/a + 6*cos(phi)*cos(theta)^3/a)/a)*E2)

T_r_phi

(1/6*(6*sin(phi)*sin(theta)^2*cos(phi)^2/a^2 - 3*(2*sin(theta)^2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(phi) - (6*sin(phi)*sin(theta)^2*cos(phi)^2/a - (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi))/a - (6*sin(phi)^3*sin(theta)^3/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi)*sin(theta) + 6*sin(phi)*sin(theta)*cos(theta)^2/a)/(a*sin(theta)))*E2)

T_theta_phi

(-1/6*((6*sin(phi)*sin(theta)^2*cos(phi)^2/a - (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi))*cot(theta)/a + (6*sin(phi)^3*sin(theta)^2*cos(theta)/a + (6*sin(theta)^2*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi)*cos(theta) - 6*sin(phi)*sin(theta)^2*cos(theta)/a)/(a*sin(theta)))*E2)

T_r_r.substitute(phi=0)

(-1/3*(3*(v - 1)*((2*sin(theta)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta) + 2*sin(theta)*cos(theta)^2/a^2) - (((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) + 6*sin(theta)*cos(theta)^2/a)/a - ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) - 6*sin(theta)^3/a)/a)*v)*E1)

T_r_r.substitute(theta=pi/2)

(-1/3*(3*(v - 1)*((2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*cos(phi) + 2*sin(phi)^2*cos(phi)/a^2) - (((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a)/a - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a - 6*cos(phi)/a)/a)*v)*E1)

T_theta_theta.substitute(phi=0)

(-1/3*((v - 1)*(((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) + 6*sin(theta)*cos(theta)^2/a)/a - ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) - 6*sin(theta)^3/a)/a) - (3*(2*sin(theta)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta) + ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*cos(theta) - 6*sin(theta)^2*cos(theta)/a)*cot(theta)/a + 6*sin(theta)*cos(theta)^2/a^2 + ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) + 6*sin(theta)*cos(theta)^2/a)/a + (3*(4*v - 3)/a - 1/a)/(a*sin(theta)))*v)*E1)

T_theta_theta.substitute(theta=pi/2)

(-1/3*((v - 1)*(((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a)/a - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a - 6*cos(phi)/a)/a) - (3*(2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*cos(phi) + 6*sin(phi)^2*cos(phi)/a^2 - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) - 6*cos(phi)^3/a)/a + ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a)/a)*v)*E1)

T_phi_phi.substitute(phi=0)

(-1/3*((v - 1)*(((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*cos(theta) - 6*sin(theta)^2*cos(theta)/a)*cot(theta)/a + ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) + 6*sin(theta)*cos(theta)^2/a)/a + (3*(4*v - 3)/a - 1/a)/(a*sin(theta))) - (3*(2*sin(theta)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(theta) + 6*sin(theta)*cos(theta)^2/a^2 + ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) + 6*sin(theta)*cos(theta)^2/a)/a - ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*sin(theta) - 6*sin(theta)^3/a)/a)*v)*E1)

T_phi_phi.substitute(theta=pi/2)

(1/3*((v - 1)*(((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) - 6*cos(phi)^3/a)/a - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a)/a) + (3*(2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*cos(phi) + 6*sin(phi)^2*cos(phi)/a^2 + ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a)/a - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*cos(phi) + 6*sin(phi)^2*cos(phi)/a - 6*cos(phi)/a)/a)*v)*E1)

T_r_theta.substitute(phi=0)

(1/6*(3*(2*sin(theta)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*cos(theta) - 6*sin(theta)^2*cos(theta)/a^2 + ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*cos(theta) + 6*cos(theta)^3/a)/a - ((6*sin(theta)^2/a - 3*(4*v - 3)/a + 1/a)*cos(theta) - 6*sin(theta)^2*cos(theta)/a)/a)*E2)

T_r_theta.substitute(theta=pi/2)

(0)

T_r_phi.substitute(phi=0)

(0)

T_r_phi.substitute(theta=pi/2)

(-1/6*(3*(2*cos(phi)^2/a^2 + (4*v - 3)/a^2 - 1/a^2)*sin(phi) - 6*sin(phi)*cos(phi)^2/a^2 - ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi) - 6*sin(phi)*cos(phi)^2/a)/a + ((6*cos(phi)^2/a - 3*(4*v - 3)/a + 1/a)*sin(phi) + 6*sin(phi)^3/a)/a)*E2)

T_theta_phi.substitute(phi=0)

(0)

T_theta_phi.substitute(theta=pi/2)

(0)