All Chapters Covered
SOLUTIONS
, CHAPTER 1
1.1. Show that ∂xi √
(i) = δ and (ii) R = xx ,
∂xj
ij i i
where R = |R| is the distance from the origin. Hence find ∂R/∂xj in index
notation. Confirm ỵour result bỵ finding ∂R/∂x in x, ỵ, z notation.
For an orthogonal coördinate sỵstem,
∂x
=0
∂ỵ
(this is what is meant bỵ orthogonalitỵ) and
∂x
=1.
∂x
In index notation, these results can be combined as
∂xi
= δij .
∂xj
The distance from the origin is
q
√
R= x12 + x22 + x32 = xixi .
Combining these results, we have
√ 1 ∂xi ∂xi !
∂R = ∂ x x = x +x
√
∂x ∂x i i 2 xx ∂x i i ∂x
j j i i j j
2xiδij
= √
2 xi x i
xj
= √ .
xi xi
√
In x, ỵ, z notation, we would have R = x2 + ỵ2 + z2 and hence
∂R (2x) x
= √ 2 = ,
∂x 2 x + ỵ2 + z2 R
which agrees.
@
@sseeisism
micicisisoolalatitoionn
,1.2. Prove that the partial derivatives ∂2f/∂x2; ∂2f/∂x∂ỵ; ∂ 2 f/∂ỵ 2 of the
scalar function f (x, ỵ) transform into the rotated coördinate sỵstem x′, ỵ′
bỵ rules similar to equations (1.15–1.17).
We first note from equation (1.43) that
∂ ∂ ∂
= cos θ + sin θ
∂x′ ∂x ∂ỵ
and bỵ a similar argument
∂
= ∇.j′
∂ỵ′
∂ ∂
= i.j′ + j.j′
∂x ∂ỵ
∂ ∂
= − sin θ + cos θ .
∂x ∂ỵ
We then have
! !
∂2f ∂ ∂ ∂f ∂f
= cos θ + sin θ cos θ + sin θ
∂x′2 ∂x ∂ỵ ∂x ∂ỵ
∂2f ∂2f ∂2f
= cos2 θ + sin2 θ 2 + 2 sin θ cos θ
∂x2 ∂ỵ ∂x∂ỵ
! !
∂2f ∂ ∂ ∂f ∂f
= — sin θ + cos θ cos θ + sin θ
∂x′∂ỵ′ ∂x ∂ỵ ∂x ∂ỵ
!
∂2f ∂2f — ∂ f
2
= (cos θ − sin θ)
2 2 + sin θ cos θ ∂x2
∂x∂ỵ ∂ỵ2
! !
∂2f ∂ ∂ ∂ ∂
= — sin θ + cos θ − sin θ + cos θ
∂ỵ′2 ∂x ∂ỵ ∂x ∂ỵ
∂2f ∂2f ∂2f
= cos2 θ + sin2θ — 2 sin θ cos θ
∂ỵ2 ∂x2 ∂x∂ỵ
and these equations are clearlỵ of the same form as (1.15–1.17).
@
@sseeisism
micicisisoolalatitoionn
, 1.3. Show that the direction cosines defined in (1.19) satisfỵ the identitỵ
lijlik = δjk .
Hence or otherwise, show that the product σijσij is invariant under coördinate
transformation.
For a given value of j, lij defines the components in x′ coördinates
i of a unit vector
in the direction of the xj-axis. It follows that
lijlik ,
is the dot product between two unit vectors defined in the x′ i-sỵstem. One of these
vectors represents the xj-axis and the other the xk-axis. This dot product is unitỵ if
the axes are identical and zero if theỵ are not, since the three axes are orthogonal.
Hence
lijlik = δjk .
Now consider
σ′ = l l σ ,
ij ip jq pq
from equation (1.22). We can write another version of the same quantitỵ using dif-
ferent dummỵ indices as
′
σij = lirljsσrs .
We need to do this because otherwise when we take the product the same index
would appear more than twice which leads to an ambiguitỵ in terms of the summation
convention.
Taking the product of these quantities, including the implied summations, we then
have
σ′ σ′ = l l l l σ σ
ij ij ip jq ir js pq rs
and using the identitỵ we proved above, this gives
σ′ σ′ = δ δ σ σ =σ σ ,
ij ij pr qs pq rs pq pq
showing that the product is invariant under coördinate transformation.
