Page 133 - Intro to Tensor Calculus

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I 46. Verify the relations
∂g ij ∂g nm
= −g mj g ni
∂x k ∂x k
∂g in mn ij ∂g jm
= −g g
∂x k ∂x k

1 ∂ √ ijk
jk
ijk
I 47. Assume that B is an absolute tensor. Is the quantity T = √ gB a tensor? Justify
g ∂x i
your answer. If your answer is “no”, explain your answer and determine if there any conditions you can
impose upon B ijk such that the above quantity will be a tensor?

I 48. The e-permutation symbol can be used to define various vector products. Let A i ,B i ,C i ,D i
i =1,... ,N denote vectors, then expand and verify the following products:
(a) In two dimensions
R =e ij A i B j a scalar determinant.

R i =e ij A j a vector (rotation).
(b) In three dimensions
S =e ijk A i B j C k a scalar determinant.
a vector cross product.
S i =e ijk B j C k
a skew-symmetric matrix
S ij =e ijk C k
(c) In four dimensions

a scalar determinant.
T =e ijkm A i B j C k D m
4-dimensional cross product.
T i =e ijkm B j C k D m
skew-symmetric matrix.
T ij =e ijkm C k D m
T ijk =e ikm D m skew-symmetric tensor.
with similar products in higher dimensions.

I 49. Expand the curl operator for:
(a) Two dimensions B = e ij A j,i
(b) Three dimensions B i = e ijk A k,j
(c) Four dimensions B ij = e ijkm A m,k