Page 37 - Intro to Tensor Calculus

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I 42. Determine if the following statement is true or false. Justify your answer. e ijk A i B j C k = e ijk A j B k C i .

I 43. Let a ij ,i, j =1, 2 denote the components of a 2 × 2 matrix A, which are functions of time t.

a 11 a 12
(a) Expand both |A| = e ij a i1 a j2 and |A| = to verify that these representations are the same.
a 21 a 22
(b) Verify the equivalence of the derivative relations

d|A| da i1 da j2 d|A| da 11 da 12 a 11 a 12
dt
= e ij a j2 + e ij a i1 and = dt + da 21 da 22
dt dt dt dt a 21 a 22 dt dt
(c) Let a ij ,i, j =1, 2, 3 denote the components of a 3 × 3 matrix A, which are functions of time t. Develop
appropriate relations, expand them and verify, similar to parts (a) and (b) above, the representation of
a determinant and its derivative.

1
2
3
I 44. For f = f(x ,x ,x )and φ = φ(f) differentiable scalar functions, use the indicial notation to find a
formula to calculate grad φ.
~
~
I 45. Use the indicial notation to prove (a) ∇×∇φ = 0 (b) ∇· ∇× A =0
I 46. If A ij is symmetric and B ij is skew-symmetric, i, j =1, 2, 3, then calculate C = A ij B ij .
1
2
3
3
1
2
I 47. Assume A ij = A ij (x , x , x )and A ij = A ij (x ,x ,x )for i, j =1, 2, 3 are related by the expression
i
∂x ∂x j ∂A mn
A mn = A ij m n . Calculate the derivative k .
∂x ∂x ∂x
I 48. Prove that if any two rows (or two columns) of a matrix are interchanged, then the value of the
determinant of the matrix is multiplied by minus one. Construct your proof using 3 × 3 matrices.
I 49. Prove that if two rows (or columns) of a matrix are proportional, then the value of the determinant
of the matrix is zero. Construct your proof using 3 × 3 matrices.

I 50. Prove that if a row (or column) of a matrix is altered by adding some constant multiple of some other
row (or column), then the value of the determinant of the matrix remains unchanged. Construct your proof
using 3 × 3 matrices.

I 51. Simplify the expression φ = e ijk e `mnA i` A jm A kn .

I 52. Let A ijk denote a third order system where i, j, k =1, 2. (a) How many components does this system
have? (b) Let A ijk be skew-symmetric in the last pair of indices, how many independent components does
the system have?

I 53. Let A ijk denote a third order system where i, j, k =1, 2, 3. (a) How many components does this
system have? (b) In addition let A ijk = A jik and A ikj = −A ijk and determine the number of distinct
nonzero components for A ijk .