Page 38 - Intro to Tensor Calculus

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I 54. Show that every second order system T ij can be expressed as the sum of a symmetric system A ij and
skew-symmetric system B ij . Find A ij and B ij in terms of the components of T ij .

I 55. Consider the system A ijk , i,j,k =1, 2, 3, 4.
(a) How many components does this system have?
(b) Assume A ijk is skew-symmetric in the last pair of indices, how many independent components does this
system have?
(c) Assume that in addition to being skew-symmetric in the last pair of indices, A ijk + A jki + A kij =0 is
satisfied for all values of i, j, and k, then how many independent components does the system have?

~
r
I 56. (a) Write the equation of a line ~ = ~ 0 + t A in indicial form. (b) Write the equation of the plane
r
~n · (~ − ~ 0 ) = 0 in indicial form. (c) Write the equation of a general line in scalar form. (d) Write the
r
r
equation of a plane in scalar form. (e) Find the equation of the line defined by the intersection of the
planes 2x +3y +6z =12 and 6x +3y + z =6. (f) Find the equation of the plane through the points
(5, 3, 2), (3, 1, 5), (1, 3, 3). Find also the normal to this plane.

I 57. The angle 0 ≤ θ ≤ π between two skew lines in space is defined as the angle between their direction
vectors when these vectors are placed at the origin. Show that for two lines with direction numbers a i and
b i i =1, 2, 3, the cosine of the angle between these lines satisfies

a i b i
cos θ = √ √
a i a i b i b i

I 58. Let a ij = −a ji for i, j =1, 2,...,N and prove that for N odd det(a ij )= 0.
2
∂λ ∂ λ
I 59. Let λ = A ij x i x j where A ij = A ji and calculate (a) (b)
∂x m ∂x m ∂x k
I 60. Given an arbitrary nonzero vector U k , k =1, 2, 3, define the matrix elements a ij = e ijk U k ,where e ijk
is the e-permutation symbol. Determine if a ij is symmetric or skew-symmetric. Suppose U k is defined by
the above equation for arbitrary nonzero a ij ,thensolve for U k in terms of the a ij .
I 61. If A ij = A i B j 6=0 for all i, j values and A ij = A ji for i, j =1, 2,...,N, show that A ij = λB i B j
where λ is a constant. State what λ is.

I 62. Assume that A ijkm ,with i, j, k, m =1, 2, 3, is completely skew-symmetric. How many independent
components does this quantity have?

I 63. Consider R ijkm , i, j, k, m =1, 2, 3, 4. (a) How many components does this quantity have? (b) If
R ijkm = −R ijmk = −R jikm then how many independent components does R ijkm have? (c) If in addition
R ijkm = R kmij determine the number of independent components.

I 64. Let x i = a ij ¯x j , i, j =1, 2, 3 denote a change of variables from a barred system of coordinates to an
¯
¯
unbarred system of coordinates and assume that A i = a ij A j where a ij are constants, A i is a function of the
¯
∂A i
¯ x j variables and A j is a function of the x j variables. Calculate .
∂¯x m