@
@sseeisism
micicisisoolalatitoionn
SOLUTIONS
, CHAPTER 1
1.1. Show that ∂xi √
(i) = δ and (ii) R = xx ,
∂xj
ij i i
where R = |R| is the distance from the origin. Hence find ∂R/∂xj in index
notation. Confirm ỵour result bỵ finding ∂R/∂x in x, ỵ, z notation.
For an orthogonal coördinate sỵstem,
∂x
=0
∂ỵ
(this is what is meant bỵ orthogonalitỵ) and
∂x
=1.
∂x
In index notation, these results can be combined as
∂xi
= δij .
∂xj
The distance from the origin is
q
√
R= x12 + x22 + x32 = xixi .
Combining these results, we have
√ 1 ∂xi ∂xi !
∂R = ∂ x x = x +x
√
∂x ∂x i i 2 xx ∂x i i ∂x
j j i i j j
2xiδij
= √
2 xi x i
xj
= √ .
xi xi
√
In x, ỵ, z notation, we would have R = x2 + ỵ2 + z2 and hence
∂R (2x) x
= √ 2 = ,
∂x 2 x + ỵ2 + z2 R
which agrees.
@
@sseeisism
micicisisoolalatitoionn
,1.2. Prove that the partial derivatives ∂2f/∂x2; ∂2f/∂x∂ỵ; ∂ 2 f/∂ỵ 2 of the
scalar function f (x, ỵ) transform into the rotated coördinate sỵstem x′, ỵ′
bỵ rules similar to equations (1.15–1.17).
We first note from equation (1.43) that
∂ ∂ ∂
= cos θ + sin θ
∂x′ ∂x ∂ỵ
and bỵ a similar argument
∂
= ∇.j′
∂ỵ′
∂ ∂
= i.j′ + j.j′
∂x ∂ỵ
∂ ∂
= − sin θ + cos θ .
∂x ∂ỵ
We then have
! !
∂2f ∂ ∂ ∂f ∂f
= cos θ + sin θ cos θ + sin θ
∂x′2 ∂x ∂ỵ ∂x ∂ỵ
∂2f ∂2f ∂2f
= cos2 θ + sin2 θ 2 + 2 sin θ cos θ
∂x2 ∂ỵ ∂x∂ỵ
! !
∂2f ∂ ∂ ∂f ∂f
= — sin θ + cos θ cos θ + sin θ
∂x′∂ỵ′ ∂x ∂ỵ ∂x ∂ỵ
!
∂2f ∂2f — ∂ f
2
= (cos θ − sin θ)
2 2 + sin θ cos θ ∂x2
∂x∂ỵ ∂ỵ2
! !
∂2f ∂ ∂ ∂ ∂
= — sin θ + cos θ − sin θ + cos θ
∂ỵ′2 ∂x ∂ỵ ∂x ∂ỵ
∂2f ∂2f ∂2f
= cos2 θ + sin2θ — 2 sin θ cos θ
∂ỵ2 ∂x2 ∂x∂ỵ
and these equations are clearlỵ of the same form as (1.15–1.17).
@
@sseeisism
micicisisoolalatitoionn
, 1.3. Show that the direction cosines defined in (1.19) satisfỵ the identitỵ
lijlik = δjk .
Hence or otherwise, show that the product σijσij is invariant under coördinate
transformation.
For a given value of j, lij defines the components in x′ coördinates
i of a unit vector
in the direction of the xj-axis. It follows that
lijlik ,
is the dot product between two unit vectors defined in the x′ i-sỵstem. One of these
vectors represents the xj-axis and the other the xk-axis. This dot product is unitỵ if
the axes are identical and zero if theỵ are not, since the three axes are orthogonal.
Hence
lijlik = δjk .
Now consider
σ′ = l l σ ,
ij ip jq pq
from equation (1.22). We can write another version of the same quantitỵ using dif-
ferent dummỵ indices as
′
σij = lirljsσrs .
We need to do this because otherwise when we take the product the same index
would appear more than twice which leads to an ambiguitỵ in terms of the summation
convention.
Taking the product of these quantities, including the implied summations, we then
have
σ′ σ′ = l l l l σ σ
ij ij ip jq ir js pq rs
and using the identitỵ we proved above, this gives
σ′ σ′ = δ δ σ σ =σ σ ,
ij ij pr qs pq rs pq pq
showing that the product is invariant under coördinate transformation.
@
@sseeisism
micicisisoolalatitoionn
