Page 129 - Intro to Tensor Calculus

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I 13.
√
(a) Show g is a relative scalar of weight +1.
(b) Use the results from problem 9(c) and problem 44, Exercise 1.4, to show that
√
√ ∂ g m √
( g) ,k = − g =0.
∂x k km

m ∂ √ 1 ∂g
(c) Show that = ln( g)= .
km ∂x k 2g ∂x k

m ∂ √ 1 ∂g
I 14. Use the result from problem 9(b) to show = ln( g)= .
km ∂x k 2g ∂x k
√
Hint: Expand the covariant derivative rst,p and then substitute rst = ge rst . Simplify by inner
e rst
multiplication with √ and note the Exercise 1.1, problem 26.
g
I 15. Calculate the covariant derivative A i ,m and then contract on m and i to show that
1 ∂ √
i i
A = √ gA .
,i i
g ∂x
1 ∂ √ ij i pq
I 16. Show √ gg + g =0. Hint: See problem 14.
g ∂x j pq
I 17. Prove that the covariant derivative of a sum equals the sum of the covariant derivatives.
Hint: Assume C i = A i + B i and write out the covariant derivative for C i,j .

i
i
I 18. Let C = A B j and prove that the covariant derivative of a product equals the first term times the
j
covariant derivative of the second term plus the second term times the covariant derivative of the first term.
α
∂x ∂x β
¯
I 19. Start with the transformation law A ij = A αβ and take an ordinary derivative of both sides
i
∂¯x ∂¯x j
k
with respect to ¯x and hence derive the relation for A ij,k given in (1.4.30).
i
∂x ∂x j
ij
I 20. Start with the transformation law A = A ¯ αβ and take an ordinary derivative of both sides
α
∂¯x ∂¯x β
k
with respect to x and hence derive the relation for A ij given in (1.4.30).
,k
I 21. Find the covariant derivatives of
(a) A ijk (b) A ij (c) A i (d)
k jk A ijk
i
i
I 22. Find the intrinsic derivative along the curve x = x (t), i =1,... ,N for
(a) A ijk (b) A ij (c) A i (d) A ijk
k jk

I 23.
i ~
k ~
~
~
(a) Assume A = A E i and show that dA = A i dx E i .
,k
~
~
k ~ i
~ i
(b) Assume A = A i E and show that dA = A i,k dx E .