PHYS 326 Problem Set #9
Problem 1 : The components of two vectors, A and B , and a second-order tensor, Τ , are given in one
 1
 0
 
 0
 0
 
 2
3 0
 0

0 2 
coordinate system by matrices A =  0  , B =  1  , T =  3 4 0  . In a second coordinate system,
 3
 –1 
obtained from the first by rotation, the components of A and B are A′ = 1  0  , B′ = 1  0  . Find the
2 1
2  3
 
 
components of Τ in this new coordinate system and hence evaluate, with a minimum of calculation, Tij Tji .
Problem 2 : A symmetric second-order Cartesian tensor is defined by Tij = δij − 3xi xj .
Evaluate the following surface integrals, each taken over the surface of the unit sphere:
(i)
S
Tij da ; (ii)
S
Tik Tkj da ; (iii)
S
xiTjk da .
Problem 3 : In a certain system of units the electromagnetic stress tensor Tij is given by
Tij = Ei Ej + Bi Bj − 1 δij(Ek Ek + Bk Bk ) = Ei Ej + Bi Bj − 1 δij(E 2 + B 2 ) , where the electric and magnetic fields,
2
2
E and B , are first-order tensors (vectors). (i) Show that Tij is a second-order tensor. [Hint: You must
show that Tij transforms as T ′ij = Lim Lin Tmn .] (ii) Consider a situation in which | E | = | B | (so that E 2 =
B 2 ) but the directions of E and B are not parallel. Show that E ± B are principal axes of the stress tensor
and find the corresponding principal values. Determine the third principal axis and its corresponding
principal value. [Hint: Define a vector υi = Ei ± Bi and show that υi is an eigenvector of Tij , that is Tij υj =
λi υi (sum over j is implied but there is no sum over i !) What are the eigenvalues (principal values) λi ? The
third principal axis must be orthogonal to both of the principal axes E + B and E − B .]
Answers for part (ii) : λ1 = E . B , λ2 = − E . B , λ3 = − E 2 (or λ3 = − B 2 because E 2 = B 2 is given.)
Problem 4 : Let the Cartesian coordinate system (x 1 = x , x 2 = y) and another coordinate system (u1 = u ,
u2 = υ) are related by x = u(1 − υ) , y = u 2υ – υ2 . Find : (i) the covariant basis vectors e 1 ≡ e u and
e 2 ≡ e υ . (Express these vectors in terms of Cartesian unit vectors x̂ and ŷ .) (ii) the scale factors h1 ≡
hu and h2 ≡ hυ , (iii) contravariant basis vectors e 1 ≡ e u and e 2 ≡ e υ (in terms of x̂ and ŷ ), (iv) the
covariant metric elements gij and contravariant metric elements, gij , (v) infinitesimal displacement vector
d
, (vi) arclength element d 2 in the (u , υ) coordinate system, (vii) the area element vector d a ,
Problem 5 : (i) Find the covariant and contravariant basis vectors e i , e i , respectively (in terms of the
Cartesian unit vectors x̂ , ŷ , and ẑ ) of the spherical coordinates (u1 , u2 , u3 ) = (r , θ , φ) .
(ii) Find the covariant and contravariant metric elements gij , gij , respectively.
(iii) Calculate the Jacobian of the transformation from Cartesian coordinates to the spherical coordinates
дr /дx дr /дy дr /дz
д(r , θ , φ )
J dx dy dz = dr dθ dφ , where J =
= дθ /дx дθ /дy дθ /дz .
д( x , y , z )
дφ /дx дφ /дy дφ /дz
Note that it is much easier to calculate the inverse Jacobian J −1 :
дx /дr дx /дθ дx /дφ
д( x , y , z )
dx dy dz = J −1dr dθ dφ , where J −1 =
= дy /дr дy /дθ дy /дφ .
д(r , θ , φ )
дz /дr дz /дθ дz /дφ