, Chapter 1
Section 1.2
1-1 Verify (1.2-12) and (1.2-13).
r − r0 = (r − r0 )i(r − r0 )
= r i(r − r0 ) − r0 i (r − r0 )
= r i r − r i r0 − r0 i r + r0 i r0
2 2
= r + r0 − 2(r i r0 )
Since r = r , r0 = r0 , and r = r r̂ , then
r − r0 = r 2 + r02 − 2r( r̂ i r0 )
⎛ r02 r̂ i r0 ⎞
= r ⎜ 1+ 2 − 2
2
⎝ r r ⎟⎠
= r 1+ b
2
⎛ r0 ⎞ r̂ i r0
where b = ⎜ ⎟ − 2
⎝r⎠ r
1
,1-2 Using Fig. P1-2, show that
u = cos α = sin θ cosψ ,
v = cos β = sin θ sinψ ,
and
w = cos γ = cosθ ,
where u , v , and w are dimensionless direction cosines with respect to the X , Y , and Z
axes, respectively.
Figure P1-2
2
,
r cos α
cosψ =
r sin θ
u = sin θ cosψ = cos α
3
Section 1.2
1-1 Verify (1.2-12) and (1.2-13).
r − r0 = (r − r0 )i(r − r0 )
= r i(r − r0 ) − r0 i (r − r0 )
= r i r − r i r0 − r0 i r + r0 i r0
2 2
= r + r0 − 2(r i r0 )
Since r = r , r0 = r0 , and r = r r̂ , then
r − r0 = r 2 + r02 − 2r( r̂ i r0 )
⎛ r02 r̂ i r0 ⎞
= r ⎜ 1+ 2 − 2
2
⎝ r r ⎟⎠
= r 1+ b
2
⎛ r0 ⎞ r̂ i r0
where b = ⎜ ⎟ − 2
⎝r⎠ r
1
,1-2 Using Fig. P1-2, show that
u = cos α = sin θ cosψ ,
v = cos β = sin θ sinψ ,
and
w = cos γ = cosθ ,
where u , v , and w are dimensionless direction cosines with respect to the X , Y , and Z
axes, respectively.
Figure P1-2
2
,
r cos α
cosψ =
r sin θ
u = sin θ cosψ = cos α
3
